# lfunc_search downloaded from the LMFDB on 27 July 2026. # Search link: https://www.lmfdb.org/L/1/1305/1305.727 # Query "{'degree': 1, 'conductor': 1305}" returned 324 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "1-1305-1305.1003-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1003" [[0, 0.0]] [] 0 true true false false -0.0072727198739914204 0 1.27858900899 ["Character/Dirichlet/1305/1003"] "1-1305-1305.1004-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1004" [[0, 0.0]] [] 0 true true false false -0.18287008837545035 0 0.616932799576 ["Character/Dirichlet/1305/1004"] "1-1305-1305.1012-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1012" [[0, 0.0]] [] 0 true true false false 0.3989077462251143 0 0.987262908621 ["Character/Dirichlet/1305/1012"] "1-1305-1305.1022-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1022" [[0, 0.0]] [] 0 true true false false 0.37716903918635186 0 1.17950443673 ["Character/Dirichlet/1305/1022"] "1-1305-1305.1024-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1024" [[0, 0.0]] [] 0 true true false false 0.28304991740847835 0 1.0587681297 ["Character/Dirichlet/1305/1024"] "1-1305-1305.1028-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1028" [[0, 0.0]] [] 0 true true false false 0.15662376478190773 0 1.19517985799 ["Character/Dirichlet/1305/1028"] "1-1305-1305.1033-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1033" [[0, 0.0]] [] 0 true true false false -0.24319640618506427 0 0.748706899222 ["Character/Dirichlet/1305/1033"] "1-1305-1305.1037-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1037" [[0, 0.0]] [] 0 true true false false 0.28383256994941003 0 1.1021072975 ["Character/Dirichlet/1305/1037"] "1-1305-1305.1039-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1039" [[0, 0.0]] [] 0 true true false false -0.33927190679802827 0 2.22459635054 ["Character/Dirichlet/1305/1039"] "1-1305-1305.104-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.104" [[0, 0.0]] [] 0 true true false false -0.16916862428659976 0 0.489593779298 ["Character/Dirichlet/1305/104"] "1-1305-1305.1042-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1042" [[0, 0.0]] [] 0 true true false false 0.07802008041525202 0 0.737417801186 ["Character/Dirichlet/1305/1042"] "1-1305-1305.1067-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1067" [[0, 0.0]] [] 0 true true false false -0.10040177539235783 0 0.992424878849 ["Character/Dirichlet/1305/1067"] "1-1305-1305.1069-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1069" [[0, 0.0]] [] 0 true true false false -0.039064943598960185 0 1.28457263428 ["Character/Dirichlet/1305/1069"] "1-1305-1305.1082-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1082" [[0, 0.0]] [] 0 true true false false 0.12115401299587003 0 1.29435721135 ["Character/Dirichlet/1305/1082"] "1-1305-1305.1087-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1087" [[0, 0.0]] [] 0 true true false false -0.12403411146269087 0 0.28099947463 ["Character/Dirichlet/1305/1087"] "1-1305-1305.1094-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1094" [[0, 0.0]] [] 0 true true false false -0.09490768940232744 0 1.20829830797 ["Character/Dirichlet/1305/1094"] "1-1305-1305.1118-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1118" [[0, 0.0]] [] 0 true true false false 0.2088485181976616 0 1.42910269434 ["Character/Dirichlet/1305/1118"] "1-1305-1305.1123-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1123" [[0, 0.0]] [] 0 true true false false -0.15523400721194133 0 0.739669585264 ["Character/Dirichlet/1305/1123"] "1-1305-1305.1127-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1127" [[0, 0.0]] [] 0 true true false false -0.20096084801156852 0 0.74943516277 ["Character/Dirichlet/1305/1127"] "1-1305-1305.1139-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1139" [[0, 0.0]] [] 0 true true false false 0.15184358334611647 0 0.970482520193 ["Character/Dirichlet/1305/1139"] "1-1305-1305.1159-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1159" [[0, 0.0]] [] 0 true true false false -0.11111111111111113 0 0.861365328361 ["Character/Dirichlet/1305/1159"] "1-1305-1305.1168-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1168" [[0, 0.0]] [] 0 true true false false 0.09151726553650258 0 0.817405016492 ["Character/Dirichlet/1305/1168"] "1-1305-1305.1177-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1177" [[0, 0.0]] [] 0 true true false false -0.05328675092143029 0 0.952614828099 ["Character/Dirichlet/1305/1177"] "1-1305-1305.119-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.119" [[0, 0.0]] [] 0 true true false false 0.4762145207390097 0 0.303971697517 ["Character/Dirichlet/1305/119"] "1-1305-1305.1192-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1192" [[0, 0.0]] [] 0 true true false false -0.46345916145137633 0 0.604703511105 ["Character/Dirichlet/1305/1192"] "1-1305-1305.1193-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1193" [[0, 0.0]] [] 0 true true false false -0.28383256994941003 0 0.637368464895 ["Character/Dirichlet/1305/1193"] "1-1305-1305.1199-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1199" [[0, 0.0]] [] 0 true true false false -0.19843674296123195 0 0.497668450843 ["Character/Dirichlet/1305/1199"] "1-1305-1305.1202-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1202" [[0, 0.0]] [] 0 true true false false 0.055054178178913324 0 1.18673355002 ["Character/Dirichlet/1305/1202"] "1-1305-1305.1217-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1217" [[0, 0.0]] [] 0 true true false false -0.2730070155237195 0 0.914585577613 ["Character/Dirichlet/1305/1217"] "1-1305-1305.122-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.122" [[0, 0.0]] [] 0 true true false false 0.4982145927057839 0 1.48516622357 ["Character/Dirichlet/1305/122"] "1-1305-1305.1228-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1228" [[0, 0.0]] [] 0 true true false false -0.3143186163264014 0 0.920961651647 ["Character/Dirichlet/1305/1228"] "1-1305-1305.1229-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1229" [[0, 0.0]] [] 0 true true false false -0.42962136112389426 0 1.33596899898 ["Character/Dirichlet/1305/1229"] "1-1305-1305.1237-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1237" [[0, 0.0]] [] 0 true true false false 0.0004773328281086889 0 1.12295906909 ["Character/Dirichlet/1305/1237"] "1-1305-1305.1238-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1238" [[0, 