# lfunc_search downloaded from the LMFDB on 25 June 2026. # Search link: https://www.lmfdb.org/L/2/92/23.18/c1-0 # Query "{'degree': 2, 'conductor': 92, 'spectral_label': 'c1-0'}" returned 132 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. "2-92-1.1-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 2.43652221488536246001833799986 ["EllipticCurve/Q/92/b", "ModularForm/GL2/Q/holomorphic/92/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/92/2/a/b"] "2-92-1.1-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.5 1 3.80206517892802911800653447979 ["EllipticCurve/Q/92/a", "ModularForm/GL2/Q/holomorphic/92/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/92/2/a/a"] "2-92-23.12-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.12" [] [[0.5, 0.0]] 1 true true false false -0.2508089205123931 0 1.38451886126928944133462712852 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/81/1"] "2-92-23.12-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.12" [] [[0.5, 0.0]] 1 true true false false 0.11999138746935732 0 2.87271157771041116109145278792 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/81/2"] "2-92-23.13-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.13" [] [[0.5, 0.0]] 1 true true false false -0.07455350440565733 0 2.35401475583982768211390930939 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/13/2"] "2-92-23.13-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.13" [] [[0.5, 0.0]] 1 true true false false 0.05150973284584014 0 2.44406715768035351040481065609 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/13/1"] "2-92-23.16-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.16" [] [[0.5, 0.0]] 1 true true false false -0.05150973284584014 0 1.99822007706356907414247232382 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/85/1"] "2-92-23.16-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.16" [] [[0.5, 0.0]] 1 true true false false 0.07455350440565733 0 3.08828830180894641792282261638 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/85/2"] "2-92-23.18-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.18" [] [[0.5, 0.0]] 1 true true false false -0.22828813782900803 0 1.28018652023636663535204959689 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/41/1"] "2-92-23.18-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.18" [] [[0.5, 0.0]] 1 true true false false 0.08208303640972804 0 2.91650208968170250628165228900 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/41/2"] "2-92-23.2-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.2" [] [[0.5, 0.0]] 1 true true false false -0.11999138746935732 0 1.88651142605817688110252411733 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/25/2"] "2-92-23.2-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.2" [] [[0.5, 0.0]] 1 true true false false 0.2508089205123931 0 3.60377068539049339525285314324 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/25/1"] "2-92-23.3-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.3" [] [[0.5, 0.0]] 1 true true false false -0.2541241675781718 0 1.22216756229914468491495340579 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/49/1"] "2-92-23.3-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.3" [] [[0.5, 0.0]] 1 true true false false 0.0450097274179254 0 2.53009620560665999772594151532 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/49/2"] "2-92-23.4-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.4" [] [[0.5, 0.0]] 1 true true false false -0.1401190652880036 0 1.50865022472740446597481635932 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/73/1"] "2-92-23.4-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.4" [] [[0.5, 0.0]] 1 true true false false 0.03691356957930115 0 2.75816976085574177475349359227 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/73/2"] "2-92-23.6-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.6" [] [[0.5, 0.0]] 1 true true false false -0.03691356957930115 0 2.39888837798450508560481424104 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/29/2"] "2-92-23.6-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.6" [] [[0.5, 0.0]] 1 true true false false 0.1401190652880036 0 2.59150141901499813509101343603 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/29/1"] "2-92-23.8-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.8" [] [[0.5, 0.0]] 1 true true false false -0.0450097274179254 0 2.09251117208412343917962359886 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/77/2"] "2-92-23.8-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.8" [] [[0.5, 0.0]] 1 true true false false 0.2541241675781718 0 3.48230188305601206118291930612 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/77/1"] "2-92-23.9-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "23.9" [] [[0.5, 0.0]] 1 true true false false -0.08208303640972804 0 2.42830682292154710101942505897 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/9/2"] "2-92-23.9-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "23.9" [] [[0.5, 0.0]] 1 true true false false 0.22828813782900803 0 3.09436552775246597526899220966 ["ModularForm/GL2/Q/holomorphic/92/2/e/a/9/1"] "2-92-92.11-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false -0.13548219021010446 0 0.924523678929361924604204680750 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/2"] "2-92-92.11-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false -0.32342113702247777 0 1.28684954387240792460417384019 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/7"] "2-92-92.11-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.009291068090380137 0 1.91474903875906894683589333909 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/3"] "2-92-92.11-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.0044702727713413605 