# lfunc_search downloaded from the LMFDB on 14 June 2026. # Search link: https://www.lmfdb.org/L/2/34^2/1156.375/c0-0 # Query "{'degree': 2, 'conductor': 1156, 'spectral_label': 'c0-0'}" returned 85 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. "2-34e2-1156.1007-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1007" [] [[0.0, 0.0]] 0 true true false false -0.17387543252595156 0 1.33075143349303764918648170264 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/1007/1"] "2-34e2-1156.1019-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1019" [] [[0.0, 0.0]] 0 true true false false -0.04498269896193772 0 1.45058883492430517636768617904 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/1019/1"] "2-34e2-1156.103-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.103" [] [[0.0, 0.0]] 0 true true false false -0.10726643598615918 0 1.42604584576712154741517710172 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/103/1"] "2-34e2-1156.1055-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1055" [] [[0.0, 0.0]] 0 true true false false 0.10726643598615918 0 1.52468738026870222163861167618 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/1055/1"] "2-34e2-1156.1067-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1067" [] [[0.0, 0.0]] 0 true true false false 0.3762975778546713 0 2.11235309736635328512070535690 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/1067/1"] "2-34e2-1156.1075-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1075" [] [[0.0, 0.0]] 0 true true false false -0.14273356401384085 0 0.957098284840174723029601516978 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/1075/1"] "2-34e2-1156.1087-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1087" [] [[0.0, 0.0]] 0 true true false false -0.051903114186851215 0 0.48680214664612358280771602990 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/1087/1"] "2-34e2-1156.1123-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1123" [] [[0.0, 0.0]] 0 true true false false 0.2906574394463668 0 2.40560666274826302324379782136 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/1123/1"] "2-34e2-1156.1135-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1135" [] [[0.0, 0.0]] 0 true true false false -0.051903114186851215 0 1.20762991033269822769312722712 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/1135/1"] "2-34e2-1156.1143-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.1143" [] [[0.0, 0.0]] 0 true true false false -0.3762975778546713 0 0.71297059867034199720794574493 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/1143/1"] "2-34e2-1156.115-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.115" [] [[0.0, 0.0]] 0 true true false false 0.20934256055363323 0 1.98349781007112786245706999438 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/115/1"] "2-34e2-1156.123-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.123" [] [[0.0, 0.0]] 0 true true false false 0.4506920415224913 0 1.75307865071205826874789685813 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/123/1"] "2-34e2-1156.135-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.135" [] [[0.0, 0.0]] 0 true true false false 0.04498269896193772 0 1.95925444809955159535992324438 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/135/1"] "2-34e2-1156.171-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.171" [] [[0.0, 0.0]] 0 true true false false 0.07612456747404846 0 1.26359413014448654490991500921 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/171/1"] "2-34e2-1156.183-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.183" [] [[0.0, 0.0]] 0 true true false false -0.027681660899653984 0 1.70256817407606003915508432947 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/183/1"] "2-34e2-1156.191-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.191" [] [[0.0, 0.0]] 0 true true false false -0.20934256055363323 0 0.73585452035323312860481590062 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/191/1"] "2-34e2-1156.203-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.203" [] [[0.0, 0.0]] 0 true true false false 0.12629757785467127 0 1.18408560457446018734734438584 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/203/1"] "2-34e2-1156.239-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.239" [] [[0.0, 0.0]] 0 true true false false 0.20069204152249134 0 1.88886315088026377400314852111 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/239/1"] "2-34e2-1156.259-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.259" [] [[0.0, 0.0]] 0 true true false false 0.027681660899653984 0 1.03961452497543225094037625698 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/259/1"] "2-34e2-1156.271-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.271" [] [[0.0, 0.0]] 0 true true false false -0.027681660899653984 0 0.76848937547880494863100160967 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/271/1"] "2-34e2-1156.307-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.307" [] [[0.0, 0.0]] 0 true true false false -0.027681660899653984 0 1.92048349028876229258633561738 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/307/1"] "2-34e2-1156.319-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.319" [] [[0.0, 0.0]] 0 true true false false 0.027681660899653984 0 1.95885873500417092811286789641 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/319/1"] "2-34e2-1156.339-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.339" [] [[0.0, 0.0]] 0 true true false false -0.2993079584775087 0 0.855291356762008806064445845852 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/339/1"] "2-34e2-1156.35-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.35" [] [[0.0, 0.0]] 0 true true false false -0.2906574394463668 0 1.28601908529817087469629756301 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/35/1"] "2-34e2-1156.375-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.375" [] [[0.0, 0.0]] 0 true true false false -0.3737024221453287 0 0.850730756124675275330105089148 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/375/1"] "2-34e2-1156.387-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.387" [] [[0.0, 0.0]] 0 true true false false -0.20934256055363323 0 1.40186473171508875261297463725 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/387/1"] "2-34e2-1156.395-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.395" [] [[0.0, 0.0]] 0 true true false false -0.027681660899653984 0 0.991797458304696327724782054036 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/395/1"] "2-34e2-1156.407-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.407" [] [[0.0, 0.0]] 0 true true false false -0.4238754325259516 0 0.54864881468592047112449689459 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/407/1"] "2-34e2-1156.443-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.443" [] [[0.0, 0.0]] 0 true true false false 0.04498269896193772 0 1.36594786777538793152864852565 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/443/1"] "2-34e2-1156.455-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.455" [] [[0.0, 0.0]] 0 true true false false -0.04930795847750865 0 1.60048085806193551603809729281 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/455/1"] "2-34e2-1156.463-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.463" [] [[0.0, 0.0]] 0 true true false false 0.20934256055363323 0 1.82597936689942816334792907441 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/463/1"] "2-34e2-1156.47-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.47" [] [[0.0, 0.0]] 0 true true false false -0.4506920415224913 0 0.15912076497982595295220522734 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/47/1"] "2-34e2-1156.475-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.475" [] [[0.0, 0.0]] 0 true true false false 0.39273356401384085 0 2.40379774017015247762345280619 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/475/1"] "2-34e2-1156.511-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.511" [] [[0.0, 0.0]] 0 true true false false 0.051903114186851215 0 1.03519128124228648072767871759 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/511/1"] "2-34e2-1156.523-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.523" [] [[0.0, 0.0]] 0 true true false false 0.051903114186851215 0 2.10460164519940506045807827662 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/523/1"] "2-34e2-1156.531-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.531" [] [[0.0, 0.0]] 0 true true false false 0.04930795847750865 0 1.51000763534474136452288407846 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/531/1"] "2-34e2-1156.543-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.543" [] [[0.0, 0.0]] 0 true true false false -0.2906574394463668 0 0.36238845056206168124369861738 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/543/1"] "2-34e2-1156.55-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.55" [] [[0.0, 0.0]] 0 true true false false 0.051903114186851215 0 1.58834062366811849672041022519 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/55/1"] "2-34e2-1156.591-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.591" [] [[0.0, 0.0]] 0 true true false false 0.12370242214532873 0 2.01798744635457746537230128243 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/591/1"] "2-34e2-1156.599-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.599" [] [[0.0, 0.0]] 0 true true false false -0.051903114186851215 0 1.77107474032880831894118929692 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/599/1"] "2-34e2-1156.611-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.611" [] [[0.0, 0.0]] 0 true true false false 0.2906574394463668 0 1.82648247560790348177308102230 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/611/1"] "2-34e2-1156.647-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.647" [] [[0.0, 0.0]] 0 true true false false -0.051903114186851215 0 1.62158317312793800906900972005 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/647/1"] "2-34e2-1156.659-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.659" [] [[0.0, 