# lfunc_search downloaded from the LMFDB on 07 April 2026. # Search link: https://www.lmfdb.org/L/1/1580/1580.67 # Query "{'degree': 1, 'conductor': 1580}" returned 231 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. "1-1580-1580.1003-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1003" [[0, 0.0]] [] 0 true true false false -0.27050788082551874 0 0.397494313725 ["Character/Dirichlet/1580/1003"] "1-1580-1580.1019-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1019" [[0, 0.0]] [] 0 true true false false -0.17648921734555253 0 0.745255267404 ["Character/Dirichlet/1580/1019"] "1-1580-1580.1039-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1039" [[0, 0.0]] [] 0 true true false false 0.3325352236365132 0 1.56537428852 ["Character/Dirichlet/1580/1039"] "1-1580-1580.1043-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1043" [[0, 0.0]] [] 0 true true false false 0.08043476126553807 0 1.25470823925 ["Character/Dirichlet/1580/1043"] "1-1580-1580.1047-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1047" [[0, 0.0]] [] 0 true true false false -0.4293015243835955 0 1.85196217205 ["Character/Dirichlet/1580/1047"] "1-1580-1580.1063-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1063" [[0, 0.0]] [] 0 true true false false -0.08777605898000276 0 0.994073579633 ["Character/Dirichlet/1580/1063"] "1-1580-1580.1067-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1067" [[0, 0.0]] [] 0 true true false false 0.22288634307877678 0 1.27024594522 ["Character/Dirichlet/1580/1067"] "1-1580-1580.1103-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1103" [[0, 0.0]] [] 0 true true false false -0.13733668600065335 0 0.161214789228 ["Character/Dirichlet/1580/1103"] "1-1580-1580.1127-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1127" [[0, 0.0]] [] 0 true true false false 0.10066603560286892 0 1.09907585527 ["Character/Dirichlet/1580/1127"] "1-1580-1580.1139-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1139" [[0, 0.0]] [] 0 true true false false -0.3325352236365132 0 0.691602866051 ["Character/Dirichlet/1580/1139"] "1-1580-1580.1159-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1159" [[0, 0.0]] [] 0 true true false false 0.09355471171685649 0 1.15388859166 ["Character/Dirichlet/1580/1159"] "1-1580-1580.1187-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1187" [[0, 0.0]] [] 0 true true false false -0.04667815190399342 0 0.372223755079 ["Character/Dirichlet/1580/1187"] "1-1580-1580.1199-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1199" [[0, 0.0]] [] 0 true true false false -0.2606817118423076 0 0.667308280949 ["Character/Dirichlet/1580/1199"] "1-1580-1580.1203-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1203" [[0, 0.0]] [] 0 true true false false 0.4801507085946984 0 0.194336958571 ["Character/Dirichlet/1580/1203"] "1-1580-1580.1207-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1207" [[0, 0.0]] [] 0 true true false false -0.4801507085946984 0 1.59495883869 ["Character/Dirichlet/1580/1207"] "1-1580-1580.1219-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1219" [[0, 0.0]] [] 0 true true false false -0.45150631072711345 0 1.55867224094 ["Character/Dirichlet/1580/1219"] "1-1580-1580.1223-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1223" [[0, 0.0]] [] 0 true true false false -0.27576423725982613 0 0.283648499721 ["Character/Dirichlet/1580/1223"] "1-1580-1580.1227-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1227" [[0, 0.0]] [] 0 true true false false 0.01922684648449237 0 1.19075696328 ["Character/Dirichlet/1580/1227"] "1-1580-1580.123-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.123" [[0, 0.0]] [] 0 true true false false -0.0478808960213761 0 1.07105418287 ["Character/Dirichlet/1580/123"] "1-1580-1580.1239-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1239" [[0, 0.0]] [] 0 true true false false 0.4103614604038897 0 1.49023975406 ["Character/Dirichlet/1580/1239"] "1-1580-1580.1247-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1247" [[0, 0.0]] [] 0 true true false false -0.1387636284286353 0 0.692696773022 ["Character/Dirichlet/1580/1247"] "1-1580-1580.1259-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1259" [[0, 0.0]] [] 0 true true false false 0.3128435597109653 0 0.837772728643 ["Character/Dirichlet/1580/1259"] "1-1580-1580.1279-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1279" [[0, 0.0]] [] 0 true true false false -0.4777209442203937 0 0.367695125826 ["Character/Dirichlet/1580/1279"] "1-1580-1580.1283-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1283" [[0, 0.0]] [] 0 true true false false 0.40861684258021386 0 1.71664792189 ["Character/Dirichlet/1580/1283"] "1-1580-1580.1287-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1287" [[0, 0.0]] [] 0 true true false false 0.27050788082551874 0 1.1681227673 ["Character/Dirichlet/1580/1287"] "1-1580-1580.1299-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1299" [[0, 0.0]] [] 0 true true false false 0.2996872530535747 0 1.04375062098 ["Character/Dirichlet/1580/1299"] "1-1580-1580.1339-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1339" [[0, 0.0]] [] 0 true true false false -0.16937958363087796 0 0.688220531104 ["Character/Dirichlet/1580/1339"] "1-1580-1580.1347-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1347" [[0, 0.0]] [] 0 true true false false -0.39449028444162115 0 1.60853550495 ["Character/Dirichlet/1580/1347"] "1-1580-1580.1363-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1363" [[0, 0.0]] [] 0 true true false false 0.39449028444162115 0 0.201339005536 ["Character/Dirichlet/1580/1363"] "1-1580-1580.1383-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1383" [[0, 0.0]] [] 0 true true false false 0.04667815190399342 0 0.814749824631 ["Character/Dirichlet/1580/1383"] "1-1580-1580.1387-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1387" [[0, 0.0]] [] 0 true true false false 0.12832729515340727 0 0.754887600025 ["Character/Dirichlet/1580/1387"] "1-1580-1580.139-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.139" [[0, 0.0]] [] 0 true true false false -0.4103614604038897 0 0.211946520847 ["Character/Dirichlet/1580/139"] "1-1580-1580.1399-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1399" [[0, 0.0]] [] 0 true true false false -0.03147423714292071 0 1.14745166247 ["Character/Dirichlet/1580/1399"] "1-1580-1580.1407-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1407" [[0, 0.0]] [] 0 true true false false 0.07554215557191445 0 0.914438992987 ["Character/Dirichlet/1580/1407"] "1-1580-1580.1427-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1427" [[0, 0.0]] [] 0 true true false false -0.08043476126553807 0 0.879424320726 ["Character/Dirichlet/1580/1427"] "1-1580-1580.143-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.143" [[0, 0.0]] [] 0 true true false false -0.10066603560286892 0 0.624298640026 ["Character/Dirichlet/1580/143"] "1-1580-1580.1439-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1439" [[0, 0.0]] [] 0 true true false false 0.2606817118423076 0 0.990625189052 ["Character/Dirichlet/1580/1439"] "1-1580-1580.1443-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1443" [[0, 0.0]] [] 0 true true false false -0.07554215557191445 0 0.66322056137 ["Character/Dirichlet/1580/1443"] "1-1580-1580.1447-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1447" [[0, 0.0]] [] 0 true true false false -0.40861684258021386 0 0.520319940913 ["Character/Dirichlet/1580/1447"] "1-1580-1580.1459-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1459" [[0, 0.0]] [] 0 true true false false 0.16939189079821224 0 1.00287643918 ["Character/Dirichlet/1580/1459"] "1-1580-1580.1467-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1467" [[0, 0.0]] [] 0 true true false false 0.4565148554023896 0 1.93378697572 ["Character/Dirichlet/1580/1467"] "1-1580-1580.1479-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1479" [[0, 0.0]] [] 0 true true false false 0.24951862699413907 0 0.91275351211 ["Character/Dirichlet/1580/1479"] "1-1580-1580.1487-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1487" [[0, 0.0]] [] 0 true true false false 0.31497181960341863 0 1.30216860218 ["Character/Dirichlet/1580/1487"] "1-1580-1580.1499-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1499" [[0, 0.0]] [] 0 true true false false 0.241931457205962 0 1.33643748491 ["Character/Dirichlet/1580/1499"] "1-1580-1580.1503-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1503" [[0, 0.0]] [] 0 true true false false -0.22288634307877678 0 0.976497475453 ["Character/Dirichlet/1580/1503"] "1-1580-1580.1523-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1523" [[0, 0.0]] [] 0 true true false false 0.3436411002305183 0 1.37217889827 ["Character/Dirichlet/1580/1523"] "1-1580-1580.1527-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1527" [[0, 0.0]] [] 0 true true false false 0.13733668600065335 0 0.857417957737 ["Character/Dirichlet/1580/1527"] "1-1580-1580.1543-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1543" [[0, 0.0]] [] 0 true true false false -0.156981344690291 0 0.524764715813 ["Character/Dirichlet/1580/1543"] "1-1580-1580.1547-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1547" [[0, 0.0]] [] 0 true true false false 0.11684851828480104 0 1.02893715984 ["Character/Dirichlet/1580/1547"] "1-1580-1580.1559-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1559" [[0, 0.0]] [] 0 true true false false 0.4777209442203937 0 1.54701955919 ["Character/Dirichlet/1580/1559"] "1-1580-1580.1563-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1563" [[0, 0.0]] [] 0 true true false false -0.31497181960341863 0 0.381829508957 ["Character/Dirichlet/1580/1563"] "1-1580-1580.1579-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.1579" [[0, 0.0]] [] 0 true true true true 0.0 0 0.483242878585 ["Character/Dirichlet/1580/1579"] "1-1580-1580.163-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.163" [[0, 0.0]] [] 0 true true false false -0.2566429524403215 0 0.137233779613 ["Character/Dirichlet/1580/163"] "1-1580-1580.167-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.167" [[0, 0.0]] [] 0 true true false false 0.0478808960213761 0 0.675291573974 ["Character/Dirichlet/1580/167"] "1-1580-1580.183-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.183" [[0, 0.0]] [] 0 true true false false 0.4151749662450029 0 1.05032389786 ["Character/Dirichlet/1580/183"] "1-1580-1580.199-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.199" [[0, 0.0]] [] 0 true true false false -0.07290235481734793 0 1.13621328288 ["Character/Dirichlet/1580/199"] "1-1580-1580.203-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.203" [[0, 0.0]] [] 0 true true false false 0.2803066642276062 