0 2.34978872640513454812384083733 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/1"] "2-92-92.11-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false -0.18690231125729304 0 2.44900881774292823826180844307 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/6"] "2-92-92.11-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false -0.14347478337846037 0 2.49094221712477891317822731626 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/9"] "2-92-92.11-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.007685764759566762 0 2.73238896738063104631392610839 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/10"] "2-92-92.11-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.24998481311310372 0 2.76555278765542763835006623241 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/4"] "2-92-92.11-c1-0-8" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.1655681520229776 0 3.15866151326135199360540584924 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/8"] "2-92-92.11-c1-0-9" 0.8571019184413134 0.7346236985957799 2 92 "92.11" [] [[0.5, 0.0]] 1 true true false false 0.4081664843334484 0 3.36161013627857491227734376323 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/11/5"] "2-92-92.15-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false 0.46161588784116675 0 0.46773010779926880147682104483 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/5"] "2-92-92.15-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false -0.2157891253796434 0 0.824850600176036401056978144698 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/2"] "2-92-92.15-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false -0.24856637078082175 0 1.52341579119179246371192410032 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/9"] "2-92-92.15-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false -0.07671586288214843 0 1.84595792902495345361599357728 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/3"] "2-92-92.15-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false -0.1520951499994639 0 2.27282752084962362412821926846 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/6"] "2-92-92.15-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false -0.12814623178223972 0 2.54361200603530803193839604631 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/7"] "2-92-92.15-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false 0.06598874839713846 0 2.67291586382326710976445311486 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/1"] "2-92-92.15-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false 0.21121007420622623 0 2.78532754246224609130821609398 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/4"] "2-92-92.15-c1-0-8" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false 0.149355319810099 0 3.02437426085628383519470042322 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/8"] "2-92-92.15-c1-0-9" 0.8571019184413134 0.7346236985957799 2 92 "92.15" [] [[0.5, 0.0]] 1 true true false false 0.044597758341831335 0 3.17059551999384193458312960891 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/15/10"] "2-92-92.19-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false -0.3147487156276082 0 1.22580483711375761620590504139 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/2"] "2-92-92.19-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false -0.3150329805327105 0 1.35540090103492915815237315310 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/5"] "2-92-92.19-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false -0.08458285029789786 0 1.53213603109817863915496226139 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/3"] "2-92-92.19-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false -0.29510382031470744 0 2.14835488434664532777484096047 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/6"] "2-92-92.19-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false -0.142956197301753 0 2.26101115351629662376998210342 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/9"] "2-92-92.19-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false 0.15063961845104148 0 2.28143666750605414929842905188 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/1"] "2-92-92.19-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false 0.033392796029874806 0 3.04481056506654525591316681333 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/8"] "2-92-92.19-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false 0.19237878052798324 0 3.13633585930559293065683010512 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/7"] "2-92-92.19-c1-0-8" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false 0.18129421649379848 0 3.76519937013946084840119954495 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/10"] "2-92-92.19-c1-0-9" 0.8571019184413134 0.7346236985957799 2 92 "92.19" [] [[0.5, 0.0]] 1 true true false false 0.4377422893518938 0 3.94489756041117878741152811430 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/19/4"] "2-92-92.43-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false -0.21121007420622623 0 1.85191991690095680505901408398 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/4"] "2-92-92.43-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false 0.07671586288214843 0 1.92322972805299230581388002583 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/3"] "2-92-92.43-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false -0.06598874839713846 0 2.06535385499939738597839132763 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/1"] "2-92-92.43-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false -0.149355319810099 0 2.07682405190389625756081851241 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/8"] "2-92-92.43-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false 