# lfunc_search downloaded from the LMFDB on 23 April 2026. # Search link: https://www.lmfdb.org/L/2/12^2/144.131/c1-0 # Query "{'degree': 2, 'conductor': 144, 'spectral_label': 'c1-0'}" returned 236 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. "2-12e2-1.1-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.92209901273574427656973400179 ["EllipticCurve/Q/144/a", "ModularForm/GL2/Q/holomorphic/144/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/144/2/a/a"] "2-12e2-1.1-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 1.97363471421369880359461961552 ["EllipticCurve/Q/144/b", "ModularForm/GL2/Q/holomorphic/144/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/144/2/a/b"] "2-12e2-12.11-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "12.11" [] [[0.5, 0.0]] 1 true true false false -0.1520433619923482 0 1.37578877750920465414474926871 ["ModularForm/GL2/Q/holomorphic/144/2/c/a/143/2"] "2-12e2-12.11-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "12.11" [] [[0.5, 0.0]] 1 true true false false 0.1520433619923482 0 2.59850145117782529263796106565 ["ModularForm/GL2/Q/holomorphic/144/2/c/a/143/1"] "2-12e2-144.11-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.4718935043589696 0 0.03615152167066825982388299659 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/3"] "2-12e2-144.11-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.3885943103334025 0 0.23353789861414415743521203947 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/13"] "2-12e2-144.11-c1-0-10" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.00024155296965125975 0 1.96082088071371502367803087194 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/10"] "2-12e2-144.11-c1-0-11" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.11914913070758776 0 2.05239593328193508966205664191 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/8"] "2-12e2-144.11-c1-0-12" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.021128109955185263 0 2.12382356680289453530598186440 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/6"] "2-12e2-144.11-c1-0-13" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.058850523677659076 0 2.12535137948752662973328731355 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/21"] "2-12e2-144.11-c1-0-14" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.10924781940598116 0 2.25950116544637645280007566158 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/16"] "2-12e2-144.11-c1-0-15" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.10936944723101912 0 2.41033730974772165315687865063 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/20"] "2-12e2-144.11-c1-0-16" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.11043918748648854 0 2.45300818335388549102721879228 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/15"] "2-12e2-144.11-c1-0-17" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.3853049117059345 0 2.69993957789130575308045979522 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/4"] "2-12e2-144.11-c1-0-18" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.09122651779784474 0 2.98881969431127000975737198929 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/22"] "2-12e2-144.11-c1-0-19" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.2813641466071868 0 3.08905815284900212354190935436 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/17"] "2-12e2-144.11-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.3265061898183975 0 0.36757874302497931361126040671 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/9"] "2-12e2-144.11-c1-0-20" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.2994464769829798 0 3.30007200009331022370235197300 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/18"] "2-12e2-144.11-c1-0-21" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.364730244099081 0 3.66484881578482014220812167983 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/14"] "2-12e2-144.11-c1-0-3" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.050634465046377024 0 1.00039119801692588905605276178 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/5"] "2-12e2-144.11-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.3459968922584117 0 1.14321988557430552495786442776 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/7"] "2-12e2-144.11-c1-0-5" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.31741620646952 0 1.42121077885755049292857623763 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/12"] "2-12e2-144.11-c1-0-6" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.12339614628192337 0 1.52596933061496290350472427046 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/1"] "2-12e2-144.11-c1-0-7" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false -0.13676111363293406 0 1.79581985841554414074372007152 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/19"] "2-12e2-144.11-c1-0-8" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.15508442647725468 0 1.80525785726359857899404836537 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/11"] "2-12e2-144.11-c1-0-9" 1.072308625865911 1.149845789106438 2 144 "144.11" [] [[0.5, 0.0]] 1 true true false false 0.19554706335271338 0 1.86300766391380955148194981557 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/11/2"] "2-12e2-144.13-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.44255678574436147 0 0.67173066330893369274861301257 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/8"] "2-12e2-144.13-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.17361111111111113 0 0.913121273494331531416946209800 ["ModularForm/GL2/Q/holomorphic/144/2/x/d/13/1"] "2-12e2-144.13-c1-0-10" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.1304394693491581 0 2.03487701293062148527152137107 