# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/2/4232 # Query "{'degree': 2, 'conductor': 4232}" returned 126 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. "2-4232-1.1-c1-0-0" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.11301769854140592958173141172 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/3"] "2-4232-1.1-c1-0-1" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.14859341265290506009171798570 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/7"] "2-4232-1.1-c1-0-10" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.36304465069762870646774909803 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/3"] "2-4232-1.1-c1-0-100" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.50348501760894694313632997160 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/6"] "2-4232-1.1-c1-0-101" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.50522161353544676434886475921 ["EllipticCurve/Q/4232/f", "ModularForm/GL2/Q/holomorphic/4232/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/f"] "2-4232-1.1-c1-0-102" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.53380455915213139797559434035 ["ModularForm/GL2/Q/holomorphic/4232/2/a/o/1/2"] "2-4232-1.1-c1-0-103" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.55426095599133298481854782958 ["ModularForm/GL2/Q/holomorphic/4232/2/a/s/1/4"] "2-4232-1.1-c1-0-104" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.56583096685929030785987702066 ["ModularForm/GL2/Q/holomorphic/4232/2/a/n/1/2"] "2-4232-1.1-c1-0-105" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.57815875676999702689557542755 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/14"] "2-4232-1.1-c1-0-106" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.59856207619618760058184258617 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/10"] "2-4232-1.1-c1-0-107" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.60092847267324975255695188707 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/9"] "2-4232-1.1-c1-0-108" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.61969086792946089852460180414 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/15"] "2-4232-1.1-c1-0-109" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.62026166754184965942299973128 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/10"] "2-4232-1.1-c1-0-11" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.37571011846328827779060800975 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/8"] "2-4232-1.1-c1-0-110" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.63176868344243567313541085783 ["ModularForm/GL2/Q/holomorphic/4232/2/a/u/1/2"] "2-4232-1.1-c1-0-111" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.65379462603776780591131879725 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/15"] "2-4232-1.1-c1-0-112" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.66666741792326794466452080210 ["ModularForm/GL2/Q/holomorphic/4232/2/a/l/1/2"] "2-4232-1.1-c1-0-113" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.69574328988234199689761978619 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/9"] "2-4232-1.1-c1-0-114" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.72448900237207768978902674922 ["EllipticCurve/Q/4232/i", "ModularForm/GL2/Q/holomorphic/4232/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/i"] "2-4232-1.1-c1-0-115" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.74139777217839552616782994163 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/14"] "2-4232-1.1-c1-0-116" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.74708163067258245097425488924 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/13"] "2-4232-1.1-c1-0-117" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.76209637763590367342151347497 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/13"] "2-4232-1.1-c1-0-118" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.78338800412453402089141130063 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/12"] "2-4232-1.1-c1-0-119" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.79008011979486974669513582231 ["ModularForm/GL2/Q/holomorphic/4232/2/a/t/1/3"] "2-4232-1.1-c1-0-12" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.43741995721352758800186104953 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/5"] "2-4232-1.1-c1-0-120" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.79949217230409809256128206539 ["ModularForm/GL2/Q/holomorphic/4232/2/a/u/1/3"] "2-4232-1.1-c1-0-121" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.80429698205386515014777856126 ["EllipticCurve/Q/4232/h", "ModularForm/GL2/Q/holomorphic/4232/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/h"] "2-4232-1.1-c1-0-122" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.83036306716282595731336402118 ["ModularForm/GL2/Q/holomorphic/4232/2/a/u/1/4"] "2-4232-1.1-c1-0-123" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.88133503390189803246768829430 ["ModularForm/GL2/Q/holomorphic/4232/2/a/t/1/4"] "2-4232-1.1-c1-0-124" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.93051844814768305172406664986 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/15"] "2-4232-1.1-c1-0-125" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.95483726278169801592174862894 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/11"] "2-4232-1.1-c1-0-13" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.47448293208521989088624572972 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/2"] "2-4232-1.1-c1-0-14" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.49399685391662444455397513362 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/3"] "2-4232-1.1-c1-0-15" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.51325867860755018422212720232 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/5"] "2-4232-1.1-c1-0-16" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.53594425065794722681186413665 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/1"] "2-4232-1.1-c1-0-17" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.54620442237640254295606961864 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/9"] "2-4232-1.1-c1-0-18" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.56132019561071802782895764419 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/5"] "2-4232-1.1-c1-0-19" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.58068238686505894696412901157 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/2"] "2-4232-1.1-c1-0-2" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.19346616891994817642180739125 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/1"] "2-4232-1.1-c1-0-20" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.58542207181325975270198684467 ["ModularForm/GL2/Q/holomorphic/4232/2/a/r/1/3"] "2-4232-1.1-c1-0-21" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.60018318571631462171877453488 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/5"] "2-4232-1.1-c1-0-22" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.60853710882646040868322592327 