0.0]] [] 0 true true false false -0.38505670937244496 0 1.71806965374 ["Character/Dirichlet/1305/1238"] "1-1305-1305.1244-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1244" [[0, 0.0]] [] 0 true true false false 0.3385814284155004 0 1.1846084728 ["Character/Dirichlet/1305/1244"] "1-1305-1305.1253-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1253" [[0, 0.0]] [] 0 true true false false -0.4002158206912217 0 1.8572726999 ["Character/Dirichlet/1305/1253"] "1-1305-1305.1258-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1258" [[0, 0.0]] [] 0 true true false false -0.4899476789335082 0 1.34724161224 ["Character/Dirichlet/1305/1258"] "1-1305-1305.1273-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1273" [[0, 0.0]] [] 0 true true false false 0.2226995550503309 0 1.52724072002 ["Character/Dirichlet/1305/1273"] "1-1305-1305.1274-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1274" [[0, 0.0]] [] 0 true true false false -0.03786179294991744 0 1.14707916064 ["Character/Dirichlet/1305/1274"] "1-1305-1305.1283-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1283" [[0, 0.0]] [] 0 true true false false 0.20096084801156852 0 1.45395012515 ["Character/Dirichlet/1305/1283"] "1-1305-1305.1292-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1292" [[0, 0.0]] [] 0 true true false false 0.10727893159466716 0 0.573173236115 ["Character/Dirichlet/1305/1292"] "1-1305-1305.1298-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1298" [[0, 0.0]] [] 0 true true false false 0.10762437877462662 0 1.12564694091 ["Character/Dirichlet/1305/1298"] "1-1305-1305.1303-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.1303" [[0, 0.0]] [] 0 true true false false -0.09818811075953135 0 1.30446814423 ["Character/Dirichlet/1305/1303"] "1-1305-1305.133-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.133" [[0, 0.0]] [] 0 true true false false -0.22949494209621366 0 2.00707366727 ["Character/Dirichlet/1305/133"] "1-1305-1305.139-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.139" [[0, 0.0]] [] 0 true true false false -0.4385058709797495 0 1.66857639177 ["Character/Dirichlet/1305/139"] "1-1305-1305.14-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.14" [[0, 0.0]] [] 0 true true false false -0.23991598482786033 0 0.573814383864 ["Character/Dirichlet/1305/14"] "1-1305-1305.142-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.142" [[0, 0.0]] [] 0 true true false false 0.17668552400289206 0 1.10489875062 ["Character/Dirichlet/1305/142"] "1-1305-1305.148-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.148" [[0, 0.0]] [] 0 true true false false 0.3603326473738403 0 1.08188637086 ["Character/Dirichlet/1305/148"] "1-1305-1305.157-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.157" [[0, 0.0]] [] 0 true true false false 0.22949494209621366 0 0.0600773733247 ["Character/Dirichlet/1305/157"] "1-1305-1305.158-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.158" [[0, 0.0]] [] 0 true true false false -0.12115401299587003 0 0.392832882915 ["Character/Dirichlet/1305/158"] "1-1305-1305.164-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.164" [[0, 0.0]] [] 0 true true false false -0.0608036506377226 0 1.10926846767 ["Character/Dirichlet/1305/164"] "1-1305-1305.167-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.167" [[0, 0.0]] [] 0 true true false false 0.006054792171632218 0 1.2124409943 ["Character/Dirichlet/1305/167"] "1-1305-1305.169-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.169" [[0, 0.0]] [] 0 true true false false 0.4385058709797495 0 0.00335173042187 ["Character/Dirichlet/1305/169"] "1-1305-1305.173-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.173" [[0, 0.0]] [] 0 true true false false -0.4492152066985028 0 0.0239077059513 ["Character/Dirichlet/1305/173"] "1-1305-1305.193-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.193" [[0, 0.0]] [] 0 true true false false -0.17668552400289206 0 0.384281015886 ["Character/Dirichlet/1305/193"] "1-1305-1305.212-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.212" [[0, 0.0]] [] 0 true true false false -0.15662376478190773 0 0.690130284666 ["Character/Dirichlet/1305/212"] "1-1305-1305.227-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.227" [[0, 0.0]] [] 0 true true false false 0.27660996656714115 0 1.40490879912 ["Character/Dirichlet/1305/227"] "1-1305-1305.23-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.23" [[0, 0.0]] [] 0 true true false false -0.27660996656714115 0 0.278480700422 ["Character/Dirichlet/1305/23"] "1-1305-1305.247-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.247" [[0, 0.0]] [] 0 true true false false 0.4285104595391751 0 1.41343514204 ["Character/Dirichlet/1305/247"] "1-1305-1305.248-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.248" [[0, 0.0]] [] 0 true true false false -0.06892925958011621 0 0.891955783619 ["Character/Dirichlet/1305/248"] "1-1305-1305.257-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.257" [[0, 0.0]] [] 0 true true false false -0.4787386257893463 0 0.190677890749 ["Character/Dirichlet/1305/257"] "1-1305-1305.274-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.274" [[0, 0.0]] [] 0 true true false false -0.28304991740847835 0 0.315369742551 ["Character/Dirichlet/1305/274"] "1-1305-1305.292-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.292" [[0, 0.0]] [] 0 true true false false -0.030080046142169505 0 1.0870507406 ["Character/Dirichlet/1305/292"] "1-1305-1305.322-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.322" [[0, 0.0]] [] 0 true true false false 0.3143186163264014 0 1.61230448907 ["Character/Dirichlet/1305/322"] "1-1305-1305.329-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.329" [[0, 0.0]] [] 0 true true false false -0.4762145207390097 0 1.46998742441 ["Character/Dirichlet/1305/329"] "1-1305-1305.337-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.337" [[0, 0.0]] [] 0 true true false false 0.15523400721194133 0 1.49482949477 ["Character/Dirichlet/1305/337"] "1-1305-1305.34-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.34" [[0, 0.0]] [] 0 true true false false -0.16011049711839223 0 0.925605718381 ["Character/Dirichlet/1305/34"] "1-1305-1305.347-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.347" [[0, 0.0]] [] 0 true true false false 0.4492152066985028 0 1.36852335268 ["Character/Dirichlet/1305/347"] "1-1305-1305.353-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.353" [[0, 