# lfunc_search downloaded from the LMFDB on 02 May 2026. # Search link: https://www.lmfdb.org/L/1/1480/1480.1427/r0-0 # Query "{'degree': 1, 'conductor': 1480, 'spectral_label': 'r0-0'}" returned 105 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. "1-1480-1480.1003-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1003" [[0, 0.0]] [] 0 true true false false 0.010103478784855936 0 0.963878217055 ["Character/Dirichlet/1480/1003"] "1-1480-1480.1013-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1013" [[0, 0.0]] [] 0 true true false false 0.047165823610768834 0 1.20066772572 ["Character/Dirichlet/1480/1013"] "1-1480-1480.1019-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1019" [[0, 0.0]] [] 0 true true false false 0.4671278464519332 0 1.39194396869 ["Character/Dirichlet/1480/1019"] "1-1480-1480.1027-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1027" [[0, 0.0]] [] 0 true true false false -0.010103478784855936 0 0.590736367149 ["Character/Dirichlet/1480/1027"] "1-1480-1480.1029-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1029" [[0, 0.0]] [] 0 true true false false 0.2764599116428185 0 1.76380450168 ["Character/Dirichlet/1480/1029"] "1-1480-1480.1043-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1043" [[0, 0.0]] [] 0 true true false false 0.45826082946601276 0 0.00637044209644 ["Character/Dirichlet/1480/1043"] "1-1480-1480.1053-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1053" [[0, 0.0]] [] 0 true true false false -0.16766091928539528 0 0.951040600956 ["Character/Dirichlet/1480/1053"] "1-1480-1480.1059-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1059" [[0, 0.0]] [] 0 true true false false -0.08844585244621275 0 0.220032032728 ["Character/Dirichlet/1480/1059"] "1-1480-1480.1069-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1069" [[0, 0.0]] [] 0 true true false false 0.2443163894632368 0 1.07445362571 ["Character/Dirichlet/1480/1069"] "1-1480-1480.107-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.107" [[0, 0.0]] [] 0 true true false false -0.1675795149493715 0 0.271633849855 ["Character/Dirichlet/1480/107"] "1-1480-1480.1083-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1083" [[0, 0.0]] [] 0 true true false false -0.06469206221141781 0 0.754500809803 ["Character/Dirichlet/1480/1083"] "1-1480-1480.1093-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1093" [[0, 0.0]] [] 0 true true false false 0.37902375086454154 0 0.00118375695298 ["Character/Dirichlet/1480/1093"] "1-1480-1480.1107-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1107" [[0, 0.0]] [] 0 true true false false 0.487501481113391 0 0.218449581087 ["Character/Dirichlet/1480/1107"] "1-1480-1480.1109-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1109" [[0, 0.0]] [] 0 true true true true 0.0 0 0.882211430633 ["Character/Dirichlet/1480/1109"] "1-1480-1480.1133-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1133" [[0, 0.0]] [] 0 true true false false -0.17654994803360446 0 0.559413192755 ["Character/Dirichlet/1480/1133"] "1-1480-1480.1139-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1139" [[0, 0.0]] [] 0 true true false false 0.08844585244621275 0 0.74773267943 ["Character/Dirichlet/1480/1139"] "1-1480-1480.1163-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1163" [[0, 0.0]] [] 0 true true false false 0.36553097935920387 0 1.5041022484 ["Character/Dirichlet/1480/1163"] "1-1480-1480.117-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.117" [[0, 0.0]] [] 0 true true false false 0.19996198140957835 0 1.23543395143 ["Character/Dirichlet/1480/117"] "1-1480-1480.1179-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1179" [[0, 0.0]] [] 0 true true false false 0.4834695787286623 0 1.67929638383 ["Character/Dirichlet/1480/1179"] "1-1480-1480.1187-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1187" [[0, 0.0]] [] 0 true true false false 5.412384628883896e-05 0 0.891508173913 ["Character/Dirichlet/1480/1187"] "1-1480-1480.1197-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1197" [[0, 0.0]] [] 0 true true false false -0.37902375086454154 0 1.8014838033 ["Character/Dirichlet/1480/1197"] "1-1480-1480.1213-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1213" [[0, 0.0]] [] 0 true true false false 0.0003417568588210862 0 1.09121651646 ["Character/Dirichlet/1480/1213"] "1-1480-1480.1219-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1219" [[0, 0.0]] [] 0 true true false false -0.015194411793787981 0 0.738733533329 ["Character/Dirichlet/1480/1219"] "1-1480-1480.123-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.123" [[0, 0.0]] [] 0 true true false false -0.487501481113391 0 1.58819501888 ["Character/Dirichlet/1480/123"] "1-1480-1480.1253-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1253" [[0, 0.0]] [] 0 true true false false 0.3953654831412706 0 1.16422612322 ["Character/Dirichlet/1480/1253"] "1-1480-1480.1267-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1267" [[0, 