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/7"] "2-4232-1.1-c1-0-23" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.64575997440847504015086996864 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/4"] "2-4232-1.1-c1-0-24" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.65536906326806390753875862888 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/11"] "2-4232-1.1-c1-0-25" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.65998037406009917389791340426 ["EllipticCurve/Q/4232/c", "ModularForm/GL2/Q/holomorphic/4232/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/c"] "2-4232-1.1-c1-0-26" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.67869560734292311864316731399 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/2"] "2-4232-1.1-c1-0-27" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.67881973729472760646099230435 ["EllipticCurve/Q/4232/b", "ModularForm/GL2/Q/holomorphic/4232/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/b"] "2-4232-1.1-c1-0-28" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.68052083036249587872546033615 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/1"] "2-4232-1.1-c1-0-29" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.68395167712277210571966003749 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/9"] "2-4232-1.1-c1-0-3" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.27417750125880409811690463192 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/4"] "2-4232-1.1-c1-0-30" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.70011938121512403580863099121 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/7"] "2-4232-1.1-c1-0-31" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.70267166236837410255085372228 ["ModularForm/GL2/Q/holomorphic/4232/2/a/p/1/1"] "2-4232-1.1-c1-0-32" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.71479664800044447057246204375 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/2"] "2-4232-1.1-c1-0-33" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.72963444537470928424503481375 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/3"] "2-4232-1.1-c1-0-34" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.73527605129130601965263563952 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/10"] "2-4232-1.1-c1-0-35" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.78777293489746843424094488711 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/2"] "2-4232-1.1-c1-0-36" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.806473064193087584911795600593 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/6"] "2-4232-1.1-c1-0-37" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.810971917054565588325840186253 ["ModularForm/GL2/Q/holomorphic/4232/2/a/p/1/2"] "2-4232-1.1-c1-0-38" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.827107142346132017844584957978 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/8"] "2-4232-1.1-c1-0-39" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.864910541179198035741653497218 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/3"] "2-4232-1.1-c1-0-4" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.28194400507159637541374760594 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/6"] "2-4232-1.1-c1-0-40" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.872082387208203732696420144908 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/10"] "2-4232-1.1-c1-0-41" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.892868954002520655119999180288 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/9"] "2-4232-1.1-c1-0-42" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.893712482833436799650090792193 ["ModularForm/GL2/Q/holomorphic/4232/2/a/s/1/1"] "2-4232-1.1-c1-0-43" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.902325711841343435592434102955 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/4"] "2-4232-1.1-c1-0-44" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.907679385378854299246195186136 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/5"] "2-4232-1.1-c1-0-45" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.920539542996875272749540144026 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/4"] "2-4232-1.1-c1-0-46" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.923853708795923884708244594394 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/3"] "2-4232-1.1-c1-0-47" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.931996709263695310929191080094 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/4"] "2-4232-1.1-c1-0-48" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.943616059233076456121247915724 ["ModularForm/GL2/Q/holomorphic/4232/2/a/r/1/2"] "2-4232-1.1-c1-0-49" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.965204595135563988441797034842 ["ModularForm/GL2/Q/holomorphic/4232/2/a/o/1/1"] "2-4232-1.1-c1-0-5" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.31577642208192868426932196748 ["ModularForm/GL2/Q/holomorphic/4232/2/a/q/1/1"] "2-4232-1.1-c1-0-50" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 0.980902911099197828524094242923 ["ModularForm/GL2/Q/holomorphic/4232/2/a/m/1/1"] "2-4232-1.1-c1-0-51" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.00063053954540200548890126126 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/2"] "2-4232-1.1-c1-0-52" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.01597401477912307902189339188 ["ModularForm/GL2/Q/holomorphic/4232/2/a/l/1/1"] "2-4232-1.1-c1-0-53" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.02299203229363560417019783271 ["ModularForm/GL2/Q/holomorphic/4232/2/a/t/1/1"] "2-4232-1.1-c1-0-54" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.02953148667658514896058995572 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/6"] "2-4232-1.1-c1-0-55" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.03162960832014507406945854718 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/11"] "2-4232-1.1-c1-0-56" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.04916092919213393146990368874 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/12"] "2-4232-1.1-c1-0-57" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.05447424687138421819906381000 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/6"] "2-4232-1.1-c1-0-58" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.08056964453480419089812764480 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/4"] "2-4232-1.1-c1-0-59" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.08379004953026440689311770431 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/3"] "2-4232-1.1-c1-0-6" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 0.33133094822572756385577275580 ["EllipticCurve/Q/4232/g", "ModularForm/GL2/Q/holomorphic/4232/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/g"] "2-4232-1.1-c1-0-60" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.09207714578435870764691477159 