0.0]] [] 0 true true false false 0.4982145927057839 0 0.513032783906 ["Character/Dirichlet/1305/353"] "1-1305-1305.358-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.358" [[0, 0.0]] [] 0 true true false false 0.46345916145137633 0 1.2744991113 ["Character/Dirichlet/1305/358"] "1-1305-1305.364-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.364" [[0, 0.0]] [] 0 true true false false -0.2308251639927245 0 0.718708260134 ["Character/Dirichlet/1305/364"] "1-1305-1305.367-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.367" [[0, 0.0]] [] 0 true true false false -0.2226995550503309 0 0.300547167886 ["Character/Dirichlet/1305/367"] "1-1305-1305.374-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.374" [[0, 0.0]] [] 0 true true false false 0.0608036506377226 0 1.38279428935 ["Character/Dirichlet/1305/374"] "1-1305-1305.38-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.38" [[0, 0.0]] [] 0 true true false false -0.055054178178913324 0 0.55010761072 ["Character/Dirichlet/1305/38"] "1-1305-1305.383-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.383" [[0, 0.0]] [] 0 true true false false 0.3220064015310005 0 0.175469624444 ["Character/Dirichlet/1305/383"] "1-1305-1305.389-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.389" [[0, 0.0]] [] 0 true true false false 0.16916862428659976 0 1.73301129651 ["Character/Dirichlet/1305/389"] "1-1305-1305.4-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.4" [[0, 0.0]] [] 0 true true false false -0.4457284743620183 0 0.505106518421 ["Character/Dirichlet/1305/4"] "1-1305-1305.403-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.403" [[0, 0.0]] [] 0 true true false false 0.0004773328281086889 0 0.663857471007 ["Character/Dirichlet/1305/403"] "1-1305-1305.418-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.418" [[0, 0.0]] [] 0 true true false false 0.05328675092143029 0 0.658439202322 ["Character/Dirichlet/1305/418"] "1-1305-1305.427-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.427" [[0, 0.0]] [] 0 true true false false 0.24319640618506427 0 0.935137411378 ["Character/Dirichlet/1305/427"] "1-1305-1305.428-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.428" [[0, 0.0]] [] 0 true true false false -0.17015339900315116 0 0.523116452155 ["Character/Dirichlet/1305/428"] "1-1305-1305.43-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.43" [[0, 0.0]] [] 0 true true false false -0.30024230263747426 0 0.804282097123 ["Character/Dirichlet/1305/43"] "1-1305-1305.439-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.439" [[0, 0.0]] [] 0 true true false false 0.33204930341575944 0 1.10071451788 ["Character/Dirichlet/1305/439"] "1-1305-1305.472-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.472" [[0, 0.0]] [] 0 true true false false 0.4899476789335082 0 0.261053645849 ["Character/Dirichlet/1305/472"] "1-1305-1305.473-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.473" [[0, 0.0]] [] 0 true true false false -0.33283195595669113 0 0.527564922082 ["Character/Dirichlet/1305/473"] "1-1305-1305.479-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.479" [[0, 0.0]] [] 0 true true false false 0.3681841417295612 0 1.45701043323 ["Character/Dirichlet/1305/479"] "1-1305-1305.484-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.484" [[0, 0.0]] [] 0 true true false false 0.2308251639927245 0 1.46621220546 ["Character/Dirichlet/1305/484"] "1-1305-1305.488-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.488" [[0, 0.0]] [] 0 true true false false 0.10040177539235783 0 0.709457811295 ["Character/Dirichlet/1305/488"] "1-1305-1305.49-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.49" [[0, 0.0]] [] 0 true true false false 0.4530473862149468 0 0.175739003229 ["Character/Dirichlet/1305/49"] "1-1305-1305.497-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.497" [[0, 0.0]] [] 0 true true false false 0.17015339900315116 0 1.16147998773 ["Character/Dirichlet/1305/497"] "1-1305-1305.499-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.499" [[0, 0.0]] [] 0 true true false false 0.16011049711839223 0 1.20072521669 ["Character/Dirichlet/1305/499"] "1-1305-1305.508-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.508" [[0, 0.0]] [] 0 true true false false 0.24769773163560826 0 1.40924073103 ["Character/Dirichlet/1305/508"] "1-1305-1305.518-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.518" [[0, 0.0]] [] 0 true true false false 0.3450531830358704 0 1.35469663194 ["Character/Dirichlet/1305/518"] "1-1305-1305.524-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.524" [[0, 0.0]] [] 0 true true false false -0.09040636395178342 0 0.808734245274 ["Character/Dirichlet/1305/524"] "1-1305-1305.527-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.527" [[0, 0.0]] [] 0 true true false false 0.4002158206912217 0 0.184977933894 ["Character/Dirichlet/1305/527"] "1-1305-1305.529-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.529" [[0, 0.0]] [] 0 true true false false -0.18315727862326206 0 0.394865152945 ["Character/Dirichlet/1305/529"] "1-1305-1305.542-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.542" [[0, 0.0]] [] 0 true true false false 0.06892925958011621 0 0.521431514791 ["Character/Dirichlet/1305/542"] "1-1305-1305.544-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.544" [[0, 0.0]] [] 0 true true false false -0.33204930341575944 0 0.688089864147 ["Character/Dirichlet/1305/544"] "1-1305-1305.553-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.553" [[0, 0.0]] [] 0 true true false false -0.20628823731695287 0 0.128144675916 ["Character/Dirichlet/1305/553"] "1-1305-1305.554-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.554" [[0, 0.0]] [] 0 true true false false 0.19843674296123195 0 1.11118686595 ["Character/Dirichlet/1305/554"] "1-1305-1305.562-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.562" [[0, 0.0]] [] 0 true true false false -0.09151726553650258 0 0.454693878909 ["Character/Dirichlet/1305/562"] "1-1305-1305.569-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.569" [[0, 0.0]] [] 0 true true false false 0.09490768940232744 0 1.26105349303 ["Character/Dirichlet/1305/569"] "1-1305-1305.574-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.574" [[0, 0.0]] [] 0 true true false false 0.33927190679802827 0 0.202186423461 ["Character/Dirichlet/1305/574"] "1-1305-1305.583-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.583" [[0, 0.0]] [] 0 true true false