0.0]] [] 0 true true false false 0.3437877061241549 0 1.27915819846 ["Character/Dirichlet/1480/1267"] "1-1480-1480.1269-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1269" [[0, 0.0]] [] 0 true true false false 0.38015740008528104 0 1.46582054669 ["Character/Dirichlet/1480/1269"] "1-1480-1480.1277-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1277" [[0, 0.0]] [] 0 true true false false 0.030257664976911642 0 0.750909144514 ["Character/Dirichlet/1480/1277"] "1-1480-1480.1283-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1283" [[0, 0.0]] [] 0 true true false false 5.412384628883896e-05 0 0.704841245763 ["Character/Dirichlet/1480/1283"] "1-1480-1480.1293-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1293" [[0, 0.0]] [] 0 true true false false -0.10329850738117966 0 0.408574895632 ["Character/Dirichlet/1480/1293"] "1-1480-1480.13-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.13" [[0, 0.0]] [] 0 true true false false 0.44476805796067514 0 0.223629757134 ["Character/Dirichlet/1480/13"] "1-1480-1480.1307-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1307" [[0, 0.0]] [] 0 true true false false -0.3112932899386076 0 0.501454232581 ["Character/Dirichlet/1480/1307"] "1-1480-1480.1317-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1317" [[0, 0.0]] [] 0 true true false false -0.3953654831412706 0 0.364000142572 ["Character/Dirichlet/1480/1317"] "1-1480-1480.1323-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1323" [[0, 0.0]] [] 0 true true false false -0.18631166995963933 0 0.606004767065 ["Character/Dirichlet/1480/1323"] "1-1480-1480.133-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.133" [[0, 0.0]] [] 0 true true false false 0.428426325683946 0 1.65660138788 ["Character/Dirichlet/1480/133"] "1-1480-1480.1403-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1403" [[0, 0.0]] [] 0 true true false false 0.3112932899386076 0 1.54336490225 ["Character/Dirichlet/1480/1403"] "1-1480-1480.1419-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1419" [[0, 0.0]] [] 0 true true false false -0.4671278464519332 0 0.061166884355 ["Character/Dirichlet/1480/1419"] "1-1480-1480.1427-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1427" [[0, 0.0]] [] 0 true true false false 0.3116441839445732 0 1.23439815321 ["Character/Dirichlet/1480/1427"] "1-1480-1480.1437-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1437" [[0, 0.0]] [] 0 true true false false -0.023753790234794965 0 1.05000514595 ["Character/Dirichlet/1480/1437"] "1-1480-1480.1469-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.1469" [[0, 0.0]] [] 0 true true false false 0.47658796662402614 0 0.0866112133605 ["Character/Dirichlet/1480/1469"] "1-1480-1480.147-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.147" [[0, 0.0]] [] 0 true true false false -0.4118959044126083 0 0.641540535522 ["Character/Dirichlet/1480/147"] "1-1480-1480.179-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.179" [[0, 0.0]] [] 0 true true false false -0.11185788582218664 0 0.85201448305 ["Character/Dirichlet/1480/179"] "1-1480-1480.189-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.189" [[0, 0.0]] [] 0 true true false false -0.4017924256277524 0 1.50500727484 ["Character/Dirichlet/1480/189"] "1-1480-1480.19-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.19" [[0, 0.0]] [] 0 true true false false -0.057846430610480046 0 0.990425585151 ["Character/Dirichlet/1480/19"] "1-1480-1480.229-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.229" [[0, 0.0]] [] 0 true true false false 0.04636492505340447 0 1.01408561338 ["Character/Dirichlet/1480/229"] "1-1480-1480.237-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.237" [[0, 0.0]] [] 0 true true false false -0.2388067673398488 0 0.556511894031 ["Character/Dirichlet/1480/237"] "1-1480-1480.243-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.243" [[0, 0.0]] [] 0 true true false false 0.13543599276978985 0 1.03818460869 ["Character/Dirichlet/1480/243"] "1-1480-1480.253-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.253" [[0, 0.0]] [] 0 true true false false -0.19996198140957835 0 0.336720528235 ["Character/Dirichlet/1480/253"] "1-1480-1480.269-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.269" [[0, 0.0]] [] 0 true true false false -0.47658796662402614 0 1.33136919527 ["Character/Dirichlet/1480/269"] "1-1480-1480.27-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.27" [[0, 0.0]] [] 0 true true false false 0.20794669550211062 0 1.53723684209 ["Character/Dirichlet/1480/27"] "1-1480-1480.277-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.277" [[0, 0.0]] [] 0 true true false false 0.10329850738117966 0 0.65168318929 ["Character/Dirichlet/1480/277"] "1-1480-1480.3-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.3" [[0, 0.0]] [] 0 true true false false -0.1762623150210722 0 0.560442892632 ["Character/Dirichlet/1480/3"] "1-1480-1480.307-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.307" [[0, 0.0]] [] 0 true true false false -0.031738504327327235 0 0.504509778857 ["Character/Dirichlet/1480/307"] "1-1480-1480.323-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.323" [[0, 0.0]] [] 0 true true false false 0.031738504327327235 0 0.810628227288 ["Character/Dirichlet/1480/323"] "1-1480-1480.339-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.339" [[0, 0.0]] [] 0 true true false false 0.11185788582218664 0 1.35304238362 ["Character/Dirichlet/1480/339"] "1-1480-1480.349-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.349" [[0, 0.0]] [] 0 true true false false -0.04636492505340447 0 1.19291056395 ["Character/Dirichlet/1480/349"] "1-1480-1480.357-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.357" [[0, 0.0]] [] 0 true true false false 0.16766091928539528 0 1.08172174743 ["Character/Dirichlet/1480/357"] "1-1480-1480.363-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.363" [[0, 0.0]] [] 0 true true false false -0.3116441839445732 0 0.254830721654 ["Character/Dirichlet/1480/363"] "1-1480-1480.403-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.403" [[0, 0.0]] [] 0 true true false false -0.3437877061241549 0 0.57209112805 ["Character/Dirichlet/1480/403"] "1-1480-1480.413-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.413" [[0, 0.0]] [] 0 true true false false 0.023753790234794965 0 0.777557191083 ["Character/Dirichlet/1480/413"] "1-1480-1480.443-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.443" [[0, 0.0]] [] 0 true true false false 0.4118959044126083 0 1.44793663607 ["Character/Dirichlet/1480/443"] "1-1480-1480.459-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.459" [[0, 0.0]] [] 0 true true false false -0.3269108629272405 0 0.62814292894 ["Character/Dirichlet/1480/459"] "1-1480-1480.469-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.469" [[0, 0.0]] [] 0 true true false false -0.4118417805663195 0 1.64550308615 ["Character/Dirichlet/1480/469"] "1-1480-1480.499-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.499" [[0, 0.0]] [] 0 true true false false 0.015194411793787981 0 1.14049500412 ["Character/Dirichlet/1480/499"] "1-1480-1480.507-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.507" [[0, 0.0]] [] 0 true true false false 0.06469206221141781 0 0.707372733746 ["Character/Dirichlet/1480/507"] "1-1480-1480.509-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.509" [[0, 0.0]] [] 0 true true false false 0.4017924256277524 0 0.0476098390279 ["Character/Dirichlet/1480/509"] "1-1480-1480.533-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.533" [[0, 0.0]] [] 0 true true false false -0.41501495851463216 0 1.95657884766 ["Character/Dirichlet/1480/533"] "1-1480-1480.557-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.557" [[0, 0.0]] [] 0 true true false false 0.14595052619787172 0 1.34454793014 ["Character/Dirichlet/1480/557"] "1-1480-1480.573-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.573" [[0, 0.0]] [] 0 true true false false -0.0729096837936037 0 0.98916050887 ["Character/Dirichlet/1480/573"] "1-1480-1480.579-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.579" [[0, 0.0]] [] 0 true true false false 0.07955682369800361 0 0.963929963962 ["Character/Dirichlet/1480/579"] "1-1480-1480.589-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.589" [[0, 0.0]] [] 0 true true false false -0.10060261447400073 0 0.833748157362 ["Character/Dirichlet/1480/589"] "1-1480-1480.59-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.59" [[0, 0.0]] [] 0 true true false false -0.4834695787286623 0 0.293830064335 ["Character/Dirichlet/1480/59"] "1-1480-1480.597-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.597" [[0, 0.0]] [] 0 true true false false 0.41501495851463216 0 0.18177639039 ["Character/Dirichlet/1480/597"] "1-1480-1480.603-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.603" [[0, 0.0]] [] 0 true true false false -0.20794669550211062 0 0.928040556886 ["Character/Dirichlet/1480/603"] "1-1480-1480.637-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.637" [[0, 0.0]] [] 0 true true false false -0.047165823610768834 0 0.824967080316 ["Character/Dirichlet/1480/637"] "1-1480-1480.653-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.653" [[0, 0.0]] [] 0 true true false false -0.00854727188938807 0 0.705793097384 ["Character/Dirichlet/1480/653"] "1-1480-1480.669-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.669" [[0, 0.0]] [] 0 true true false false 0.4118417805663195 0 0.167051015861 ["Character/Dirichlet/1480/669"] "1-1480-1480.67-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.67" [[0, 0.0]] [] 0 true true false false -0.13543599276978985 0 0.862511418445 ["Character/Dirichlet/1480/67"] "1-1480-1480.707-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.707" [[0, 0.0]] [] 0 true true