0.2157891253796434 0 2.66868175406329073296215558032 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/2"] "2-92-92.43-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false 0.1520951499994639 0 2.88266816260284670210028246568 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/6"] "2-92-92.43-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false -0.044597758341831335 0 3.05005575841345277946111409194 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/10"] "2-92-92.43-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.43" [] [[0.5, 0.0]] 1 true true false false 0.12814623178223972 0 3.07505866406294034293060458606 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/43/7"] "2-92-92.43-c1-0-8" 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92 "92.79" [] [[0.5, 0.0]] 1 true true false false -0.46933445895825154 0 0.12680133304378270428802310664 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/79/1"] "2-92-92.79-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.79" [] [[0.5, 0.0]] 1 true true false false -0.0727713743340165 0 1.36423525698628737634096644846 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/79/4"] "2-92-92.79-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.79" [] [[0.5, 0.0]] 1 true true false false -0.3301131733791867 0 1.45736051763148093939928166508 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/79/7"] "2-92-92.79-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.79" [] [[0.5, 0.0]] 1 true true false false -0.1156006115051112 0 1.65340137349609571101403669089 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/79/2"] "2-92-92.79-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.79" [] [[0.5, 0.0]] 1 true true false false -0.11725242435515408 0 2.12100440311816263783149729957 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true false false 0.3465903807500515 0 3.94988951240019232446269822972 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/79/6"] "2-92-92.83-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.36710201291457284 0 0.19758216342831310333309869596 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/6"] "2-92-92.83-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.3194513633585151 0 0.76165969864400421179886609773 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/1"] "2-92-92.83-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.3407758597337004 0 1.39626097774918193075998262489 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/7"] "2-92-92.83-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false 0.06234250505829847 0 1.82535007817656941880357046885 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/3"] "2-92-92.83-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.1235134955618087 0 2.25001175290149908267419341796 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/4"] "2-92-92.83-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false 0.04299533048088751 0 2.41241073198961727709381566804 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/2"] "2-92-92.83-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.05985404958583899 0 2.53827152034917089062402967159 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/10"] "2-92-92.83-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false -0.0656150132865608 0 3.00838517635751579404423280591 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/8"] "2-92-92.83-c1-0-8" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false 0.25870052226416845 0 3.38209494579856589638569237182 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/5"] "2-92-92.83-c1-0-9" 0.8571019184413134 0.7346236985957799 2 92 "92.83" [] [[0.5, 0.0]] 1 true true false false 0.15950199806112988 0 3.61325073772554286282474796347 ["ModularForm/GL2/Q/holomorphic/92/2/h/a/83/9"] "2-92-92.91-c1-0-0" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false -0.14243869138692497 0 1.17512742785760637790603110853 ["ModularForm/GL2/Q/holomorphic/92/2/b/a/91/2"] "2-92-92.91-c1-0-1" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false -0.3573559997892315 0 1.21001969050650181210683268297 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/2"] "2-92-92.91-c1-0-2" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false -0.3093106668774352 0 1.62670093299634777105663758010 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/4"] "2-92-92.91-c1-0-3" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false -0.14243869138692497 0 1.62806963452952433286508874001 ["ModularForm/GL2/Q/holomorphic/92/2/b/a/91/4"] "2-92-92.91-c1-0-4" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false 0.14243869138692497 0 2.23664913918173950709089547972 ["ModularForm/GL2/Q/holomorphic/92/2/b/a/91/1"] "2-92-92.91-c1-0-5" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false 0.024022666455898158 0 2.56747103653534940794669672036 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/5"] "2-92-92.91-c1-0-6" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false -0.024022666455898158 0 2.83392605673296513729713089253 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/6"] "2-92-92.91-c1-0-7" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false 0.14243869138692497 0 2.88379729438543074857811845954 ["ModularForm/GL2/Q/holomorphic/92/2/b/a/91/3"] "2-92-92.91-c1-0-8" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false 0.3573559997892315 0 3.33978666997632839420065090311 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/1"] "2-92-92.91-c1-0-9" 0.8571019184413134 0.7346236985957799 2 92 "92.91" [] [[0.5, 0.0]] 1 true true false false 0.3093106668774352 0 3.74303582627021454666040648183 ["ModularForm/GL2/Q/holomorphic/92/2/b/b/91/3"] # 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).