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/11"] "2-4232-1.1-c1-0-61" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.10495002710607792705436134967 ["EllipticCurve/Q/4232/a", "ModularForm/GL2/Q/holomorphic/4232/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/a"] "2-4232-1.1-c1-0-62" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.11317389572375369556147991032 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/13"] "2-4232-1.1-c1-0-63" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.12060603206109047069198111601 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/5"] "2-4232-1.1-c1-0-64" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.12595790299866945065869493777 ["ModularForm/GL2/Q/holomorphic/4232/2/a/k/1/2"] "2-4232-1.1-c1-0-65" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.13034072062787753744301249711 ["EllipticCurve/Q/4232/j", "ModularForm/GL2/Q/holomorphic/4232/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/j"] "2-4232-1.1-c1-0-66" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.14644583961839750929191711161 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/11"] "2-4232-1.1-c1-0-67" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.14796285619962485411544033116 ["ModularForm/GL2/Q/holomorphic/4232/2/a/k/1/1"] "2-4232-1.1-c1-0-68" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.15717606700273143679368932365 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/8"] "2-4232-1.1-c1-0-69" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.15793543775604427001474439638 ["ModularForm/GL2/Q/holomorphic/4232/2/a/s/1/2"] "2-4232-1.1-c1-0-7" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.34045213405961695054907080134 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/1"] "2-4232-1.1-c1-0-70" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.16122487866165607841646531964 ["ModularForm/GL2/Q/holomorphic/4232/2/a/r/1/4"] "2-4232-1.1-c1-0-71" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.16952296049376472131707578439 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/13"] "2-4232-1.1-c1-0-72" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.17461094001747730469015874270 ["EllipticCurve/Q/4232/e", "ModularForm/GL2/Q/holomorphic/4232/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/e"] "2-4232-1.1-c1-0-73" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.17474831820021114013273191022 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/6"] "2-4232-1.1-c1-0-74" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.19004835784414914984101205616 ["ModularForm/GL2/Q/holomorphic/4232/2/a/n/1/1"] "2-4232-1.1-c1-0-75" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.20507596545394901889088580641 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/14"] "2-4232-1.1-c1-0-76" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.20677403481020638843826571440 ["ModularForm/GL2/Q/holomorphic/4232/2/a/w/1/7"] "2-4232-1.1-c1-0-77" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.26186231846982987286930291921 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/1"] "2-4232-1.1-c1-0-78" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.26346131015850082697749261034 ["ModularForm/GL2/Q/holomorphic/4232/2/a/t/1/2"] "2-4232-1.1-c1-0-79" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.26875286583462086548450788431 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/7"] "2-4232-1.1-c1-0-8" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.34545999562453121947626447955 ["ModularForm/GL2/Q/holomorphic/4232/2/a/r/1/1"] "2-4232-1.1-c1-0-80" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.28230927094082958975904356713 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/8"] "2-4232-1.1-c1-0-81" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.28942611930565601018090779195 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/10"] "2-4232-1.1-c1-0-82" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.29715090699822229637221949127 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/2"] "2-4232-1.1-c1-0-83" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.30145306334386218476653972893 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/12"] "2-4232-1.1-c1-0-84" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.32424076463810419210362053649 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/4"] "2-4232-1.1-c1-0-85" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.32529523601093230209424204704 ["ModularForm/GL2/Q/holomorphic/4232/2/a/u/1/1"] "2-4232-1.1-c1-0-86" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.33897236937056561831204889491 ["ModularForm/GL2/Q/holomorphic/4232/2/a/z/1/14"] "2-4232-1.1-c1-0-87" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.35580916776763394871368172209 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/5"] "2-4232-1.1-c1-0-88" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.37891908120084316198339800922 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/12"] "2-4232-1.1-c1-0-89" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.39476024529286609505595219818 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/8"] "2-4232-1.1-c1-0-9" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 0.35336444923563368525439558407 ["ModularForm/GL2/Q/holomorphic/4232/2/a/x/1/1"] "2-4232-1.1-c1-0-90" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.41059479618211545946334417824 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/1"] "2-4232-1.1-c1-0-91" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 1.41136839608173212498999778853 ["EllipticCurve/Q/4232/d", "ModularForm/GL2/Q/holomorphic/4232/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/4232/2/a/d"] "2-4232-1.1-c1-0-92" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.41341794776757882465034739326 ["ModularForm/GL2/Q/holomorphic/4232/2/a/v/1/6"] "2-4232-1.1-c1-0-93" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.42905997454007367838358075211 ["ModularForm/GL2/Q/holomorphic/4232/2/a/s/1/3"] "2-4232-1.1-c1-0-94" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.42939555400733453358451497408 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/12"] "2-4232-1.1-c1-0-95" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.43203746107204954243244158227 ["ModularForm/GL2/Q/holomorphic/4232/2/a/bb/1/15"] "2-4232-1.1-c1-0-96" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.44951433902820413650524524863 ["ModularForm/GL2/Q/holomorphic/4232/2/a/q/1/2"] "2-4232-1.1-c1-0-97" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.47314251883802644759138886178 ["ModularForm/GL2/Q/holomorphic/4232/2/a/y/1/7"] "2-4232-1.1-c1-0-98" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.49347615551611498430571112290 ["ModularForm/GL2/Q/holomorphic/4232/2/a/ba/1/8"] "2-4232-1.1-c1-0-99" 5.813148040038708 33.79269013540588 2 4232 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 1.49437037896534227898197973393 ["ModularForm/GL2/Q/holomorphic/4232/2/a/m/1/2"] # 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).