# lfunc_search downloaded from the LMFDB on 26 May 2026. # Search link: https://www.lmfdb.org/L/2/85/17.2/c1-0 # Query "{'degree': 2, 'conductor': 85, 'spectral_label': 'c1-0'}" returned 215 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. "2-85-1.1-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 2.17582772087516134866370577695 ["ModularForm/GL2/Q/holomorphic/85/2/a/c/1/1"] "2-85-1.1-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.0 0 2.93032858041166400177262840994 ["ModularForm/GL2/Q/holomorphic/85/2/a/c/1/2"] "2-85-1.1-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 2.95634545112088224055144157853 ["ModularForm/GL2/Q/holomorphic/85/2/a/b/1/1"] "2-85-1.1-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "1.1" [] [[0.5, 0.0]] 1 true true true true 0.0 0 3.13757525842324917581154074415 ["EllipticCurve/Q/85/a", "ModularForm/GL2/Q/holomorphic/85/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/85/2/a/a"] "2-85-1.1-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "1.1" [] [[0.5, 0.0]] 1 true true false true 0.5 1 4.27247643895843496960279170082 ["ModularForm/GL2/Q/holomorphic/85/2/a/b/1/2"] "2-85-17.13-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.34363166905800074 0 1.72159546765021364890788823991 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/6"] "2-85-17.13-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false -0.1214988995323004 0 2.26741207881119474684057189381 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/5"] "2-85-17.13-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.03276830934711976 0 2.46116072350025363969141529904 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/4"] "2-85-17.13-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.1717781399892859 0 2.86124431411402519785983983850 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/2"] "2-85-17.13-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.34889927364212614 0 3.76214853329508148560640814744 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/3"] "2-85-17.13-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.13" [] [[0.5, 0.0]] 1 true true false false 0.450674410800454 0 4.40170198694495347286724613915 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/81/1"] "2-85-17.15-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false -0.08471208355423293 0 1.93543693104800046984715487414 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/2"] "2-85-17.15-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false -0.1696675596182316 0 2.47672759645299743449605434089 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/5"] "2-85-17.15-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false 0.12207575189305513 0 2.78746953348926650965759202979 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/3"] "2-85-17.15-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false -0.1524254205881274 0 2.82759175112767052115383124372 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/6"] "2-85-17.15-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false 0.12393748115918918 0 3.42465488065181914391997182381 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/4"] "2-85-17.15-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.15" [] [[0.5, 0.0]] 1 true true false false -0.4790192203812254 0 4.12106162840519913050752844776 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/66/1"] "2-85-17.16-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.14440762491464432 0 1.21563611534517583450234751434 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/2"] "2-85-17.16-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.2582766314779518 0 1.25744698738074957478225882982 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/4"] "2-85-17.16-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.14440762491464432 0 2.07961361918891517013062975765 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/1"] "2-85-17.16-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false -0.07487944137462285 0 2.56351429255536338061609757318 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/6"] "2-85-17.16-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.07487944137462285 0 3.44635277789694562052469751827 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/5"] "2-85-17.16-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.16" [] [[0.5, 0.0]] 1 true true false false 0.2582766314779518 0 3.48120839069573025536433432123 ["ModularForm/GL2/Q/holomorphic/85/2/d/a/16/3"] "2-85-17.2-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false -0.10525535383393528 0 0.69569041440203107906335757608 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/1"] "2-85-17.2-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false -0.2463420702729366 0 2.18474600623475618755788602905 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/6"] "2-85-17.2-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false -0.11982718497523819 0 2.49820052709460728138734361082 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/5"] "2-85-17.2-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false 0.308188838209872 0 2.99659937610804764127536307363 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/2"] "2-85-17.2-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false 0.20236892085168331 0 3.06045011009649777364574448665 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/3"] "2-85-17.2-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.2" [] [[0.5, 0.0]] 1 true true false false 0.06168833592144314 0 3.18196813149302897793044728956 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/36/4"] "2-85-17.4-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false -0.34889927364212614 0 1.31216762820195459865791169534 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/4"] "2-85-17.4-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false -0.450674410800454 0 1.36563827234338205544906405839 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/6"] "2-85-17.4-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false -0.03276830934711976 0 1.90901283877240005840362122233 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/3"] "2-85-17.4-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false -0.1717781399892859 0 2.50606871569465001846473175196 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/5"] "2-85-17.4-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false 0.1214988995323004 0 2.82052032323703283600060930000 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/2"] "2-85-17.4-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.4" [] [[0.5, 0.0]] 1 true true false false 0.34363166905800074 0 3.65478180832305816676565542493 ["ModularForm/GL2/Q/holomorphic/85/2/e/a/21/1"] "2-85-17.8-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false 0.4790192203812254 0 0.987172650165626962243973530006 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/1"] "2-85-17.8-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false -0.12393748115918918 0 1.75996424768553766619335775501 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/4"] "2-85-17.8-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false -0.12207575189305513 0 1.82640033083294669724485013956 