# lfunc_search downloaded from the LMFDB on 06 May 2026. # Search link: https://www.lmfdb.org/L/2/3870 # Query "{'degree': 2, 'conductor': 3870}" returned 222 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. "2-3870-1.1-c1-0-0" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.21000511431460651128875797403 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bp/1/1"] "2-3870-1.1-c1-0-1" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.40755224664390953414477364100 ["EllipticCurve/Q/3870/a", "ModularForm/GL2/Q/holomorphic/3870/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/a"] "2-3870-1.1-c1-0-10" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.71037599851234790570812809138 ["ModularForm/GL2/Q/holomorphic/3870/2/a/ba/1/2"] "2-3870-1.1-c1-0-11" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.74326626027804977047596165779 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bp/1/2"] "2-3870-1.1-c1-0-12" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.75304764582842998859369057359 ["EllipticCurve/Q/3870/n", "ModularForm/GL2/Q/holomorphic/3870/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/n"] "2-3870-1.1-c1-0-13" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.77617113516744728202405103641 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bh/1/1"] "2-3870-1.1-c1-0-14" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.77754418575142143397773359814 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bj/1/1"] "2-3870-1.1-c1-0-15" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.78835179387016024002027523859 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bp/1/5"] "2-3870-1.1-c1-0-16" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.800660285015148661781170479977 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bi/1/1"] "2-3870-1.1-c1-0-17" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.845030637156503738700964951450 ["ModularForm/GL2/Q/holomorphic/3870/2/a/be/1/2"] "2-3870-1.1-c1-0-18" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.872607638507278072608186150816 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bf/1/2"] "2-3870-1.1-c1-0-19" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.880013299064967561218289301905 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/2"] "2-3870-1.1-c1-0-2" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.42662781191717845990673199664 ["ModularForm/GL2/Q/holomorphic/3870/2/a/ba/1/1"] "2-3870-1.1-c1-0-20" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.890940811796379477327058742995 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/1"] "2-3870-1.1-c1-0-21" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.894406358981433353452153957829 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bj/1/2"] "2-3870-1.1-c1-0-22" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.897817399788257947504327421108 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bh/1/2"] "2-3870-1.1-c1-0-23" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.910888461676690457535636346468 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bo/1/2"] "2-3870-1.1-c1-0-24" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.976368800959382641191357436754 ["EllipticCurve/Q/3870/k", "ModularForm/GL2/Q/holomorphic/3870/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/k"] "2-3870-1.1-c1-0-25" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.979264501029070962266385473240 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bl/1/1"] "2-3870-1.1-c1-0-26" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.990101204987624459755574484830 ["EllipticCurve/Q/3870/h", "ModularForm/GL2/Q/holomorphic/3870/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/h"] "2-3870-1.1-c1-0-27" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.994629269792957959503019271601 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bl/1/2"] "2-3870-1.1-c1-0-28" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.00796227881335689970531424032 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bo/1/1"] "2-3870-1.1-c1-0-29" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.01334633012071426076652562989 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bp/1/6"] "2-3870-1.1-c1-0-3" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.51406677593355030712430276302 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bf/1/1"] "2-3870-1.1-c1-0-30" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.02153912935430409981072622490 ["EllipticCurve/Q/3870/t", "ModularForm/GL2/Q/holomorphic/3870/2/a/t/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/t"] "2-3870-1.1-c1-0-31" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.03953745023110355800911209023 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bm/1/1"] "2-3870-1.1-c1-0-32" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.05003507327214663885926555682 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bd/1/2"] "2-3870-1.1-c1-0-33" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.08707090313213128612585936333 ["EllipticCurve/Q/3870/s", "ModularForm/GL2/Q/holomorphic/3870/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/s"] "2-3870-1.1-c1-0-34" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.09320085129117620908703533017 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/4"] "2-3870-1.1-c1-0-35" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.11641784642073869342281311014 ["EllipticCurve/Q/3870/y", "ModularForm/GL2/Q/holomorphic/3870/2/a/y/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/y"] "2-3870-1.1-c1-0-36" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.13560483362698952190295613170 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bi/1/2"] "2-3870-1.1-c1-0-37" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.14814097317476177740082992620 ["EllipticCurve/Q/3870/z", "ModularForm/GL2/Q/holomorphic/3870/2/a/z/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/z"] "2-3870-1.1-c1-0-38" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.15863951134604777143460029973 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bb/1/1"] "2-3870-1.1-c1-0-39" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.16667102321887920979007288298 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/5"] "2-3870-1.1-c1-0-4" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.52551825956255616209965461358 ["ModularForm/GL2/Q/holomorphic/3870/2/a/be/1/1"] "2-3870-1.1-c1-0-40" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.16916968615876657557569283120 ["EllipticCurve/Q/3870/x", "ModularForm/GL2/Q/holomorphic/3870/2/a/x/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/x"] "2-3870-1.1-c1-0-41" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.21131856486192879183088786982 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/6"] "2-3870-1.1-c1-0-42" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.21905729867656325872124572113 