false 0.13811042515161806 0 1.02110571683 ["Character/Dirichlet/1305/583"] "1-1305-1305.587-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.587" [[0, 0.0]] [] 0 true true false false -0.3450531830358704 0 0.776816820758 ["Character/Dirichlet/1305/587"] "1-1305-1305.589-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.589" [[0, 0.0]] [] 0 true true false false -0.49472786036929944 0 1.49177824301 ["Character/Dirichlet/1305/589"] "1-1305-1305.592-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.592" [[0, 0.0]] [] 0 true true false false 0.0072727198739914204 0 0.626222399045 ["Character/Dirichlet/1305/592"] "1-1305-1305.598-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.598" [[0, 0.0]] [] 0 true true false false -0.020974183962842025 0 0.970758034705 ["Character/Dirichlet/1305/598"] "1-1305-1305.599-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.599" [[0, 0.0]] [] 0 true true false false -0.3385814284155004 0 0.716546805628 ["Character/Dirichlet/1305/599"] "1-1305-1305.607-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.607" [[0, 0.0]] [] 0 true true false false 0.30024230263747426 0 1.35429638664 ["Character/Dirichlet/1305/607"] "1-1305-1305.608-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.608" [[0, 0.0]] [] 0 true true false false 0.2730070155237195 0 1.47006419672 ["Character/Dirichlet/1305/608"] "1-1305-1305.628-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.628" [[0, 0.0]] [] 0 true true false false -0.3989077462251143 0 0.307595335432 ["Character/Dirichlet/1305/628"] "1-1305-1305.632-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.632" [[0, 0.0]] [] 0 true true false false 0.17737600238541995 0 0.918187078153 ["Character/Dirichlet/1305/632"] "1-1305-1305.634-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.634" [[0, 0.0]] [] 0 true true false false 0.18315727862326206 0 0.773482980604 ["Character/Dirichlet/1305/634"] "1-1305-1305.652-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.652" [[0, 0.0]] [] 0 true true false false 0.09818811075953135 0 1.22282766142 ["Character/Dirichlet/1305/652"] "1-1305-1305.659-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.659" [[0, 0.0]] [] 0 true true false false 0.18287008837545035 0 0.612351784454 ["Character/Dirichlet/1305/659"] "1-1305-1305.662-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.662" [[0, 0.0]] [] 0 true true false false 0.0011678112106365938 0 0.744336190996 ["Character/Dirichlet/1305/662"] "1-1305-1305.682-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.682" [[0, 0.0]] [] 0 true true false false 0.20628823731695287 0 1.31896000256 ["Character/Dirichlet/1305/682"] "1-1305-1305.688-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.688" [[0, 0.0]] [] 0 true true false false 0.06698821501028089 0 0.933163128313 ["Character/Dirichlet/1305/688"] "1-1305-1305.704-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.704" [[0, 0.0]] [] 0 true true false false 0.42962136112389426 0 0.247629035308 ["Character/Dirichlet/1305/704"] "1-1305-1305.709-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.709" [[0, 0.0]] [] 0 true true false false 0.49472786036929944 0 0.509508975553 ["Character/Dirichlet/1305/709"] "1-1305-1305.724-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.724" [[0, 0.0]] [] 0 true true false false 0.11111111111111113 0 0.952707841043 ["Character/Dirichlet/1305/724"] "1-1305-1305.727-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.727" [[0, 0.0]] [] 0 true true false false -0.24769773163560826 0 0.659074470067 ["Character/Dirichlet/1305/727"] "1-1305-1305.733-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.733" [[0, 0.0]] [] 0 true true false false 0.31373948775872484 0 1.01774769783 ["Character/Dirichlet/1305/733"] "1-1305-1305.742-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.742" [[0, 0.0]] [] 0 true true false false 0.16893547130079195 0 0.864582694684 ["Character/Dirichlet/1305/742"] "1-1305-1305.758-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.758" [[0, 0.0]] [] 0 true true false false -0.006054792171632218 0 1.17535019789 ["Character/Dirichlet/1305/758"] "1-1305-1305.767-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.767" [[0, 0.0]] [] 0 true true false false 0.33283195595669113 0 1.51424883014 ["Character/Dirichlet/1305/767"] "1-1305-1305.772-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.772" [[0, 0.0]] [] 0 true true false false -0.06698821501028089 0 0.410602783747 ["Character/Dirichlet/1305/772"] "1-1305-1305.788-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.788" [[0, 0.0]] [] 0 true true false false 0.22400762951643832 0 0.781412598443 ["Character/Dirichlet/1305/788"] "1-1305-1305.794-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.794" [[0, 0.0]] [] 0 true true false false -0.15184358334611647 0 0.770381416555 ["Character/Dirichlet/1305/794"] "1-1305-1305.799-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.799" [[0, 0.0]] [] 0 true true false false -0.4530473862149468 0 1.40887123116 ["Character/Dirichlet/1305/799"] "1-1305-1305.803-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.803" [[0, 0.0]] [] 0 true true false false -0.10727893159466716 0 0.97737356639 ["Character/Dirichlet/1305/803"] "1-1305-1305.823-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.823" [[0, 0.0]] [] 0 true true false false -0.26772545671128595 0 0.738753223283 ["Character/Dirichlet/1305/823"] "1-1305-1305.824-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.824" [[0, 0.0]] [] 0 true true false false -0.10860915349117806 0 0.842179697692 ["Character/Dirichlet/1305/824"] "1-1305-1305.83-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.83" [[0, 0.0]] [] 0 true true false false -0.37716903918635186 0 0.0599753706994 ["Character/Dirichlet/1305/83"] "1-1305-1305.839-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.839" [[0, 0.0]] [] 0 true true false false 0.23991598482786033 0 0.72827545269 ["Character/Dirichlet/1305/839"] "1-1305-1305.848-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.848" [[0, 0.0]] [] 0 true true false false 0.4787386257893463 0 1.16944275618 ["Character/Dirichlet/1305/848"] "1-1305-1305.853-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.853" [[0, 0.0]] [] 0 true true false false -0.16893547130079195 0 0.713503875582 ["Character/Dirichlet/1305/853"] "1-1305-1305.857-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.857" [[0, 0.0]] [] 0 true true