0 1.2829086037 ["Character/Dirichlet/1580/203"] "1-1580-1580.207-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.207" [[0, 0.0]] [] 0 true true false false -0.3929029176030569 0 0.4941550184 ["Character/Dirichlet/1580/207"] "1-1580-1580.219-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.219" [[0, 0.0]] [] 0 true true false false -0.24951862699413907 0 0.512280782963 ["Character/Dirichlet/1580/219"] "1-1580-1580.223-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.223" [[0, 0.0]] [] 0 true true false false 0.1387636284286353 0 1.21355814556 ["Character/Dirichlet/1580/223"] "1-1580-1580.23-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.23" [[0, 0.0]] [] 0 true true false false 0.09429968965073533 0 1.12236055059 ["Character/Dirichlet/1580/23"] "1-1580-1580.247-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.247" [[0, 0.0]] [] 0 true true false false 0.15494604902654718 0 1.10594124994 ["Character/Dirichlet/1580/247"] "1-1580-1580.263-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.263" [[0, 0.0]] [] 0 true true false false -0.038871505174130024 0 0.581789998763 ["Character/Dirichlet/1580/263"] "1-1580-1580.283-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.283" [[0, 0.0]] [] 0 true true false false -0.05935967288998232 0 0.589280025414 ["Character/Dirichlet/1580/283"] "1-1580-1580.287-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.287" [[0, 0.0]] [] 0 true true false false -0.4308888912221598 0 0.429350719986 ["Character/Dirichlet/1580/287"] "1-1580-1580.319-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.319" [[0, 0.0]] [] 0 true true false false -0.09355471171685649 0 0.849379405542 ["Character/Dirichlet/1580/319"] "1-1580-1580.327-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.327" [[0, 0.0]] [] 0 true true false false 0.08777605898000276 0 1.02775293886 ["Character/Dirichlet/1580/327"] "1-1580-1580.347-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.347" [[0, 0.0]] [] 0 true true false false -0.026546186903941917 0 1.10557493405 ["Character/Dirichlet/1580/347"] "1-1580-1580.359-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.359" [[0, 0.0]] [] 0 true true false false 0.042908186854484755 0 0.811190840144 ["Character/Dirichlet/1580/359"] "1-1580-1580.367-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.367" [[0, 0.0]] [] 0 true true false false 0.2027543780787253 0 0.939091729547 ["Character/Dirichlet/1580/367"] "1-1580-1580.379-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.379" [[0, 0.0]] [] 0 true true false false -0.3128435597109653 0 0.260098106008 ["Character/Dirichlet/1580/379"] "1-1580-1580.383-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.383" [[0, 0.0]] [] 0 true true false false -0.11684851828480104 0 0.476565982728 ["Character/Dirichlet/1580/383"] "1-1580-1580.39-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.39" [[0, 0.0]] [] 0 true true false false -0.241931457205962 0 0.610325769036 ["Character/Dirichlet/1580/39"] "1-1580-1580.403-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.403" [[0, 0.0]] [] 0 true true false false -0.15494604902654718 0 0.785861702563 ["Character/Dirichlet/1580/403"] "1-1580-1580.419-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.419" [[0, 0.0]] [] 0 true true false false 0.03147423714292071 0 0.903543023239 ["Character/Dirichlet/1580/419"] "1-1580-1580.427-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.427" [[0, 0.0]] [] 0 true true false false 0.156981344690291 0 0.852147535734 ["Character/Dirichlet/1580/427"] "1-1580-1580.447-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.447" [[0, 0.0]] [] 0 true true false false 0.27576423725982613 0 1.58036772685 ["Character/Dirichlet/1580/447"] "1-1580-1580.467-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.467" [[0, 0.0]] [] 0 true true false false -0.2803066642276062 0 0.276128093035 ["Character/Dirichlet/1580/467"] "1-1580-1580.483-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.483" [[0, 0.0]] [] 0 true true false false -0.12832729515340727 0 0.779956648801 ["Character/Dirichlet/1580/483"] "1-1580-1580.487-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.487" [[0, 0.0]] [] 0 true true false false 0.47731195192464654 0 0.208772372265 ["Character/Dirichlet/1580/487"] "1-1580-1580.523-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.523" [[0, 0.0]] [] 0 true true false false 0.4308888912221598 0 1.71316548518 ["Character/Dirichlet/1580/523"] "1-1580-1580.547-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.547" [[0, 0.0]] [] 0 true true false false -0.3011037607498632 0 0.863035174573 ["Character/Dirichlet/1580/547"] "1-1580-1580.559-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.559" [[0, 0.0]] [] 0 true true false false -0.12328461987586904 0 0.598294201059 ["Character/Dirichlet/1580/559"] "1-1580-1580.563-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.563" [[0, 0.0]] [] 0 true true false false -0.021262142148236185 0 0.94576538771 ["Character/Dirichlet/1580/563"] "1-1580-1580.59-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.59" [[0, 0.0]] [] 0 true true false false 0.16937958363087796 0 0.938084586647 ["Character/Dirichlet/1580/59"] "1-1580-1580.603-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.603" [[0, 0.0]] [] 0 true true false false 0.3929029176030569 0 1.3144270572 ["Character/Dirichlet/1580/603"] "1-1580-1580.619-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.619" [[0, 0.0]] [] 0 true true false false 0.12328461987586904 0 1.28486657037 ["Character/Dirichlet/1580/619"] "1-1580-1580.639-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.639" [[0, 0.0]] [] 0 true true false false 0.45150631072711345 0 0.19888403013 ["Character/Dirichlet/1580/639"] "1-1580-1580.643-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.643" [[0, 0.0]] [] 0 true true false false -0.08843213219478059 0 0.73341185678 ["Character/Dirichlet/1580/643"] "1-1580-1580.659-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.659" [[0, 0.0]] [] 0 true true false false 0.07290235481734793 0 1.26171585095 ["Character/Dirichlet/1580/659"] "1-1580-1580.663-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.663" [[0, 0.0]] [] 0 true true false false -0.2027543780787253 0 0.267753771194 ["Character/Dirichlet/1580/663"] "1-1580-1580.67-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.67" [[0, 0.0]] [] 0 true true false false 0.05935967288998232 0 1.01495617439 ["Character/Dirichlet/1580/67"] "1-1580-1580.679-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.679" [[0, 0.0]] [] 0 true true false false -0.16939189079821224 0 0.786039670896 ["Character/Dirichlet/1580/679"] "1-1580-1580.683-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.683" [[0, 0.0]] [] 0 true true false false 0.026546186903941917 0 1.02234702879 ["Character/Dirichlet/1580/683"] "1-1580-1580.687-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.687" [[0, 0.0]] [] 0 true true false false -0.09429968965073533 0 0.962229713607 ["Character/Dirichlet/1580/687"] "1-1580-1580.727-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.727" [[0, 0.0]] [] 0 true true false false 0.2566429524403215 0 0.729766422403 ["Character/Dirichlet/1580/727"] "1-1580-1580.739-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.739" [[0, 0.0]] [] 0 true true false false 0.23973160476115513 0 1.15906483415 ["Character/Dirichlet/1580/739"] "1-1580-1580.743-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.743" [[0, 0.0]] [] 0 true true false false -0.01922684648449237 0 1.31217560656 ["Character/Dirichlet/1580/743"] "1-1580-1580.747-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.747" [[0, 0.0]] [] 0 true true false false 0.08843213219478059 0 0.6982940551 ["Character/Dirichlet/1580/747"] "1-1580-1580.759-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.759" [[0, 0.0]] [] 0 true true false false -0.23973160476115513 0 0.141658457077 ["Character/Dirichlet/1580/759"] "1-1580-1580.763-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.763" [[0, 0.0]] [] 0 true true false false 0.09955604608504276 0 0.796805221834 ["Character/Dirichlet/1580/763"] "1-1580-1580.779-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.779" [[0, 0.0]] [] 0 true true false false -0.042908186854484755 0 0.585268154157 ["Character/Dirichlet/1580/779"] "1-1580-1580.783-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.783" [[0, 0.0]] [] 0 true true false false -0.4565148554023896 0 0.338491346051 ["Character/Dirichlet/1580/783"] "1-1580-1580.787-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.787" [[0, 0.0]] [] 0 true true false false 0.038871505174130024 0 1.17160501031 ["Character/Dirichlet/1580/787"] "1-1580-1580.803-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.803" [[0, 0.0]] [] 0 true true false false 0.3011037607498632 0 1.26106916202 ["Character/Dirichlet/1580/803"] "1-1580-1580.819-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.819" [[0, 0.0]] [] 0 true true false false -0.49456185890424315 0 1.69506151875 ["Character/Dirichlet/1580/819"] "1-1580-1580.83-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.83" [[0, 0.0]] [] 0 true true false false 0.4293015243835955 0 0.236151245752 ["Character/Dirichlet/1580/83"] "1-1580-1580.859-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.859" [[0, 0.0]] [] 0 true true false false 0.17648921734555253 0 0.803465061148 ["Character/Dirichlet/1580/859"] "1-1580-1580.863-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.863" [[0, 0.0]] [] 0 true true false false -0.47731195192464654 0 1.42762262079 ["Character/Dirichlet/1580/863"] "1-1580-1580.87-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.87" [[0, 0.0]] [] 0 true true false false 0.021262142148236185 0 0.902994294833 ["Character/Dirichlet/1580/87"] "1-1580-1580.887-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.887" [[0, 0.0]] [] 0 true true false false -0.3436411002305183 0 0.175958171534 ["Character/Dirichlet/1580/887"] "1-1580-1580.899-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.899" [[0, 0.0]] [] 0 true true false false 0.49456185890424315 0 0.303926622397 ["Character/Dirichlet/1580/899"] "1-1580-1580.907-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.907" [[0, 0.0]] [] 0 true true false false -0.09955604608504276 0 0.785598804275 ["Character/Dirichlet/1580/907"] "1-1580-1580.939-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.939" [[0, 0.0]] [] 0 true true false false -0.2996872530535747 0 0.300151720355 ["Character/Dirichlet/1580/939"] "1-1580-1580.967-r0-0-0" 7.337489152549826 7.337489152549826 1 1580 "1580.967" [[0, 0.0]] [] 0 true true false false -0.4151749662450029 0 0.272476265609 ["Character/Dirichlet/1580/967"] "1-1580-1580.1007-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1007" [[1, 0.0]] [] 0 true true false false 0.25748367921826965 0 0.0892850279949 ["Character/Dirichlet/1580/1007"] "1-1580-1580.1023-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1023" [[1, 0.0]] [] 0 true true false false -0.25748367921826965 0 1.38872520041 ["Character/Dirichlet/1580/1023"] "1-1580-1580.103-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.103" [[1, 0.0]] [] 0 true true false false -0.056629858444470965 0 0.957456159314 ["Character/Dirichlet/1580/103"] "1-1580-1580.1059-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1059" [[1, 0.0]] [] 0 true true false false -0.4311227508971007 0 0.00645619630306 ["Character/Dirichlet/1580/1059"] "1-1580-1580.107-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.107" [[1, 0.0]] [] 0 true true false false 0.3278357003485468 0 1.18028739028 ["Character/Dirichlet/1580/107"] "1-1580-1580.1079-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1079" [[1, 0.0]] [] 0 true true false false -0.3123398583275656 0 0.40721405114 ["Character/Dirichlet/1580/1079"] "1-1580-1580.1083-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1083" [[1, 0.0]] [] 0 true true false false -0.1195783327303124 0 0.199384659972 ["Character/Dirichlet/1580/1083"] "1-1580-1580.1087-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1087" [[1, 0.0]] [] 0 true true false false -0.322257364816498 0 1.60467903588 ["Character/Dirichlet/1580/1087"] "1-1580-1580.1099-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1099" [[1, 0.0]] [] 0 true true false false 0.13158924018500215 0 0.461063114757 ["Character/Dirichlet/1580/1099"] "1-1580-1580.1119-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1119" [[1, 0.0]] [] 0 true true false false -0.11079214366274512 0 0.286733559547 ["Character/Dirichlet/1580/1119"] "1-1580-1580.1123-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1123" [[1, 0.0]] [] 0 true true false false 0.17257761625491588 0 0.796444726223 ["Character/Dirichlet/1580/1123"] "1-1580-1580.1143-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1143" [[1, 0.0]] [] 0 true true false false 0.08128779521082057 0 0.986607405661 ["Character/Dirichlet/1580/1143"] "1-1580-1580.1147-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1147" [[1, 0.0]] [] 0 true true false false 0.015201740770043761 0 0.436971965102 ["Character/Dirichlet/1580/1147"] "1-1580-1580.1163-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1163" [[1, 0.0]] [] 0 true true false false 0.16237727741846925 0 0.566915737731 ["Character/Dirichlet/1580/1163"] "1-1580-1580.1167-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1167" [[1, 0.0]] [] 0 true true false false -0.16237727741846925 0 0.417138630582 ["Character/Dirichlet/1580/1167"] "1-1580-1580.1179-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1179" [[1, 0.0]] [] 0 true true false false 0.11079214366274512 0 0.789983258083 ["Character/Dirichlet/1580/1179"] "1-1580-1580.1183-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1183" [[1, 0.0]] [] 0 true true false false 0.15382736161857036 0 0.540535676342 ["Character/Dirichlet/1580/1183"] "1-1580-1580.119-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.119" [[1, 0.0]] [] 0 true true false false -0.3652177525086149 0 0.262987619888 ["Character/Dirichlet/1580/119"] "1-1580-1580.1243-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1243" [[1, 0.0]] [] 0 true true false false 0.389616848633002 0 0.0329676894991 ["Character/Dirichlet/1580/1243"] "1-1580-1580.1263-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1263" [[1, 0.0]] [] 0 true true false false -0.0881040955873917 0 0.272682668975 ["Character/Dirichlet/1580/1263"] "1-1580-1580.1267-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1267" [[1, 0.0]] [] 0 true true false false -0.00545061612946481 0 0.0558409992726 ["Character/Dirichlet/1580/1267"] "1-1580-1580.127-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.127" [[1, 0.0]] [] 0 true true false false -0.15162750917376344 0 0.27048229417 ["Character/Dirichlet/1580/127"] "1-1580-1580.1303-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1303" [[1, 0.0]] [] 0 true true false false -0.3300355527933537 0 1.64381731921 ["Character/Dirichlet/1580/1303"] "1-1580-1580.1307-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1307" [[1, 0.0]] [] 0 true true false false 0.13101228244187643 0 1.15091655483 ["Character/Dirichlet/1580/1307"] "1-1580-1580.1319-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1319" [[1, 0.0]] [] 0 true true false false 0.317596214761873 0 0.960613347673 ["Character/Dirichlet/1580/1319"] "1-1580-1580.1323-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1323" [[1, 0.0]] [] 0 true true false false 0.08127548804348626 0 0.530404197104 ["Character/Dirichlet/1580/1323"] "1-1580-1580.1327-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1327" [[1, 0.0]] [] 0 true true false false -0.2247394641235736 0 0.464089251088 ["Character/Dirichlet/1580/1327"] "1-1580-1580.1359-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1359" [[1, 0.0]] [] 0 true true false false -0.3314611431470703 0 0.00438511907521 ["Character/Dirichlet/1580/1359"] "1-1580-1580.1367-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1367" [[1, 0.0]] [] 0 true true false false 0.1195783327303124 0 0.397076089649 ["Character/Dirichlet/1580/1367"] "1-1580-1580.1379-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1379" [[1, 0.0]] [] 0 true true false false 0.4996719633926111 0 1.36083671197 ["Character/Dirichlet/1580/1379"] "1-1580-1580.1403-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1403" [[1, 0.0]] [] 0 true true false false 0.4984655559912814 0 1.15950278567 ["Character/Dirichlet/1580/1403"] "1-1580-1580.1419-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1419" [[1, 0.0]] [] 0 true true false false 0.45076740958673833 0 1.1664038867 ["Character/Dirichlet/1580/1419"] "1-1580-1580.1463-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1463" [[1, 0.0]] [] 0 true true false false -0.16100645040473963 0 0.715837619114 ["Character/Dirichlet/1580/1463"] "1-1580-1580.147-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.147" [[1, 0.0]] [] 0 true true false false 0.04519590873290693 0 0.578073438056 ["Character/Dirichlet/1580/147"] "1-1580-1580.1483-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1483" [[1, 0.0]] [] 0 true true false false -0.3385854685932526 0 0.385848833089 ["Character/Dirichlet/1580/1483"] "1-1580-1580.1507-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1507" [[1, 0.0]] [] 0 true true false false -0.035180524288477344 0 0.482628773469 ["Character/Dirichlet/1580/1507"] "1-1580-1580.1519-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1519" [[1, 0.0]] [] 0 true true false false 0.06825480418209004 0 0.914674929353 ["Character/Dirichlet/1580/1519"] "1-1580-1580.1539-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1539" [[1, 0.0]] [] 0 true true false false 0.3123398583275656 0 0.892030470102 ["Character/Dirichlet/1580/1539"] "1-1580-1580.1567-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.1567" [[1, 0.0]] [] 0 true true false false 0.2113887154632607 0 0.695624008455 ["Character/Dirichlet/1580/1567"] "1-1580-1580.179-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.179" [[1, 0.0]] [] 0 true true false false -0.48743805998452283 0 0.363090576958 ["Character/Dirichlet/1580/179"] "1-1580-1580.187-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.187" [[1, 0.0]] [] 0 true true false false -0.40645776331685146 0 1.19606636671 ["Character/Dirichlet/1580/187"] "1-1580-1580.19-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.19" [[1, 0.0]] [] 0 true true false false 0.003279061832394511 0 0.319165708944 ["Character/Dirichlet/1580/19"] "1-1580-1580.227-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.227" [[1, 0.0]] [] 0 true true false false 0.2645933129329442 0 1.01850121117 ["Character/Dirichlet/1580/227"] "1-1580-1580.239-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.239" [[1, 0.0]] [] 0 true true false false 0.3652177525086149 0 1.30892859768 ["Character/Dirichlet/1580/239"] "1-1580-1580.243-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.243" [[1, 0.0]] [] 0 true true false false -0.2113887154632607 0 0.666000960887 ["Character/Dirichlet/1580/243"] "1-1580-1580.259-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.259" [[1, 0.0]] [] 0 true true false false -0.06825480418209004 0 1.10387485571 ["Character/Dirichlet/1580/259"] "1-1580-1580.267-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.267" [[1, 0.0]] [] 0 true true false false -0.41733404550836517 0 1.55133019076 ["Character/Dirichlet/1580/267"] "1-1580-1580.27-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.27" [[1, 0.0]] [] 0 true true false false 0.16100645040473963 0 0.803922463375 ["Character/Dirichlet/1580/27"] "1-1580-1580.279-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.279" [[1, 0.0]] [] 0 true true false false 0.4311227508971007 0 1.22241692496 ["Character/Dirichlet/1580/279"] "1-1580-1580.299-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.299" [[1, 0.0]] [] 0 true true false false 0.273132275983973 0 0.186656941339 ["Character/Dirichlet/1580/299"] "1-1580-1580.3-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.3" [[1, 0.0]] [] 0 true true false false -0.1816588073042482 0 0.619146194926 ["Character/Dirichlet/1580/3"] "1-1580-1580.303-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.303" [[1, 0.0]] [] 0 true true false false 0.035180524288477344 0 0.965929299814 ["Character/Dirichlet/1580/303"] "1-1580-1580.307-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.307" [[1, 0.0]] [] 0 true true false false -0.21158315746618298 0 0.487745278997 ["Character/Dirichlet/1580/307"] "1-1580-1580.323-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.323" [[1, 0.0]] [] 0 true true false false 0.36340221513972176 0 0.784140093119 ["Character/Dirichlet/1580/323"] "1-1580-1580.339-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.339" [[1, 0.0]] [] 0 true true false false -0.317596214761873 0 0.146228561179 ["Character/Dirichlet/1580/339"] "1-1580-1580.343-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.343" [[1, 0.0]] [] 0 true true false false -0.015201740770043761 0 0.602262774464 ["Character/Dirichlet/1580/343"] "1-1580-1580.363-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.363" [[1, 0.0]] [] 0 true true false false -0.25749598638560395 0 0.267354632693 ["Character/Dirichlet/1580/363"] "1-1580-1580.387-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.387" [[1, 0.0]] [] 0 true true false false -0.08838512175816085 0 0.826634161615 ["Character/Dirichlet/1580/387"] "1-1580-1580.399-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.399" [[1, 0.0]] [] 0 true true false false 0.017405619970987173 0 0.478085937621 ["Character/Dirichlet/1580/399"] "1-1580-1580.407-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.407" [[1, 0.0]] [] 0 true true false false 0.420639319223905 0 0.777588657525 ["Character/Dirichlet/1580/407"] "1-1580-1580.423-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.423" [[1, 0.0]] [] 0 true true false false 0.15162750917376344 0 0.863732749082 ["Character/Dirichlet/1580/423"] "1-1580-1580.43-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.43" [[1, 0.0]] [] 0 true true false false -0.04519590873290693 0 0.320188794031 ["Character/Dirichlet/1580/43"] "1-1580-1580.439-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.439" [[1, 0.0]] [] 0 true true false false -0.4597768004339844 0 0.332510194298 ["Character/Dirichlet/1580/439"] "1-1580-1580.443-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.443" [[1, 0.0]] [] 0 true true false false -0.3278357003485468 0 0.387676319479 ["Character/Dirichlet/1580/443"] "1-1580-1580.459-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.459" [[1, 0.0]] [] 0 true true false false 0.48743805998452283 0 1.0033605995 ["Character/Dirichlet/1580/459"] "1-1580-1580.463-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.463" [[1, 0.0]] [] 0 true true false false -0.13101228244187643 0 0.512670423826 ["Character/Dirichlet/1580/463"] "1-1580-1580.47-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.47" [[1, 0.0]] [] 0 true true false false -0.08128779521082057 0 0.815143335096 ["Character/Dirichlet/1580/47"] "1-1580-1580.479-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.479" [[1, 0.0]] [] 0 true true false false 0.3314611431470703 0 1.03211655835 ["Character/Dirichlet/1580/479"] "1-1580-1580.499-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.499" [[1, 0.0]] [] 0 true true false false 0.003279061832394511 0 0.372437856156 ["Character/Dirichlet/1580/499"] "1-1580-1580.503-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.503" [[1, 0.0]] [] 0 true true false false 0.41733404550836517 0 0.135608345104 ["Character/Dirichlet/1580/503"] "1-1580-1580.507-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.507" [[1, 0.0]] [] 0 true true false false -0.2444311280491216 0 0.257378313356 ["Character/Dirichlet/1580/507"] "1-1580-1580.519-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.519" [[1, 0.0]] [] 0 true true false false -0.13158924018500215 0 0.428339880405 ["Character/Dirichlet/1580/519"] "1-1580-1580.527-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.527" [[1, 0.0]] [] 0 true true false false 0.1816588073042482 0 0.668339707088 ["Character/Dirichlet/1580/527"] "1-1580-1580.539-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.539" [[1, 0.0]] [] 0 true true false false -0.273132275983973 0 1.65596700373 ["Character/Dirichlet/1580/539"] "1-1580-1580.543-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.543" [[1, 0.0]] [] 0 true true false false 0.08838512175816085 0 0.84216894672 ["Character/Dirichlet/1580/543"] "1-1580-1580.567-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.567" [[1, 0.0]] [] 0 true true false false -0.17257761625491588 0 0.373623967671 ["Character/Dirichlet/1580/567"] "1-1580-1580.579-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.579" [[1, 0.0]] [] 0 true true false false -0.45076740958673833 0 0.27451997738 ["Character/Dirichlet/1580/579"] "1-1580-1580.583-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.583" [[1, 0.0]] [] 0 true true false false 0.40645776331685146 0 0.167588532842 ["Character/Dirichlet/1580/583"] "1-1580-1580.587-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.587" [[1, 0.0]] [] 0 true true false false -0.36340221513972176 0 0.236410326697 ["Character/Dirichlet/1580/587"] "1-1580-1580.599-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.599" [[1, 0.0]] [] 0 true true false false -0.47125557730259066 0 0.0514802191522 ["Character/Dirichlet/1580/599"] "1-1580-1580.607-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.607" [[1, 0.0]] [] 0 true true false false 0.4984655559912814 0 0.0661733613803 ["Character/Dirichlet/1580/607"] "1-1580-1580.623-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.623" [[1, 0.0]] [] 0 true true false false -0.3877913486409663 0 1.35355071425 ["Character/Dirichlet/1580/623"] "1-1580-1580.627-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.627" [[1, 0.0]] [] 0 true true false false 0.40094765529835696 0 0.748133993171 ["Character/Dirichlet/1580/627"] "1-1580-1580.63-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.63" [[1, 0.0]] [] 0 true true false false -0.40094765529835696 0 0.012806420321 ["Character/Dirichlet/1580/63"] "1-1580-1580.647-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.647" [[1, 0.0]] [] 0 true true false false -0.389616848633002 0 1.53203869469 ["Character/Dirichlet/1580/647"] "1-1580-1580.667-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.667" [[1, 0.0]] [] 0 true true false false 0.3877913486409663 0 0.172206174535 ["Character/Dirichlet/1580/667"] "1-1580-1580.699-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.699" [[1, 0.0]] [] 0 true true false false 0.47125557730259066 0 1.03615937283 ["Character/Dirichlet/1580/699"] "1-1580-1580.7-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.7" [[1, 