0.0]] 0 true true false false 0.3572664359861592 0 2.61054270770460349266109043411 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/659/1"] "2-34e2-1156.667-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.667" [] [[0.0, 0.0]] 0 true true false false -0.12370242214532873 0 1.48239984381710691509668821097 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/667/1"] "2-34e2-1156.67-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.67" [] [[0.0, 0.0]] 0 true true false false 0.051903114186851215 0 1.37454153953959900617315380185 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/67/1"] "2-34e2-1156.679-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.679" [] [[0.0, 0.0]] 0 true true false false -0.39273356401384085 0 0.897721046801917854359794359452 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/679/1"] "2-34e2-1156.715-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.715" [] [[0.0, 0.0]] 0 true true false false -0.04498269896193772 0 1.46638673108833553140581375472 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/715/1"] "2-34e2-1156.727-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.727" [] [[0.0, 0.0]] 0 true true false false 0.3261245674740485 0 2.21412855389388690522458511680 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/727/1"] "2-34e2-1156.735-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.735" [] [[0.0, 0.0]] 0 true true false false -0.3572664359861592 0 1.04121157962032752093557134549 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/735/1"] "2-34e2-1156.747-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.747" [] [[0.0, 0.0]] 0 true true false false 0.4238754325259516 0 2.46460298854711158918327676560 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/747/1"] "2-34e2-1156.783-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.783" [] [[0.0, 0.0]] 0 true true false false 0.3737024221453287 0 1.92383587300622588798351419297 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/783/1"] "2-34e2-1156.795-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.795" [] [[0.0, 0.0]] 0 true true false false -0.4550173010380623 0 2.32352555586976874793133225946 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/795/1"] "2-34e2-1156.803-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.803" [] [[0.0, 0.0]] 0 true true false false -0.3261245674740485 0 1.22589710632426478956440652909 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/803/1"] "2-34e2-1156.815-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.815" [] [[0.0, 0.0]] 0 true true false false 0.2993079584775087 0 1.54470289932736364512309431062 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/815/1"] "2-34e2-1156.851-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.851" [] [[0.0, 0.0]] 0 true true false false 0.027681660899653984 0 1.64028543105429784179796503169 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/851/1"] "2-34e2-1156.863-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.863" [] [[0.0, 0.0]] 0 true true false false 0.4550173010380623 0 2.04236172445205968428905094443 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/863/1"] "2-34e2-1156.871-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.871" [] [[0.0, 0.0]] 0 true true false false 0.4550173010380623 0 0.38774192925476810274851058731 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/871/1"] "2-34e2-1156.883-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.883" [] [[0.0, 0.0]] 0 true true false false 0.027681660899653984 0 1.14852130125772970838857768532 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/883/1"] "2-34e2-1156.919-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.919" [] [[0.0, 0.0]] 0 true true false false -0.20069204152249134 0 1.65715191141002781406647668283 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/919/1"] "2-34e2-1156.931-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.931" [] [[0.0, 0.0]] 0 true true false false 0.17387543252595156 0 1.42542333014341118783297989622 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/931/1"] "2-34e2-1156.939-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.939" [] [[0.0, 0.0]] 0 true true false false -0.4550173010380623 0 0.848557480611422703948010146077 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/939/1"] "2-34e2-1156.951-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.951" [] [[0.0, 0.0]] 0 true true false false -0.12629757785467127 0 1.29975557571345099482002491333 ["ModularForm/GL2/Q/holomorphic/1156/1/l/a/951/1"] "2-34e2-1156.987-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.987" [] [[0.0, 0.0]] 0 true true false false -0.07612456747404846 0 1.19409123053474964611097160587 ["ModularForm/GL2/Q/holomorphic/1156/1/m/a/987/1"] "2-34e2-1156.999-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "1156.999" [] [[0.0, 0.0]] 0 true true false false 0.14273356401384085 0 1.16001575000915243741753499127 ["ModularForm/GL2/Q/holomorphic/1156/1/o/a/999/1"] "2-34e2-4.3-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "4.3" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.01712437104286264984381406238 ["ModularForm/GL2/Q/holomorphic/1156/1/c/b/579/1", "ArtinRepresentation/2.1156.8t6.b.b"] "2-34e2-4.3-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "4.3" [] [[0.0, 0.0]] 0 true true false true 0.0 0 1.49415910324576107748984374736 ["ModularForm/GL2/Q/holomorphic/1156/1/c/b/579/2", "ArtinRepresentation/2.1156.8t6.b.a"] "2-34e2-4.3-c0-0-2" 0.7595519433216867 0.5769191546037508 2 1156 "4.3" [] [[0.0, 0.0]] 0 true true true true 0.0 0 1.81128345542631433367044257082 ["ModularForm/GL2/Q/holomorphic/1156/1/c/a/579/1", "ModularForm/GL2/Q/holomorphic/1156/1/c/a", "ArtinRepresentation/2.1156.4t3.a", "ArtinRepresentation/2.1156.4t3.a.a"] "2-34e2-68.15-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.15" [] [[0.0, 0.0]] 0 true true false false -0.45354125351689395 0 0.63200268437517724297310985726 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/423/1"] "2-34e2-68.15-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.15" [] [[0.0, 0.0]] 0 true true false false -0.11759447445521348 0 0.68720107346924350493028983890 ["ModularForm/GL2/Q/holomorphic/1156/1/g/a/423/1"] "2-34e2-68.15-c0-0-2" 0.7595519433216867 0.5769191546037508 2 1156 "68.15" [] [[0.0, 0.0]] 0 true true false false -0.19095352907744845 0 1.49818981190896659507726285148 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/423/2"] "2-34e2-68.19-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.19" [] [[0.0, 0.0]] 0 true true false false -0.014077889985382725 0 1.15265089224734115874199161799 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/155/2"] "2-34e2-68.19-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.19" [] [[0.0, 0.0]] 0 true true false false -0.23791074295044418 0 1.27108921017610214870066629693 ["ModularForm/GL2/Q/holomorphic/1156/1/g/a/155/1"] "2-34e2-68.19-c0-0-2" 0.7595519433216867 0.5769191546037508 2 1156 "68.19" [] [[0.0, 0.0]] 0 true true false false 0.3695831073910404 0 1.76431244579556931700635892351 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/155/1"] "2-34e2-68.43-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.43" [] [[0.0, 0.0]] 0 true true false false -0.3695831073910404 0 0.74323389126179422601511388332 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/179/1"] "2-34e2-68.43-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.43" [] [[0.0, 0.0]] 0 true true false false 0.014077889985382725 0 1.49805676899469575772130237740 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/179/2"] "2-34e2-68.43-c0-0-2" 0.7595519433216867 0.5769191546037508 2 1156 "68.43" [] [[0.0, 0.0]] 0 true true false false 0.23791074295044418 0 1.94000668090253920983587136554 ["ModularForm/GL2/Q/holomorphic/1156/1/g/a/179/1"] "2-34e2-68.47-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.47" [] [[0.0, 0.0]] 0 true true false false -0.35550521740565766 0 1.06916387095544160553343369755 ["ModularForm/GL2/Q/holomorphic/1156/1/f/b/251/1", "ArtinRepresentation/2.1156.8t17.a.b"] "2-34e2-68.47-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.47" [] [[0.0, 0.0]] 0 true true false false 0.14449478259434237 0 1.20800603005854348550064001621 ["ModularForm/GL2/Q/holomorphic/1156/1/f/a/251/1", "ArtinRepresentation/2.1156.8t7.a.b"] "2-34e2-68.55-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.55" [] [[0.0, 0.0]] 0 true true false false -0.14449478259434237 0 1.34928060120166883895863755008 ["ModularForm/GL2/Q/holomorphic/1156/1/f/a/327/1", "ArtinRepresentation/2.1156.8t7.a.a"] "2-34e2-68.55-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.55" [] [[0.0, 0.0]] 0 true true false false 0.35550521740565766 0 1.97516299636307899317861391759 ["ModularForm/GL2/Q/holomorphic/1156/1/f/b/327/1", "ArtinRepresentation/2.1156.8t17.a.a"] "2-34e2-68.59-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.59" [] [[0.0, 0.0]] 0 true true false false 0.11759447445521348 0 1.68419321045449156319243582591 ["ModularForm/GL2/Q/holomorphic/1156/1/g/a/399/1"] "2-34e2-68.59-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.59" [] [[0.0, 0.0]] 0 true true false false 0.19095352907744845 0 2.00114764386466823104203924190 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/399/2"] "2-34e2-68.59-c0-0-2" 0.7595519433216867 0.5769191546037508 2 1156 "68.59" [] [[0.0, 0.0]] 0 true true false false 0.45354125351689395 0 2.56956287288719737630987979015 ["ModularForm/GL2/Q/holomorphic/1156/1/g/b/399/1"] "2-34e2-68.67-c0-0-0" 0.7595519433216867 0.5769191546037508 2 1156 "68.67" [] [[0.0, 0.0]] 0 true true false false -0.12031626849523071 0 1.41958524975628106160777037168 ["ModularForm/GL2/Q/holomorphic/1156/1/d/a/1155/2", "ArtinRepresentation/2.1156.8t8.a.a"] "2-34e2-68.67-c0-0-1" 0.7595519433216867 0.5769191546037508 2 1156 "68.67" [] [[0.0, 0.0]] 0 true true false false 0.12031626849523071 0 2.20994499824788278960940488802 ["ModularForm/GL2/Q/holomorphic/1156/1/d/a/1155/1", "ArtinRepresentation/2.1156.8t8.a.b"] # 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).