false false 0.38505670937244496 0 0.157930132888 ["Character/Dirichlet/1305/857"] "1-1305-1305.862-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.862" [[0, 0.0]] [] 0 true true false false 0.020974183962842025 0 0.982836770685 ["Character/Dirichlet/1305/862"] "1-1305-1305.868-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.868" [[0, 0.0]] [] 0 true true false false 0.12403411146269087 0 0.882002670643 ["Character/Dirichlet/1305/868"] "1-1305-1305.884-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.884" [[0, 0.0]] [] 0 true true false false 0.03786179294991744 0 0.724021202575 ["Character/Dirichlet/1305/884"] "1-1305-1305.893-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.893" [[0, 0.0]] [] 0 true true false false 0.0011678112106365938 0 1.10029476653 ["Character/Dirichlet/1305/893"] "1-1305-1305.904-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.904" [[0, 0.0]] [] 0 true true false false 0.06211172510383 0 1.07593946749 ["Character/Dirichlet/1305/904"] "1-1305-1305.907-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.907" [[0, 0.0]] [] 0 true true false false 0.26772545671128595 0 1.75468078494 ["Character/Dirichlet/1305/907"] "1-1305-1305.913-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.913" [[0, 0.0]] [] 0 true true false false -0.07802008041525202 0 0.723939743686 ["Character/Dirichlet/1305/913"] "1-1305-1305.914-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.914" [[0, 0.0]] [] 0 true true false false 0.09040636395178342 0 0.572753890465 ["Character/Dirichlet/1305/914"] "1-1305-1305.92-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.92" [[0, 0.0]] [] 0 true true false false -0.3220064015310005 0 1.65347898669 ["Character/Dirichlet/1305/92"] "1-1305-1305.923-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.923" [[0, 0.0]] [] 0 true true false false -0.17737600238541995 0 0.981663580818 ["Character/Dirichlet/1305/923"] "1-1305-1305.932-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.932" [[0, 0.0]] [] 0 true true false false -0.10762437877462662 0 0.229716391437 ["Character/Dirichlet/1305/932"] "1-1305-1305.934-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.934" [[0, 0.0]] [] 0 true true false false -0.06211172510383 0 0.745658069295 ["Character/Dirichlet/1305/934"] "1-1305-1305.94-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.94" [[0, 0.0]] [] 0 true true false false 0.039064943598960185 0 0.725393392418 ["Character/Dirichlet/1305/94"] "1-1305-1305.943-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.943" [[0, 0.0]] [] 0 true true false false 0.030080046142169505 0 0.930657865555 ["Character/Dirichlet/1305/943"] "1-1305-1305.959-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.959" [[0, 0.0]] [] 0 true true false false -0.3681841417295612 0 0.0383793354375 ["Character/Dirichlet/1305/959"] "1-1305-1305.967-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.967" [[0, 0.0]] [] 0 true true false false -0.13811042515161806 0 0.396920422708 ["Character/Dirichlet/1305/967"] "1-1305-1305.97-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.97" [[0, 0.0]] [] 0 true true false false -0.3603326473738403 0 0.623195081638 ["Character/Dirichlet/1305/97"] "1-1305-1305.974-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.974" [[0, 0.0]] [] 0 true true false false 0.10860915349117806 0 0.862296072134 ["Character/Dirichlet/1305/974"] "1-1305-1305.977-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.977" [[0, 0.0]] [] 0 true true false false -0.2088485181976616 0 1.00012285653 ["Character/Dirichlet/1305/977"] "1-1305-1305.979-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.979" [[0, 0.0]] [] 0 true true false false 0.4457284743620183 0 1.65453485126 ["Character/Dirichlet/1305/979"] "1-1305-1305.988-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.988" [[0, 0.0]] [] 0 true true false false -0.4285104595391751 0 0.116353658652 ["Character/Dirichlet/1305/988"] "1-1305-1305.992-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.992" [[0, 0.0]] [] 0 true true false false -0.22400762951643832 0 0.474593397457 ["Character/Dirichlet/1305/992"] "1-1305-1305.997-r0-0-0" 6.060394521568052 6.060394521568052 1 1305 "1305.997" [[0, 0.0]] [] 0 true true false false -0.31373948775872484 0 0.758229500849 ["Character/Dirichlet/1305/997"] "1-1305-1305.1013-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1013" [[1, 0.0]] [] 0 true true false false -0.12596588853730914 0 0.0323172557429 ["Character/Dirichlet/1305/1013"] "1-1305-1305.1019-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1019" [[1, 0.0]] [] 0 true true false false -0.41795069658424056 0 0.37341653034 ["Character/Dirichlet/1305/1019"] "1-1305-1305.103-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.103" [[1, 0.0]] [] 0 true true false false -0.3189292595801162 0 0.216068026665 ["Character/Dirichlet/1305/103"] "1-1305-1305.1048-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1048" [[1, 0.0]] [] 0 true true false false 0.46616743005059 0 1.2779411653 ["Character/Dirichlet/1305/1048"] "1-1305-1305.1049-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1049" [[1, 0.0]] [] 0 true true false false -0.18788827489617 0 1.98513902073 ["Character/Dirichlet/1305/1049"] "1-1305-1305.1057-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1057" [[1, 0.0]] [] 0 true true false false -0.19494582182108666 0 1.43879143948 ["Character/Dirichlet/1305/1057"] "1-1305-1305.1058-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1058" [[1, 0.0]] [] 0 true true false false -0.328020080415252 0 0.120599991024 ["Character/Dirichlet/1305/1058"] "1-1305-1305.1064-r1-0-0" 140.24172685698565 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0.0]] [] 0 true true false false -0.38811042515161803 0 0.305735337182 ["Character/Dirichlet/1305/1112"] "1-1305-1305.1114-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1114" [[1, 0.0]] [] 0 true true false false 0.41916862428659973 0 1.05489045892 ["Character/Dirichlet/1305/1114"] "1-1305-1305.112-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.112" [[1, 0.0]] [] 0 true true false false 0.2712613742106537 0 0.538287780579 ["Character/Dirichlet/1305/112"] "1-1305-1305.1129-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1129" [[1, 0.0]] [] 0 true true false false -0.2878617929499174 0 0.237903329938 ["Character/Dirichlet/1305/1129"] "1-1305-1305.113-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.113" [[1, 