# lfunc_search downloaded from the LMFDB on 25 May 2026.
# Search link: https://www.lmfdb.org/L/2/85/17.4
# Query "{'degree': 2, 'conductor': 85}" returned 17317 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"]
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"2-85-1.1-c19-0-64"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.03654690843488615960719041364	["ModularForm/GL2/Q/holomorphic/85/20/a/b/1/24"]
"2-85-1.1-c19-0-65"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.05310448255660991666930654188	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/20"]
"2-85-1.1-c19-0-66"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.06171453189995096001960433807	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/13"]
"2-85-1.1-c19-0-67"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.07015858188214721060394887354	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/8"]
"2-85-1.1-c19-0-68"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.08628352658294596817806030204	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/4"]
"2-85-1.1-c19-0-69"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.10200471636502365685666759777	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/27"]
"2-85-1.1-c19-0-7"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	0.22915007333843381989611914791	["ModularForm/GL2/Q/holomorphic/85/20/a/b/1/21"]
"2-85-1.1-c19-0-70"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.10225917383066137981774878088	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/12"]
"2-85-1.1-c19-0-71"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.11589249643866673055206544038	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/10"]
"2-85-1.1-c19-0-72"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.12403095366534101308072652758	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/25"]
"2-85-1.1-c19-0-73"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.12528477158146701461016356947	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/21"]
"2-85-1.1-c19-0-74"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.13063437509789131413500215565	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/20"]
"2-85-1.1-c19-0-75"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.14809920570957748001209104467	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/6"]
"2-85-1.1-c19-0-76"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.15538072117357684282559190992	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/8"]
"2-85-1.1-c19-0-77"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.15781071187172891809899902610	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/7"]
"2-85-1.1-c19-0-78"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.17830665628787461498589723474	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/17"]
"2-85-1.1-c19-0-79"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.23512392327947146208690853288	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/3"]
"2-85-1.1-c19-0-8"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	0.26560409717026313710015062749	["ModularForm/GL2/Q/holomorphic/85/20/a/b/1/8"]
"2-85-1.1-c19-0-80"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.27692517881418691785941200996	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/12"]
"2-85-1.1-c19-0-81"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.32981196531232871591825158710	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/19"]
"2-85-1.1-c19-0-82"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.33003521536556852507898160040	["ModularForm/GL2/Q/holomorphic/85/20/a/b/1/22"]
"2-85-1.1-c19-0-83"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.35431515416986123602980043375	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/16"]
"2-85-1.1-c19-0-84"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.36252006385618773805554857590	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/24"]
"2-85-1.1-c19-0-85"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.38466011072653875908140502472	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/5"]
"2-85-1.1-c19-0-86"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	1.44107181658755983887865989590	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/26"]
"2-85-1.1-c19-0-87"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.45304361788219035384464491718	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/23"]
"2-85-1.1-c19-0-88"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.47117882428054544816749539439	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/24"]
"2-85-1.1-c19-0-89"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.48832230479756707616446800380	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/19"]
"2-85-1.1-c19-0-9"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.0	0	0.27725833535981616124959115331	["ModularForm/GL2/Q/holomorphic/85/20/a/d/1/5"]
"2-85-1.1-c19-0-90"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.50652891242471625821413913379	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/21"]
"2-85-1.1-c19-0-91"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.54375842023508936582493020175	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/2"]
"2-85-1.1-c19-0-92"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.54599612443384931804456897609	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/11"]
"2-85-1.1-c19-0-93"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.56930619931403269685399448202	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/15"]
"2-85-1.1-c19-0-94"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.57229689376325411527075542820	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/18"]
"2-85-1.1-c19-0-95"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.84770796261143726447656313019	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/24"]
"2-85-1.1-c19-0-96"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.85946068783086789424464171831	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/22"]
"2-85-1.1-c19-0-97"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.90243283462047203707226376004	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/23"]
"2-85-1.1-c19-0-98"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.95925046450807681914155939305	["ModularForm/GL2/Q/holomorphic/85/20/a/c/1/22"]
"2-85-1.1-c19-0-99"	13.946117455153015	194.49419207292362	2	85	"1.1"	[]	[[9.5, 0.0]]	19	true	true	false	true	0.5	1	1.97671483833466149135111214185	["ModularForm/GL2/Q/holomorphic/85/20/a/a/1/20"]


# 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 <b>primitive</b> 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).