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/3"] "2-85-17.8-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false 0.08471208355423293 0 2.80548896473677538216553205172 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/2"] "2-85-17.8-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false 0.1524254205881274 0 3.38528560722189804279852706497 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/6"] "2-85-17.8-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.8" [] [[0.5, 0.0]] 1 true true false false 0.1696675596182316 0 3.69077418884048571776617518432 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/76/5"] "2-85-17.9-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false -0.308188838209872 0 1.33827655602701050561650597032 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/2"] "2-85-17.9-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false -0.06168833592144314 0 2.20005399702633186438631002513 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/4"] "2-85-17.9-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false 0.10525535383393528 0 2.21195337879718829362405162580 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/1"] "2-85-17.9-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false -0.20236892085168331 0 2.26527285271161702860616254555 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/3"] "2-85-17.9-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false 0.11982718497523819 0 3.05967928487064226286831040859 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/5"] "2-85-17.9-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "17.9" [] [[0.5, 0.0]] 1 true true false false 0.2463420702729366 0 4.26034397746175427542557155407 ["ModularForm/GL2/Q/holomorphic/85/2/l/a/26/6"] "2-85-5.4-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.37745718048439825 0 1.48594716639625440615454634913 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/8"] "2-85-5.4-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.010429506524413644 0 1.93659467977269682117792101212 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/3"] "2-85-5.4-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.362499510465432 0 2.04510466587025927639381628107 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/7"] "2-85-5.4-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false -0.09788821885501411 0 2.13063556655759572533834427822 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/5"] "2-85-5.4-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false 0.09788821885501411 0 2.81598290500285312517941002531 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/4"] "2-85-5.4-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false 0.010429506524413644 0 3.05165080139244125986926659785 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/6"] "2-85-5.4-c1-0-6" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false 0.37745718048439825 0 3.84281891265127316846266739828 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/1"] "2-85-5.4-c1-0-7" 0.8238497540091176 0.6787284171808836 2 85 "5.4" [] [[0.5, 0.0]] 1 true true false false 0.362499510465432 0 4.19908745218452527558406723933 ["ModularForm/GL2/Q/holomorphic/85/2/b/a/69/2"] "2-85-85.12-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false 0.46181342542255105 0 0.947631052391969716386303843595 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/1"] "2-85-85.12-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false -0.3335501992891152 0 1.51948776319029971010023738533 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/2"] "2-85-85.12-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false -0.152024316201003 0 1.71973649967092596749557870477 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/4"] "2-85-85.12-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false -0.017669560610728545 0 2.49153187484814964508226159558 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/3"] "2-85-85.12-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false 0.0771971826159631 0 2.49343601014541969849876696624 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/6"] "2-85-85.12-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false 0.16264230256056264 0 3.23325194747238886738028742305 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/5"] "2-85-85.12-c1-0-6" 0.8238497540091176 0.6787284171808836 2 85 "85.12" [] [[0.5, 0.0]] 1 true true false false 0.3142633802049118 0 4.33412815257241265501197217646 ["ModularForm/GL2/Q/holomorphic/85/2/r/a/12/7"] "2-85-85.19-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "85.19" [] [[0.5, 0.0]] 1 true true false false -0.17773074040340184 0 1.55870439526399366638624616916 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/19/3"] "2-85-85.19-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "85.19" [] [[0.5, 0.0]] 1 true true false false -0.28057731431132565 0 1.59222109787563151241681109913 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/19/2"] "2-85-85.19-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "85.19" [] [[0.5, 0.0]] 1 true true false false 0.0007386218045303303 0 2.10453185786772475966856917971 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/19/1"] "2-85-85.19-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "85.19" [] [[0.5, 0.0]] 1 true true false false -0.019469393676976417 0 2.28553238167153591848681809117 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["ModularForm/GL2/Q/holomorphic/85/2/c/a/84/4"] "2-85-85.84-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "85.84" [] [[0.5, 0.0]] 1 true true false false 0.21491169553067885 0 3.02729573562606429559942500866 ["ModularForm/GL2/Q/holomorphic/85/2/c/a/84/3"] "2-85-85.84-c1-0-6" 0.8238497540091176 0.6787284171808836 2 85 "85.84" [] [[0.5, 0.0]] 1 true true false false 0.31187260346646734 0 3.77999412062202688487984034550 ["ModularForm/GL2/Q/holomorphic/85/2/c/a/84/2"] "2-85-85.84-c1-0-7" 0.8238497540091176 0.6787284171808836 2 85 "85.84" [] [[0.5, 0.0]] 1 true true false false -0.47312799393499494 0 4.67277822664562071728128567343 ["ModularForm/GL2/Q/holomorphic/85/2/c/a/84/1"] "2-85-85.9-c1-0-0" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false -0.0007386218045303303 0 1.43469049720523850406707992968 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/1"] "2-85-85.9-c1-0-1" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false -0.2378405482445273 0 1.73048144185212838484214803637 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/5"] "2-85-85.9-c1-0-2" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false 0.019469393676976417 0 2.80434187622316532833073440321 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/4"] "2-85-85.9-c1-0-3" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false 0.17773074040340184 0 2.96464705557942530118479011697 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/3"] "2-85-85.9-c1-0-4" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false -0.09938722725307331 0 3.17106105126081210182067186649 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/6"] "2-85-85.9-c1-0-5" 0.8238497540091176 0.6787284171808836 2 85 "85.9" [] [[0.5, 0.0]] 1 true true false false 0.28057731431132565 0 3.65846467383102519858048977969 ["ModularForm/GL2/Q/holomorphic/85/2/m/a/9/2"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).