0.0]] [] 0 true true false false -0.4603895936854949 0 0.494525627784 ["Character/Dirichlet/1580/7"] "1-1580-1580.703-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.703" [[1, 0.0]] [] 0 true true false false -0.2645933129329442 0 0.333136390166 ["Character/Dirichlet/1580/703"] "1-1580-1580.707-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.707" [[1, 0.0]] [] 0 true true false false -0.08127548804348626 0 0.721652016956 ["Character/Dirichlet/1580/707"] "1-1580-1580.719-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.719" [[1, 0.0]] [] 0 true true false false 0.4331580465608445 0 1.20397797448 ["Character/Dirichlet/1580/719"] "1-1580-1580.723-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.723" [[1, 0.0]] [] 0 true true false false 0.2444311280491216 0 1.00212270437 ["Character/Dirichlet/1580/723"] "1-1580-1580.767-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.767" [[1, 0.0]] [] 0 true true false false 0.056629858444470965 0 0.881271847945 ["Character/Dirichlet/1580/767"] "1-1580-1580.799-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.799" [[1, 0.0]] [] 0 true true false false 0.4597768004339844 0 1.44693389362 ["Character/Dirichlet/1580/799"] "1-1580-1580.807-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.807" [[1, 0.0]] [] 0 true true false false 0.3487858074296992 0 1.04989929637 ["Character/Dirichlet/1580/807"] "1-1580-1580.823-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.823" [[1, 0.0]] [] 0 true true false false -0.420639319223905 0 0.0126658339448 ["Character/Dirichlet/1580/823"] "1-1580-1580.827-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.827" [[1, 0.0]] [] 0 true true false false 0.25749598638560395 0 0.896464759465 ["Character/Dirichlet/1580/827"] "1-1580-1580.839-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.839" [[1, 0.0]] [] 0 true true false false 0.01899298680955146 0 0.661901784088 ["Character/Dirichlet/1580/839"] "1-1580-1580.843-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.843" [[1, 0.0]] [] 0 true true false false 0.00545061612946481 0 0.426844592357 ["Character/Dirichlet/1580/843"] "1-1580-1580.847-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.847" [[1, 0.0]] [] 0 true true false false 0.3385854685932526 0 1.27427172014 ["Character/Dirichlet/1580/847"] "1-1580-1580.867-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.867" [[1, 0.0]] [] 0 true true false false 0.3300355527933537 0 0.0743335925358 ["Character/Dirichlet/1580/867"] "1-1580-1580.879-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.879" [[1, 0.0]] [] 0 true true false false -0.4331580465608445 0 0.328922412414 ["Character/Dirichlet/1580/879"] "1-1580-1580.883-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.883" [[1, 0.0]] [] 0 true true false false -0.3487858074296992 0 0.0947719239067 ["Character/Dirichlet/1580/883"] "1-1580-1580.903-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.903" [[1, 0.0]] [] 0 true true false false 0.4603895936854949 0 1.5439094901 ["Character/Dirichlet/1580/903"] "1-1580-1580.919-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.919" [[1, 0.0]] [] 0 true true false false -0.01899298680955146 0 0.815937173666 ["Character/Dirichlet/1580/919"] "1-1580-1580.923-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.923" [[1, 0.0]] [] 0 true true false false 0.322257364816498 0 0.00408288231289 ["Character/Dirichlet/1580/923"] "1-1580-1580.927-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.927" [[1, 0.0]] [] 0 true true false false -0.43417496019221463 0 1.21948936045 ["Character/Dirichlet/1580/927"] "1-1580-1580.943-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.943" [[1, 0.0]] [] 0 true true false false 0.2247394641235736 0 1.07515067035 ["Character/Dirichlet/1580/943"] "1-1580-1580.947-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.947" [[1, 0.0]] [] 0 true true false false 0.0881040955873917 0 0.709957973333 ["Character/Dirichlet/1580/947"] "1-1580-1580.959-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.959" [[1, 0.0]] [] 0 true true false false 0.4996719633926111 0 0.150700930037 ["Character/Dirichlet/1580/959"] "1-1580-1580.963-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.963" [[1, 0.0]] [] 0 true true false false 0.43417496019221463 0 0.0613462353476 ["Character/Dirichlet/1580/963"] "1-1580-1580.979-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.979" [[1, 0.0]] [] 0 true true false false 0.3853497175086665 0 0.0262294493069 ["Character/Dirichlet/1580/979"] "1-1580-1580.983-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.983" [[1, 0.0]] [] 0 true true false false 0.21158315746618298 0 0.573924414824 ["Character/Dirichlet/1580/983"] "1-1580-1580.987-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.987" [[1, 0.0]] [] 0 true true false false -0.15382736161857036 0 0.530335474123 ["Character/Dirichlet/1580/987"] "1-1580-1580.99-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.99" [[1, 0.0]] [] 0 true true false false -0.017405619970987173 0 0.69944408403 ["Character/Dirichlet/1580/99"] "1-1580-1580.999-r1-0-0" 169.79458117550755 169.79458117550755 1 1580 "1580.999" [[1, 0.0]] [] 0 true true false false -0.3853497175086665 0 1.23558137483 ["Character/Dirichlet/1580/999"] # 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).