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/6"] "2-12e2-144.13-c1-0-11" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.0011788826067379932 0 2.30662752398928025661270380539 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/17"] "2-12e2-144.13-c1-0-12" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.15280589044883086 0 2.37935441028735065629684248108 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/1"] "2-12e2-144.13-c1-0-13" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.4930555555555556 1 2.50903389517737029009421037721 ["ModularForm/GL2/Q/holomorphic/144/2/x/b/13/1"] "2-12e2-144.13-c1-0-14" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.09282907575364927 0 2.65176030441082794634860919120 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/14"] "2-12e2-144.13-c1-0-15" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.11155013671236955 0 2.65220811412533319378174478660 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/7"] "2-12e2-144.13-c1-0-16" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.303851336242357 0 2.68695137991744985013307939998 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/10"] "2-12e2-144.13-c1-0-17" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.11291387649504755 0 2.73509517102630894336564437286 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/12"] "2-12e2-144.13-c1-0-18" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.07152989429388465 0 2.87699363983037632456249618391 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/18"] "2-12e2-144.13-c1-0-19" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.2728037385588813 0 3.35074161569206822159831767250 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/16"] "2-12e2-144.13-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.2920016328401735 0 1.19783216764023036779871761904 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/2"] "2-12e2-144.13-c1-0-20" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.32577922341706733 0 3.36013875664827886000245817875 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/11"] "2-12e2-144.13-c1-0-21" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.4749463818204737 0 3.36631809577162202491171509547 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/4"] "2-12e2-144.13-c1-0-3" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.006944444444444444 0 1.27186660900970486748029768107 ["ModularForm/GL2/Q/holomorphic/144/2/x/c/13/1"] "2-12e2-144.13-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.31201458446255964 0 1.37937119597792766440501865919 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/13"] "2-12e2-144.13-c1-0-5" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.221583229335055 0 1.56013491937843888351242037489 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/5"] "2-12e2-144.13-c1-0-6" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.006944444444444444 0 1.58469767340781759911411983391 ["ModularForm/GL2/Q/holomorphic/144/2/x/a/13/1"] "2-12e2-144.13-c1-0-7" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.31856476319683585 0 1.63219223543811628058494970752 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/9"] "2-12e2-144.13-c1-0-8" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false -0.03310704167207768 0 1.83887080298154488061909788628 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/15"] "2-12e2-144.13-c1-0-9" 1.072308625865911 1.149845789106438 2 144 "144.13" [] [[0.5, 0.0]] 1 true true false false 0.08210881191432762 0 1.93730393185029525247924250472 ["ModularForm/GL2/Q/holomorphic/144/2/x/e/13/3"] "2-12e2-144.131-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.19554706335271338 0 0.60872501127766903055681199334 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/2"] "2-12e2-144.131-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.3853049117059345 0 0.897307741402226691609561730906 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/4"] "2-12e2-144.131-c1-0-10" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.11043918748648854 0 2.20246958531668556763664102473 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/15"] "2-12e2-144.131-c1-0-11" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.11914913070758776 0 2.23010514496125058762942438519 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/8"] "2-12e2-144.131-c1-0-12" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.09122651779784474 0 2.27121385717602140832234663856 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/22"] "2-12e2-144.131-c1-0-13" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.4718935043589696 0 2.46228724836530983931570670800 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/3"] "2-12e2-144.131-c1-0-14" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.10924781940598116 0 2.49396301961513652882474500168 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/16"] "2-12e2-144.131-c1-0-15" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.3459968922584117 0 2.50278069165753680827078407150 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/7"] "2-12e2-144.131-c1-0-16" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.13676111363293406 0 2.52192766952375161552322216932 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/19"] "2-12e2-144.131-c1-0-17" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.058850523677659076 0 2.72645727321152744346519573415 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/21"] "2-12e2-144.131-c1-0-18" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.10936944723101912 0 2.97727661325286391428365372281 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/20"] "2-12e2-144.131-c1-0-19" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.3265061898183975 