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bq/1/3"] "2-3870-1.1-c1-0-43" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.22576960287231942082154307194 ["EllipticCurve/Q/3870/b", "ModularForm/GL2/Q/holomorphic/3870/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/b"] "2-3870-1.1-c1-0-44" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.23004969029106209547200796703 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bm/1/2"] "2-3870-1.1-c1-0-45" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.23772208431137089909348168934 ["EllipticCurve/Q/3870/c", "ModularForm/GL2/Q/holomorphic/3870/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/c"] "2-3870-1.1-c1-0-46" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.25220072523866641439396141055 ["EllipticCurve/Q/3870/f", "ModularForm/GL2/Q/holomorphic/3870/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/f"] "2-3870-1.1-c1-0-47" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.27036891596683652802584619705 ["EllipticCurve/Q/3870/e", "ModularForm/GL2/Q/holomorphic/3870/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/e"] "2-3870-1.1-c1-0-48" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.29815046478113059533226265104 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bc/1/2"] "2-3870-1.1-c1-0-49" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.30931584673592249335107008659 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bc/1/1"] "2-3870-1.1-c1-0-5" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53221703231498026268252832259 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bd/1/1"] "2-3870-1.1-c1-0-50" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.35468288861878469024251602523 ["EllipticCurve/Q/3870/g", "ModularForm/GL2/Q/holomorphic/3870/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/g"] "2-3870-1.1-c1-0-51" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.39056762545021354456166675646 ["EllipticCurve/Q/3870/i", "ModularForm/GL2/Q/holomorphic/3870/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/i"] "2-3870-1.1-c1-0-52" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.41003774800264438876082245457 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bg/1/1"] "2-3870-1.1-c1-0-53" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.44259388252548348849377509891 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bo/1/3"] "2-3870-1.1-c1-0-54" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.47279046569573052993068388951 ["EllipticCurve/Q/3870/j", "ModularForm/GL2/Q/holomorphic/3870/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/j"] "2-3870-1.1-c1-0-55" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.48295753689410127647498893499 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bb/1/2"] "2-3870-1.1-c1-0-56" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.50914869207913737312699996086 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bg/1/2"] "2-3870-1.1-c1-0-57" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.52459919406758677741083694609 ["EllipticCurve/Q/3870/l", "ModularForm/GL2/Q/holomorphic/3870/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/l"] "2-3870-1.1-c1-0-58" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.59349972275665488519900166934 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bk/1/1"] "2-3870-1.1-c1-0-59" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.59496713924787939012981895199 ["EllipticCurve/Q/3870/o", "ModularForm/GL2/Q/holomorphic/3870/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/o"] "2-3870-1.1-c1-0-6" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.61453011688113596872235771878 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bp/1/4"] "2-3870-1.1-c1-0-60" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.60648775244345002228949754277 ["EllipticCurve/Q/3870/p", "ModularForm/GL2/Q/holomorphic/3870/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/p"] "2-3870-1.1-c1-0-61" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.61936463039180774510171006893 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bn/1/1"] "2-3870-1.1-c1-0-62" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.63110901992609325411383107007 ["EllipticCurve/Q/3870/q", "ModularForm/GL2/Q/holomorphic/3870/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/q"] "2-3870-1.1-c1-0-63" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.65432668113079297419751568145 ["EllipticCurve/Q/3870/r", "ModularForm/GL2/Q/holomorphic/3870/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/r"] "2-3870-1.1-c1-0-64" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.72325756871934914070885352380 ["EllipticCurve/Q/3870/u", "ModularForm/GL2/Q/holomorphic/3870/2/a/u/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/u"] "2-3870-1.1-c1-0-65" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.76967905770166065871918754646 ["EllipticCurve/Q/3870/v", "ModularForm/GL2/Q/holomorphic/3870/2/a/v/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/v"] "2-3870-1.1-c1-0-66" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.79556781925727990335298059867 ["EllipticCurve/Q/3870/w", "ModularForm/GL2/Q/holomorphic/3870/2/a/w/1/1", "ModularForm/GL2/Q/holomorphic/3870/2/a/w"] "2-3870-1.1-c1-0-67" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.79806162690780310729354871055 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bn/1/2"] "2-3870-1.1-c1-0-68" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.82867123022624285959568474280 ["ModularForm/GL2/Q/holomorphic/3870/2/a/bn/1/3"] "2-3870-1.1-c1-0-69" 5.558966233233974 30.902105582235524 2 3870 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.95795442751144047369011811559 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["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/46"] "2-3870-645.644-c1-0-83" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false 0.49543411741601456 0 1.63379763219909685537619439556 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/68"] "2-3870-645.644-c1-0-84" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false -0.29863131837941953 0 1.65537809129191439987446162211 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/53"] "2-3870-645.644-c1-0-85" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false 0.2633218224069776 0 1.70986893502471169643580131669 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/40"] "2-3870-645.644-c1-0-86" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false -0.2734939529616213 0 1.71673489175983874846132618368 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/49"] "2-3870-645.644-c1-0-87" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false -0.4345483713306757 0 1.95704364530723617283321822294 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/47"] "2-3870-645.644-c1-0-9" 5.558966233233974 30.902105582235524 2 3870 "645.644" [] [[0.5, 0.0]] 1 true true false false 0.39728195763588414 0 0.19125085157660774354387835005 ["ModularForm/GL2/Q/holomorphic/3870/2/f/a/3869/52"] # 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).