0.0]] [] 0 true true false false -0.027300444949669087 0 0.286054504794 ["Character/Dirichlet/1305/113"] "1-1305-1305.1138-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.1138" [[1, 0.0]] [] 0 true true false false 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true true false false 0.24394520782836782 0 0.821191682382 ["Character/Dirichlet/1305/178"] "1-1305-1305.182-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.182" [[1, 0.0]] [] 0 true true false false -0.4822745432887141 0 0.0237992983003 ["Character/Dirichlet/1305/182"] "1-1305-1305.184-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.184" [[1, 0.0]] [] 0 true true false false 0.2737854792609903 0 0.0937920259499 ["Character/Dirichlet/1305/184"] "1-1305-1305.187-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.187" [[1, 0.0]] [] 0 true true false false -0.4171680440433089 0 0.442239193986 ["Character/Dirichlet/1305/187"] "1-1305-1305.194-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.194" [[1, 0.0]] [] 0 true true false false 0.20304738621494672 0 0.598948289304 ["Character/Dirichlet/1305/194"] "1-1305-1305.2-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.2" [[1, 0.0]] [] 0 true true false false 0.002302268364391729 0 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140.24172685698565 1 1305 "1305.259" [[1, 0.0]] [] 0 true true false false 0.4899159848278603 0 0.2705994048 ["Character/Dirichlet/1305/259"] "1-1305-1305.263-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.263" [[1, 0.0]] [] 0 true true false false -0.17851045953917508 0 0.906761270391 ["Character/Dirichlet/1305/263"] "1-1305-1305.268-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.268" [[1, 0.0]] [] 0 true true false false -0.2712613742106537 0 0.394760239828 ["Character/Dirichlet/1305/268"] "1-1305-1305.272-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.272" [[1, 0.0]] [] 0 true true false false -0.0637394877587248 0 0.180461548453 ["Character/Dirichlet/1305/272"] "1-1305-1305.277-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.277" [[1, 0.0]] [] 0 true true false false -0.3649432906275551 0 0.035758409791 ["Character/Dirichlet/1305/277"] "1-1305-1305.283-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.283" [[1, 0.0]] [] 0 true true false false -0.4201533990031512 0 0.0150863458033 ["Character/Dirichlet/1305/283"] "1-1305-1305.284-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.284" [[1, 0.0]] [] 0 true true false false 0.3114941290202505 0 1.1294258936 ["Character/Dirichlet/1305/284"] "1-1305-1305.293-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.293" [[1, 0.0]] [] 0 true true false false 0.11033264737384027 0 1.08185513347 ["Character/Dirichlet/1305/293"] "1-1305-1305.299-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.299" [[1, 0.0]] [] 0 true true false false -0.46695008259152165 0 1.69649926593 ["Character/Dirichlet/1305/299"] "1-1305-1305.302-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.302" [[1, 0.0]] [] 0 true true false false -0.02050505790378636 0 0.59124117276 ["Character/Dirichlet/1305/302"] "1-1305-1305.304-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.304" [[1, 0.0]] [] 0 true true false false 0.2878617929499174 0 1.17692850431 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140.24172685698565 140.24172685698565 1 1305 "1305.338" [[1, 0.0]] [] 0 true true false false -0.42668552400289206 0 0.604477973905 ["Character/Dirichlet/1305/338"] "1-1305-1305.344-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.344" [[1, 0.0]] [] 0 true true false false 0.21093505640103982 0 0.703334578075 ["Character/Dirichlet/1305/344"] "1-1305-1305.362-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.362" [[1, 0.0]] [] 0 true true false false 0.12596588853730914 0 0.772818627246 ["Character/Dirichlet/1305/362"] "1-1305-1305.373-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.373" [[1, 0.0]] [] 0 true true false false 0.09505318303587036 0 0.840657279217 ["Character/Dirichlet/1305/373"] "1-1305-1305.382-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.382" [[1, 0.0]] [] 0 true true false false 0.1502158206912217 0 0.719103430532 ["Character/Dirichlet/1305/382"] "1-1305-1305.392-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.392" [[1, 0.0]] [] 0 true true false false 0.17851045953917508 0 1.12762167102 ["Character/Dirichlet/1305/392"] "1-1305-1305.394-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.394" [[1, 0.0]] [] 0 true true false false 0.3586091534911781 0 1.4535225926 ["Character/Dirichlet/1305/394"] "1-1305-1305.398-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.398" [[1, 0.0]] [] 0 true true false false 0.09476599278805867 0 0.664777841373 ["Character/Dirichlet/1305/398"] "1-1305-1305.409-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.409" [[1, 0.0]] [] 0 true true false false -0.05156325703876805 0 0.809653726808 ["Character/Dirichlet/1305/409"] "1-1305-1305.412-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.412" [[1, 0.0]] [] 0 true true false false 0.025992370483561685 0 0.392207260661 ["Character/Dirichlet/1305/412"] "1-1305-1305.419-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.419" [[1, 0.0]] [] 0 true true false false 0.46695008259152165 0 0.16231088375 ["Character/Dirichlet/1305/419"] "1-1305-1305.434-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.434" [[1, 0.0]] [] 0 true true false false 0.1388888888888889 0 0.47848121588 ["Character/Dirichlet/1305/434"] "1-1305-1305.437-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.437" [[1, 0.0]] [] 0 true true false false -0.2800800461421695 0 0.0742606751306 ["Character/Dirichlet/1305/437"] "1-1305-1305.443-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.443" [[1, 0.0]] [] 0 true true false false 0.3415172655365026 0 1.20395211285 ["Character/Dirichlet/1305/443"] "1-1305-1305.448-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.448" [[1, 0.0]] [] 0 true true false false 0.4066237647819077 0 1.48915663215 ["Character/Dirichlet/1305/448"] "1-1305-1305.452-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.452" [[1, 0.0]] [] 0 true true false false 0.19671324907856974 0 0.678904532513 ["Character/Dirichlet/1305/452"] "1-1305-1305.454-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.454" [[1, 0.0]] [] 0 true true false false 0.4114185715844996 0 0.908128132865 ["Character/Dirichlet/1305/454"] "1-1305-1305.457-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.457" [[1, 