0 3.01516186031423621831627930050 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/9"] "2-12e2-144.131-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.2994464769829798 0 1.20320517485560631221117053512 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/18"] "2-12e2-144.131-c1-0-20" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false 0.31741620646952 0 3.07381775016345092638191665480 ["ModularForm/GL2/Q/holomorphic/144/2/u/a/131/12"] "2-12e2-144.131-c1-0-21" 1.072308625865911 1.149845789106438 2 144 "144.131" [] [[0.5, 0.0]] 1 true true false false -0.3885943103334025 0 3.90636923596638983549079528261 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0.0]] 1 true true false false 0.1388888888888889 0 2.41336522341264974174290916616 ["ModularForm/GL2/Q/holomorphic/144/2/s/c/95/1"] "2-12e2-36.23-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "36.23" [] [[0.5, 0.0]] 1 true true false false 0.1388888888888889 0 2.72349697245102963239984558617 ["ModularForm/GL2/Q/holomorphic/144/2/s/d/95/1"] "2-12e2-36.23-c1-0-5" 1.072308625865911 1.149845789106438 2 144 "36.23" [] [[0.5, 0.0]] 1 true true false false 0.47222222222222227 1 3.10905564533405997568619502780 ["ModularForm/GL2/Q/holomorphic/144/2/s/a/95/1"] "2-12e2-48.11-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false -0.35670773164002845 0 0.34622736850952760303374964206 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/1"] "2-12e2-48.11-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false -0.40462611738292853 0 0.947942240431247221091139007976 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/5"] "2-12e2-48.11-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false -0.1161621418318913 0 1.60107677076249520353303756797 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/3"] "2-12e2-48.11-c1-0-3" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false 0.12749599085484828 0 1.80752032818315695640285654960 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/2"] "2-12e2-48.11-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false -0.17659073312984813 0 1.85786522019020539428296435516 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/7"] "2-12e2-48.11-c1-0-5" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false -0.05262100765533209 0 2.52957698623826832091819826160 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/8"] "2-12e2-48.11-c1-0-6" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false 0.29128715863237514 0 2.58912882997263270749218077864 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/4"] "2-12e2-48.11-c1-0-7" 1.072308625865911 1.149845789106438 2 144 "48.11" [] [[0.5, 0.0]] 1 true true false false 0.18792458215280508 0 2.75380374741549645899048178440 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/107/6"] "2-12e2-48.35-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "48.35" [] [[0.5, 0.0]] 1 true true false false -0.29128715863237514 0 1.36942597756664329621759116926 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/35/4"] "2-12e2-48.35-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "48.35" [] [[0.5, 0.0]] 1 true true false false -0.12749599085484828 0 1.50493045485174487212830001669 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/35/2"] "2-12e2-48.35-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "48.35" [] [[0.5, 0.0]] 1 true true false false 0.1161621418318913 0 1.73326008034501168371744012335 ["ModularForm/GL2/Q/holomorphic/144/2/l/a/35/3"] "2-12e2-48.35-c1-0-3" 1.072308625865911 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["ModularForm/GL2/Q/holomorphic/144/2/l/a/35/5"] "2-12e2-9.4-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.20432586044977977 0 0.945217057579748622313714488255 ["ModularForm/GL2/Q/holomorphic/144/2/i/d/49/1"] "2-12e2-9.4-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.2777777777777778 0 1.10309822942172362190392132849 ["ModularForm/GL2/Q/holomorphic/144/2/i/a/49/1"] "2-12e2-9.4-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "9.4" [] [[0.5, 0.0]] 1 true true false false -0.01789636177244246 0 2.30975243910824142642966673921 ["ModularForm/GL2/Q/holomorphic/144/2/i/d/49/2"] "2-12e2-9.4-c1-0-3" 1.072308625865911 1.149845789106438 2 144 "9.4" [] [[0.5, 0.0]] 1 true true false false 0.05555555555555555 0 2.35108135638940597967821092355 ["ModularForm/GL2/Q/holomorphic/144/2/i/c/49/1"] "2-12e2-9.4-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "9.4" [] [[0.5, 0.0]] 1 true true false false 0.2222222222222222 0 2.83772857287000543069694881152 ["ModularForm/GL2/Q/holomorphic/144/2/i/b/49/1"] "2-12e2-9.7-c1-0-0" 1.072308625865911 1.149845789106438 2 144 "9.7" [] [[0.5, 0.0]] 1 true true false false -0.2222222222222222 0 1.31259399580731222414321464551 ["ModularForm/GL2/Q/holomorphic/144/2/i/b/97/1"] "2-12e2-9.7-c1-0-1" 1.072308625865911 1.149845789106438 2 144 "9.7" [] [[0.5, 0.0]] 1 true true false false -0.05555555555555555 0 2.04901078629379074055897362975 ["ModularForm/GL2/Q/holomorphic/144/2/i/c/97/1"] "2-12e2-9.7-c1-0-2" 1.072308625865911 1.149845789106438 2 144 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.01789636177244246 0 2.11686393901895015657619816444 ["ModularForm/GL2/Q/holomorphic/144/2/i/d/97/2"] "2-12e2-9.7-c1-0-3" 1.072308625865911 1.149845789106438 2 144 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.20432586044977977 0 2.32745663651917742497329581627 ["ModularForm/GL2/Q/holomorphic/144/2/i/d/97/1"] "2-12e2-9.7-c1-0-4" 1.072308625865911 1.149845789106438 2 144 "9.7" [] [[0.5, 0.0]] 1 true true false false 0.2777777777777778 0 3.12456873246481322824104634200 ["ModularForm/GL2/Q/holomorphic/144/2/i/a/97/1"] # 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).