# lfunc_search downloaded from the LMFDB on 02 May 2026. # Search link: https://www.lmfdb.org/L/2/87/87.17/c1-0 # Query "{'degree': 2, 'conductor': 87, 'spectral_label': 'c1-0'}" returned 181 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. "2-87-1.1-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 1.52848030486985068602653439662 ["ModularForm/GL2/Q/holomorphic/87/2/a/b/1/1"] "2-87-1.1-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 2.20009654735353763797367045973 ["ModularForm/GL2/Q/holomorphic/87/2/a/a/1/1"] "2-87-1.1-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 2.73387108539709217330115468964 ["ModularForm/GL2/Q/holomorphic/87/2/a/b/1/2"] "2-87-1.1-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 3.30630143514452427642507298744 ["ModularForm/GL2/Q/holomorphic/87/2/a/b/1/3"] "2-87-1.1-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 3.33284344106940796359913592368 ["ModularForm/GL2/Q/holomorphic/87/2/a/a/1/2"] "2-87-29.13-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.13" [] [[0.5, 0.0]] 1 true true false false -0.2415196836075747 0 1.80689044405937911001205286325 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/13/1"] "2-87-29.13-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.13" [] [[0.5, 0.0]] 1 true true false false 0.02431099297058303 0 2.30387443474146191356411231388 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/13/2"] "2-87-29.13-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.13" [] [[0.5, 0.0]] 1 true true false false 0.1848901509510199 0 3.30280658399391151816377016859 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/13/3"] "2-87-29.13-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.13" [] [[0.5, 0.0]] 1 true true false false 0.30344383944481546 0 4.03184616537391272655102077259 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/13/4"] "2-87-29.16-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false -0.31947766185930704 0 0.33663499338349850951616216263 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/16/1"] "2-87-29.16-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false 0.16182342771676841 0 1.91455088683630914974446783842 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/16/1"] "2-87-29.16-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false 0.048361618425384875 0 1.93482905611807858650630086516 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/16/2"] "2-87-29.16-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false -0.1376339710411093 0 2.54215679615060119698579005786 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/16/3"] "2-87-29.16-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false 0.03182533609900125 0 2.93429556732389874933229058579 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/16/3"] "2-87-29.16-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "29.16" [] [[0.5, 0.0]] 1 true true false false 0.3209402221396724 0 3.33691846232215542560157743956 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/16/2"] "2-87-29.20-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false -0.3209402221396724 0 1.03469816379415901820362250377 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/49/2"] "2-87-29.20-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false -0.16182342771676841 0 1.71581533436500632589262428423 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/49/1"] "2-87-29.20-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false -0.048361618425384875 0 1.80351932704876151773841839122 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/49/2"] "2-87-29.20-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false -0.03182533609900125 0 2.38999204025373528884186425916 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/49/3"] "2-87-29.20-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false 0.31947766185930704 0 2.69930477157737553204957959417 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/49/1"] "2-87-29.20-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "29.20" [] [[0.5, 0.0]] 1 true true false false 0.1376339710411093 0 3.66740160253107730561586249301 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/49/3"] "2-87-29.22-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.22" [] [[0.5, 0.0]] 1 true true false false -0.198663737519503 0 1.39822617267249514659316740078 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/22/2"] "2-87-29.22-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.22" [] [[0.5, 0.0]] 1 true true false false 0.20676675792586038 0 2.72446716534536972901693521289 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/22/1"] "2-87-29.22-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.22" [] [[0.5, 0.0]] 1 true true false false 0.011614895392857313 0 2.99387542425803434543901231548 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/22/4"] "2-87-29.22-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.22" [] [[0.5, 0.0]] 1 true true false false 0.07413984105064705 0 3.21812271479302537499477625860 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/22/3"] "2-87-29.23-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false -0.4217620479497884 0 0.840602669496891280351210769258 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/52/1"] "2-87-29.23-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false -0.2890881962645032 0 1.40518222025409769858113908078 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/52/1"] "2-87-29.23-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false -0.17126020523539132 0 1.59293336240047390911440003872 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/52/2"] "2-87-29.23-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false 0.01642868544374206 0 2.42845099258208896765894669285 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/52/2"] "2-87-29.23-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false 0.20302886285088081 0 3.64903650112333742707224865203 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/52/3"] "2-87-29.23-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "29.23" [] [[0.5, 0.0]] 1 true true false false 0.2116522871623411 0 3.75731764435968774470892816384 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/52/3"] "2-87-29.24-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false -0.01642868544374206 0 2.32574764180977286338758692199 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/82/2"] "2-87-29.24-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false -0.20302886285088081 0 2.42521494632738723457307893262 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/82/3"] "2-87-29.24-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false -0.2116522871623411 0 2.54866113577040171331492330518 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/82/3"] "2-87-29.24-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false 0.2890881962645032 0 3.11329208469633293741539300682 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/82/1"] "2-87-29.24-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false 0.4217620479497884 0 3.18949509739808649516602472934 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/82/1"] "2-87-29.24-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "29.24" [] [[0.5, 0.0]] 1 true true false false 0.17126020523539132 0 3.60024090728967690911123052478 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/82/2"] "2-87-29.25-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false 0.47790902555928166 0 0.938251382334138438353262608340 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/25/1"] "2-87-29.25-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false -0.24406311696228292 0 1.91113329301265944928779355277 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/25/1"] "2-87-29.25-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false -0.07373465139043911 0 1.92334256256028243820663637251 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/25/2"] "2-87-29.25-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false 0.10912698461971578 0 2.86944808053454088208169353824 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/25/2"] "2-87-29.25-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false 0.3325405255381621 0 3.66412296839382104671714134181 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/25/3"] "2-87-29.25-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "29.25" [] [[0.5, 0.0]] 1 true true false false 0.3413816471624331 0 4.27790468720207800230934456506 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/25/3"] "2-87-29.28-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.28" [] [[0.5, 0.0]] 1 true true false false -0.15595968107605412 0 2.00718633194260351255256283593 ["ModularForm/GL2/Q/holomorphic/87/2/c/a/28/3"] "2-87-29.28-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.28" [] [[0.5, 0.0]] 1 true true false false -0.15595968107605412 0 2.28633013347696907907297447780 ["ModularForm/GL2/Q/holomorphic/87/2/c/a/28/4"] "2-87-29.28-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.28" [] [[0.5, 0.0]] 1 true true false false 0.15595968107605412 0 2.65736448253604769807169716639 ["ModularForm/GL2/Q/holomorphic/87/2/c/a/28/1"] "2-87-29.28-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.28" [] [[0.5, 0.0]] 1 true true false false 0.15595968107605412 0 2.97038936806824430773598937284 ["ModularForm/GL2/Q/holomorphic/87/2/c/a/28/2"] "2-87-29.4-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.4" [] [[0.5, 0.0]] 1 true true false false -0.20676675792586038 0 1.08084547614026677837850972094 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/4/1"] "2-87-29.4-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.4" [] [[0.5, 0.0]] 1 true true false false -0.07413984105064705 0 2.59533023758488096287048152454 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/4/3"] "2-87-29.4-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.4" [] [[0.5, 0.0]] 1 true true false false 0.198663737519503 0 3.11036998320454956327061987260 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/4/2"] "2-87-29.4-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.4" [] [[0.5, 0.0]] 1 true true false false -0.011614895392857313 0 3.18677978205172165909607287098 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/4/4"] "2-87-29.5-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.5" [] [[0.5, 0.0]] 1 true true false false -0.21767486006710626 0 1.31335840720314956086786272610 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/34/3"] "2-87-29.5-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.5" [] [[0.5, 0.0]] 1 true true false false -0.24440938373554014 0 1.53171059161865618191000690707 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/34/2"] "2-87-29.5-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.5" [] [[0.5, 0.0]] 1 true true false false 0.1708266266282152 0 2.32208091884222823149020789333 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/34/1"] "2-87-29.5-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.5" [] [[0.5, 0.0]] 1 true true false false 0.1139900742654492 0 3.36303401425244278562343086529 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/34/4"] "2-87-29.6-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.6" [] [[0.5, 0.0]] 1 true true false false -0.1708266266282152 0 0.50178101746188341919141276171 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/64/1"] "2-87-29.6-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.6" [] [[0.5, 0.0]] 1 true true false false 0.24440938373554014 0 2.48671069793583341086589471620 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/64/2"] "2-87-29.6-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "29.6" [] [[0.5, 0.0]] 1 true true false false -0.1139900742654492 0 2.52909503681193402078837800486 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/64/4"] "2-87-29.6-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "29.6" [] [[0.5, 0.0]] 1 true true false false 0.21767486006710626 0 2.99194967560821617722478232466 ["ModularForm/GL2/Q/holomorphic/87/2/i/a/64/3"] "2-87-29.7-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "29.7" [] [[0.5, 0.0]] 1 true true false false -0.3325405255381621 0 1.62846935687453105587633689092 ["ModularForm/GL2/Q/holomorphic/87/2/g/a/7/3"] "2-87-29.7-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "29.7" [] [[0.5, 0.0]] 1 true true false false -0.3413816471624331 0 2.09730961705240444748345141327 ["ModularForm/GL2/Q/holomorphic/87/2/g/b/7/3"] "2-87-29.7-c1-0-2" 0.8334857512790124 0.6946984975851397 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true false false -0.11241049501019999 0 2.03662275311693180454804896980 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/77/5"] "2-87-87.77-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "87.77" [] [[0.5, 0.0]] 1 true true false false -0.006498930787173226 0 2.44813506957358672005672214898 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/77/1"] "2-87-87.77-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "87.77" [] [[0.5, 0.0]] 1 true true false false 0.0810298094973997 0 3.06291828523594077580366334083 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/77/7"] "2-87-87.77-c1-0-6" 0.8334857512790124 0.6946984975851397 2 87 "87.77" [] [[0.5, 0.0]] 1 true true false false 0.16956762401590936 0 3.18175743090256080957916935282 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/77/6"] "2-87-87.77-c1-0-7" 0.8334857512790124 0.6946984975851397 2 87 "87.77" [] [[0.5, 0.0]] 1 true true false false 0.18949140605236603 0 4.15814765533475231537322701546 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/77/8"] "2-87-87.8-c1-0-0" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false -0.45763264107796214 0 1.35856568382415853413260106615 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/1"] "2-87-87.8-c1-0-1" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false -0.35144650498249824 0 1.39970147431330591064486739715 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/3"] "2-87-87.8-c1-0-2" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false -0.12332622053305617 0 2.23884508489589701261872095309 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/2"] "2-87-87.8-c1-0-3" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false 0.05050149302189005 0 2.43711618079084057681567153410 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/6"] "2-87-87.8-c1-0-4" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false -0.04207270943138271 0 2.46823073051655576352990342447 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/4"] "2-87-87.8-c1-0-5" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false 0.08518012832361095 0 2.50567441044073934578688261677 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/5"] "2-87-87.8-c1-0-6" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false 0.3000932298189406 0 4.02024501125438837456155527259 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/7"] "2-87-87.8-c1-0-7" 0.8334857512790124 0.6946984975851397 2 87 "87.8" [] [[0.5, 0.0]] 1 true true false false 0.3858547299690765 0 4.22085681287125348250918757935 ["ModularForm/GL2/Q/holomorphic/87/2/k/a/8/8"] # 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).