0.0]] [] 0 true true false false -0.46616743005059 0 0.540515518304 ["Character/Dirichlet/1305/457"] "1-1305-1305.463-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.463" [[1, 0.0]] [] 0 true true false false 0.02300701552371943 0 0.973345016672 ["Character/Dirichlet/1305/463"] "1-1305-1305.47-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.47" [[1, 0.0]] [] 0 true true false false 0.18301178498971912 0 1.12009491468 ["Character/Dirichlet/1305/47"] "1-1305-1305.482-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.482" [[1, 0.0]] [] 0 true true false false -0.09476599278805867 0 0.359061463386 ["Character/Dirichlet/1305/482"] "1-1305-1305.502-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.502" [[1, 0.0]] [] 0 true true false false 0.37115401299587003 0 0.654626967447 ["Character/Dirichlet/1305/502"] "1-1305-1305.509-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.509" [[1, 0.0]] [] 0 true true false false -0.4808251639927245 0 0.0408866155908 ["Character/Dirichlet/1305/509"] "1-1305-1305.517-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.517" [[1, 0.0]] [] 0 true true false false -0.24883218878936342 0 0.629191820174 ["Character/Dirichlet/1305/517"] "1-1305-1305.52-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.52" [[1, 0.0]] [] 0 true true false false 0.42737600238541995 0 0.00133170890497 ["Character/Dirichlet/1305/52"] "1-1305-1305.533-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.533" [[1, 0.0]] [] 0 true true false false -0.23994767893350818 0 0.531485232673 ["Character/Dirichlet/1305/533"] "1-1305-1305.538-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.538" [[1, 0.0]] [] 0 true true false false 0.45884851819766165 0 1.29650258981 ["Character/Dirichlet/1305/538"] "1-1305-1305.547-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.547" [[1, 0.0]] [] 0 true true false false 0.049039151988431506 0 0.666425841106 ["Character/Dirichlet/1305/547"] "1-1305-1305.563-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.563" [[1, 0.0]] [] 0 true true false false -0.19671324907856974 0 0.63179014619 ["Character/Dirichlet/1305/563"] "1-1305-1305.572-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.572" [[1, 0.0]] [] 0 true true false false -0.006803593814935753 0 0.921403397334 ["Character/Dirichlet/1305/572"] "1-1305-1305.578-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.578" [[1, 0.0]] [] 0 true true false false 0.15181188924046868 0 0.657425997434 ["Character/Dirichlet/1305/578"] "1-1305-1305.614-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.614" [[1, 0.0]] [] 0 true true false false 0.08988950288160778 0 0.721916566143 ["Character/Dirichlet/1305/614"] "1-1305-1305.617-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.617" [[1, 0.0]] [] 0 true true false false 0.23994767893350818 0 0.642423577737 ["Character/Dirichlet/1305/617"] "1-1305-1305.619-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.619" [[1, 0.0]] [] 0 true true false false 0.05156325703876805 0 1.04603472275 ["Character/Dirichlet/1305/619"] "1-1305-1305.623-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.623" [[1, 0.0]] [] 0 true true false false -0.05024230263747425 0 0.655565450583 ["Character/Dirichlet/1305/623"] "1-1305-1305.637-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.637" [[1, 0.0]] [] 0 true true false false -0.02300701552371943 0 0.688286273411 ["Character/Dirichlet/1305/637"] "1-1305-1305.643-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.643" [[1, 0.0]] [] 0 true true false false -0.025992370483561685 0 0.84401820622 ["Character/Dirichlet/1305/643"] "1-1305-1305.644-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.644" [[1, 0.0]] [] 0 true true false false -0.08988950288160778 0 0.4641004362 ["Character/Dirichlet/1305/644"] "1-1305-1305.653-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.653" [[1, 0.0]] [] 0 true true false false 0.002302268364391729 0 0.601484744027 ["Character/Dirichlet/1305/653"] "1-1305-1305.664-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.664" [[1, 0.0]] [] 0 true true false false -0.4114185715844996 0 0.350458341537 ["Character/Dirichlet/1305/664"] "1-1305-1305.67-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.67" [[1, 0.0]] [] 0 true true false false -0.4066237647819077 0 0.497043087801 ["Character/Dirichlet/1305/67"] "1-1305-1305.673-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.673" [[1, 0.0]] [] 0 true true false false -0.1502158206912217 0 0.559776808629 ["Character/Dirichlet/1305/673"] "1-1305-1305.677-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.677" [[1, 0.0]] [] 0 true true false false -0.11033264737384027 0 0.529347226017 ["Character/Dirichlet/1305/677"] "1-1305-1305.679-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.679" [[1, 0.0]] [] 0 true true false false -0.3586091534911781 0 0.0821262276558 ["Character/Dirichlet/1305/679"] "1-1305-1305.68-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.68" [[1, 0.0]] [] 0 true true false false 0.4356813836735986 0 0.944869913323 ["Character/Dirichlet/1305/68"] "1-1305-1305.689-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.689" [[1, 0.0]] [] 0 true true false false 0.41795069658424056 0 1.01886885733 ["Character/Dirichlet/1305/689"] "1-1305-1305.698-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.698" [[1, 0.0]] [] 0 true true false false -0.4562882373169529 0 0.297905820371 ["Character/Dirichlet/1305/698"] "1-1305-1305.7-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.7" [[1, 0.0]] [] 0 true true false false -0.09505318303587036 0 0.267470470989 ["Character/Dirichlet/1305/7"] "1-1305-1305.707-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.707" [[1, 0.0]] [] 0 true true false false -0.3415172655365026 0 0.208340697246 ["Character/Dirichlet/1305/707"] "1-1305-1305.713-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.713" [[1, 0.0]] [] 0 true true false false 0.02050505790378636 0 0.864027974776 ["Character/Dirichlet/1305/713"] "1-1305-1305.718-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.718" [[1, 0.0]] [] 0 true true false false 0.3576243787746266 0 0.810085619213 ["Character/Dirichlet/1305/718"] "1-1305-1305.722-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.722" [[1, 0.0]] [] 0 true true false false 0.42668552400289206 0 1.4287208662 ["Character/Dirichlet/1305/722"] "1-1305-1305.734-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.734" [[1, 