false false 0.18631166995963933 0 0.856416199658 ["Character/Dirichlet/1480/707"] "1-1480-1480.717-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.717" [[0, 0.0]] [] 0 true true false false 0.2233740147855522 0 1.25217898633 ["Character/Dirichlet/1480/717"] "1-1480-1480.747-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.747" [[0, 0.0]] [] 0 true true false false -0.36553097935920387 0 0.427136296087 ["Character/Dirichlet/1480/747"] "1-1480-1480.749-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.749" [[0, 0.0]] [] 0 true true false false -0.2443163894632368 0 0.720471420487 ["Character/Dirichlet/1480/749"] "1-1480-1480.757-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.757" [[0, 0.0]] [] 0 true true false false 0.00854727188938807 0 0.392994582523 ["Character/Dirichlet/1480/757"] "1-1480-1480.779-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.779" [[0, 0.0]] [] 0 true true false false 0.057846430610480046 0 0.831611428675 ["Character/Dirichlet/1480/779"] "1-1480-1480.787-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.787" [[0, 0.0]] [] 0 true true false false 0.11151612896336556 0 1.21392789622 ["Character/Dirichlet/1480/787"] "1-1480-1480.789-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.789" [[0, 0.0]] [] 0 true true false false 0.10060261447400073 0 0.712093030093 ["Character/Dirichlet/1480/789"] "1-1480-1480.797-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.797" [[0, 0.0]] [] 0 true true false false -0.44476805796067514 0 1.41298332482 ["Character/Dirichlet/1480/797"] "1-1480-1480.803-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.803" [[0, 0.0]] [] 0 true true false false -0.11151612896336556 0 1.10973779849 ["Character/Dirichlet/1480/803"] "1-1480-1480.819-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.819" [[0, 0.0]] [] 0 true true false false 0.3269108629272405 0 1.45043506275 ["Character/Dirichlet/1480/819"] "1-1480-1480.83-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.83" [[0, 0.0]] [] 0 true true false false 0.1675795149493715 0 1.07428864638 ["Character/Dirichlet/1480/83"] "1-1480-1480.837-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.837" [[0, 0.0]] [] 0 true true false false 0.0003417568588210862 0 0.882319588206 ["Character/Dirichlet/1480/837"] "1-1480-1480.853-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.853" [[0, 0.0]] [] 0 true true false false -0.030257664976911642 0 0.671100126328 ["Character/Dirichlet/1480/853"] "1-1480-1480.859-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.859" [[0, 0.0]] [] 0 true true false false -0.13526991919816053 0 0.774548701422 ["Character/Dirichlet/1480/859"] "1-1480-1480.867-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.867" [[0, 0.0]] [] 0 true true false false -0.45826082946601276 0 1.63662448643 ["Character/Dirichlet/1480/867"] "1-1480-1480.893-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.893" [[0, 0.0]] [] 0 true true false false 0.2388067673398488 0 0.921402237796 ["Character/Dirichlet/1480/893"] "1-1480-1480.909-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.909" [[0, 0.0]] [] 0 true true false false -0.2764599116428185 0 0.768167784165 ["Character/Dirichlet/1480/909"] "1-1480-1480.917-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.917" [[0, 0.0]] [] 0 true true false false 0.17654994803360446 0 1.0059355469 ["Character/Dirichlet/1480/917"] "1-1480-1480.93-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.93" [[0, 0.0]] [] 0 true true false false -0.14595052619787172 0 0.754750340133 ["Character/Dirichlet/1480/93"] "1-1480-1480.933-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.933" [[0, 0.0]] [] 0 true true false false -0.2233740147855522 0 0.642253868679 ["Character/Dirichlet/1480/933"] "1-1480-1480.939-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.939" [[0, 0.0]] [] 0 true true false false 0.13526991919816053 0 0.716653411081 ["Character/Dirichlet/1480/939"] "1-1480-1480.957-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.957" [[0, 0.0]] [] 0 true true false false -0.428426325683946 0 0.570568799474 ["Character/Dirichlet/1480/957"] "1-1480-1480.979-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.979" [[0, 0.0]] [] 0 true true false false -0.07955682369800361 0 0.729176410812 ["Character/Dirichlet/1480/979"] "1-1480-1480.987-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.987" [[0, 0.0]] [] 0 true true false false 0.1762623150210722 0 1.34005857387 ["Character/Dirichlet/1480/987"] "1-1480-1480.989-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.989" [[0, 0.0]] [] 0 true true false false -0.38015740008528104 0 0.140316433629 ["Character/Dirichlet/1480/989"] "1-1480-1480.997-r0-0-0" 6.87309110492009 6.87309110492009 1 1480 "1480.997" [[0, 0.0]] [] 0 true true false false 0.0729096837936037 0 1.27242791978 ["Character/Dirichlet/1480/997"] # 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).