0.0]] [] 0 true true false false 0.25527213963070056 0 0.992510237425 ["Character/Dirichlet/1305/734"] "1-1305-1305.74-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.74" [[1, 0.0]] [] 0 true true false false -0.20304738621494672 0 0.747529363068 ["Character/Dirichlet/1305/74"] "1-1305-1305.743-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.743" [[1, 0.0]] [] 0 true true false false -0.270974183962842 0 0.646893451128 ["Character/Dirichlet/1305/743"] "1-1305-1305.749-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.749" [[1, 0.0]] [] 0 true true false false -0.3114941290202505 0 0.496183848663 ["Character/Dirichlet/1305/749"] "1-1305-1305.752-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.752" [[1, 0.0]] [] 0 true true false false 0.05024230263747425 0 0.935338191315 ["Character/Dirichlet/1305/752"] "1-1305-1305.763-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.763" [[1, 0.0]] [] 0 true true false false 0.19494582182108666 0 0.0987040384125 ["Character/Dirichlet/1305/763"] "1-1305-1305.769-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.769" [[1, 0.0]] [] 0 true true false false -0.15959363604821658 0 0.300303612162 ["Character/Dirichlet/1305/769"] "1-1305-1305.77-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.77" [[1, 0.0]] [] 0 true true false false -0.2495226671718913 0 0.0453109005834 ["Character/Dirichlet/1305/77"] "1-1305-1305.778-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.778" [[1, 0.0]] [] 0 true true false false -0.42737600238541995 0 0.927737604083 ["Character/Dirichlet/1305/778"] "1-1305-1305.779-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.779" [[1, 0.0]] [] 0 true true false false -0.06684272137673795 0 0.965473006234 ["Character/Dirichlet/1305/779"] "1-1305-1305.787-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.787" [[1, 0.0]] [] 0 true true false false -0.3576243787746266 0 0.293657853476 ["Character/Dirichlet/1305/787"] "1-1305-1305.79-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.79" [[1, 0.0]] [] 0 true true false false 0.4328700883754503 0 0.208047662937 ["Character/Dirichlet/1305/79"] "1-1305-1305.797-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.797" [[1, 0.0]] [] 0 true true false false -0.15181188924046868 0 0.342126332276 ["Character/Dirichlet/1305/797"] "1-1305-1305.808-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.808" [[1, 0.0]] [] 0 true true false false -0.12716903918635186 0 0.710890658459 ["Character/Dirichlet/1305/808"] "1-1305-1305.814-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.814" [[1, 0.0]] [] 0 true true false false 0.3818158582704388 0 0.00697603662003 ["Character/Dirichlet/1305/814"] "1-1305-1305.817-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.817" [[1, 0.0]] [] 0 true true false false -0.07200640153100055 0 0.498830318145 ["Character/Dirichlet/1305/817"] "1-1305-1305.832-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.832" [[1, 0.0]] [] 0 true true false false -0.45884851819766165 0 0.135242001218 ["Character/Dirichlet/1305/832"] "1-1305-1305.833-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.833" [[1, 0.0]] [] 0 true true false false -0.18301178498971912 0 0.0795914958972 ["Character/Dirichlet/1305/833"] "1-1305-1305.844-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.844" [[1, 0.0]] [] 0 true true false false -0.2737854792609903 0 1.42114198038 ["Character/Dirichlet/1305/844"] "1-1305-1305.859-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.859" [[1, 0.0]] [] 0 true true false false -0.4328700883754503 0 0.975889317426 ["Character/Dirichlet/1305/859"] "1-1305-1305.869-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.869" [[1, 0.0]] [] 0 true true false false -0.1388888888888889 0 0.694633077044 ["Character/Dirichlet/1305/869"] "1-1305-1305.877-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.877" [[1, 0.0]] [] 0 true true false false 0.12716903918635186 0 1.18222097733 ["Character/Dirichlet/1305/877"] "1-1305-1305.878-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.878" [[1, 0.0]] [] 0 true true false false 0.0637394877587248 0 0.545833962209 ["Character/Dirichlet/1305/878"] "1-1305-1305.887-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.887" [[1, 0.0]] [] 0 true true false false -0.08106452869920806 0 0.636287080446 ["Character/Dirichlet/1305/887"] "1-1305-1305.889-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.889" [[1, 0.0]] [] 0 true true false false 0.18919634936227742 0 0.809872877363 ["Character/Dirichlet/1305/889"] "1-1305-1305.898-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.898" [[1, 0.0]] [] 0 true true false false -0.19921520669850282 0 0.563815592085 ["Character/Dirichlet/1305/898"] "1-1305-1305.902-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.902" [[1, 0.0]] [] 0 true true false false -0.4356813836735986 0 0.0883929298228 ["Character/Dirichlet/1305/902"] "1-1305-1305.922-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.922" [[1, 0.0]] [] 0 true true false false -0.35040177539235784 0 1.36869500577 ["Character/Dirichlet/1305/922"] "1-1305-1305.938-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.938" [[1, 0.0]] [] 0 true true false false -0.28654083854862367 0 0.29132119918 ["Character/Dirichlet/1305/938"] "1-1305-1305.947-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.947" [[1, 0.0]] [] 0 true true false false 0.027300444949669087 0 0.869548856013 ["Character/Dirichlet/1305/947"] "1-1305-1305.949-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.949" [[1, 0.0]] [] 0 true true false false -0.34490768940232747 0 1.49043918039 ["Character/Dirichlet/1305/949"] "1-1305-1305.952-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.952" [[1, 0.0]] [] 0 true true false false -0.47339003343285885 0 1.02107562516 ["Character/Dirichlet/1305/952"] "1-1305-1305.968-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.968" [[1, 0.0]] [] 0 true true false false 0.4822745432887141 0 1.26536962927 ["Character/Dirichlet/1305/968"] "1-1305-1305.983-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.983" [[1, 0.0]] [] 0 true true false false 0.2495226671718913 0 0.743474637239 ["Character/Dirichlet/1305/983"] "1-1305-1305.994-r1-0-0" 140.24172685698565 140.24172685698565 1 1305 "1305.994" [[1, 0.0]] [] 0 true true false false -0.09815641665388351 0 0.984343125329 ["Character/Dirichlet/1305/994"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).