# lfunc_search downloaded from the LMFDB on 02 May 2026.
# Search link: https://www.lmfdb.org/L/2/3^4/1.1
# Query "{'degree': 2, 'conductor': 81}" returned 13098 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.



"2-3e4-1.1-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.78651121087012653708241175100	["ModularForm/GL2/Q/holomorphic/81/2/a/a/1/1"]
"2-3e4-1.1-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	3.12234009529020877746547298161	["ModularForm/GL2/Q/holomorphic/81/2/a/a/1/2"]
"2-3e4-27.13-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3316693287550542	0	1.86540728154967913402049276079	["ModularForm/GL2/Q/holomorphic/81/2/e/a/10/2"]
"2-3e4-27.13-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14648414356986902	0	2.78558948457901225601709292024	["ModularForm/GL2/Q/holomorphic/81/2/e/a/10/1"]
"2-3e4-27.16-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0076250761276583185	0	2.66954639704832544177155038989	["ModularForm/GL2/Q/holomorphic/81/2/e/a/46/1"]
"2-3e4-27.16-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1557732242758065	0	2.73169922799128222426771254845	["ModularForm/GL2/Q/holomorphic/81/2/e/a/46/2"]
"2-3e4-27.22-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0076250761276583185	0	2.37098045680480032471313580067	["ModularForm/GL2/Q/holomorphic/81/2/e/a/37/1"]
"2-3e4-27.22-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1557732242758065	0	3.79087644557024191915964516651	["ModularForm/GL2/Q/holomorphic/81/2/e/a/37/2"]
"2-3e4-27.25-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14648414356986902	0	2.16068297297437154876442698701	["ModularForm/GL2/Q/holomorphic/81/2/e/a/73/1"]
"2-3e4-27.25-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3316693287550542	0	3.93545098214888768680473724978	["ModularForm/GL2/Q/holomorphic/81/2/e/a/73/2"]
"2-3e4-27.4-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.20122961156900118	0	2.89633575768505488749041643482	["ModularForm/GL2/Q/holomorphic/81/2/e/a/64/1"]
"2-3e4-27.4-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05308146342085302	0	3.05982508720670982471144215024	["ModularForm/GL2/Q/holomorphic/81/2/e/a/64/2"]
"2-3e4-27.7-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"27.7"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.20122961156900118	0	1.36238949171048255313363429519	["ModularForm/GL2/Q/holomorphic/81/2/e/a/19/1"]
"2-3e4-27.7-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"27.7"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.05308146342085302	0	3.16761777815462144155723855216	["ModularForm/GL2/Q/holomorphic/81/2/e/a/19/2"]
"2-3e4-81.13-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.20785892248542162	0	0.938576788538077135468355605828	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/3"]
"2-3e4-81.13-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06650965375774724	0	1.78469531244498435919842016086	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/1"]
"2-3e4-81.13-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.11210962536339189	0	2.39507635433655029372911158548	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/6"]
"2-3e4-81.13-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.15764778128667667	0	2.48247281721473824599371340703	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/8"]
"2-3e4-81.13-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3565080035226371	0	2.68332026846822166023518666647	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/2"]
"2-3e4-81.13-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.12920385874574278	0	3.10803761860740475084359717354	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/5"]
"2-3e4-81.13-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.007624531691340872	0	3.19372032015702834254008307962	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/7"]
"2-3e4-81.13-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.26344929376370124	0	3.35952941383070042113607964458	["ModularForm/GL2/Q/holomorphic/81/2/g/a/13/4"]
"2-3e4-81.16-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1502560712953098	0	1.28430773702086755205563708449	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/2"]
"2-3e4-81.16-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.02031361533983531	0	1.55715856088049282814925727753	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/3"]
"2-3e4-81.16-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0642697469003151	0	2.56756167427398023149752262738	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/6"]
"2-3e4-81.16-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.10733554521941431	0	2.57677004160925351122849971394	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/8"]
"2-3e4-81.16-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.38600577002354336	0	2.86623895812375183736753856668	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/1"]
"2-3e4-81.16-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0034903014450860127	0	3.27648327818016003597296667282	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/7"]
"2-3e4-81.16-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.18031285687014761	0	3.41707088627495908795218529587	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/5"]
"2-3e4-81.16-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.495848478607969	0	3.97426229494947667271562729036	["ModularForm/GL2/Q/holomorphic/81/2/g/a/16/4"]
"2-3e4-81.22-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08252473515519798	0	1.76992595113163963232113421820	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/1"]
"2-3e4-81.22-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.24010781829740943	0	1.96312216619758705253530953510	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/7"]
"2-3e4-81.22-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.09757580115352525	0	2.52615192963002824340235134436	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/4"]
"2-3e4-81.22-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17463488992778306	0	2.61001728858544505879515016203	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/8"]
"2-3e4-81.22-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13356321790192932	0	2.66775315272243787726809911436	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/6"]
"2-3e4-81.22-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.176146623204738	0	2.98079443159265985579104046154	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/3"]
"2-3e4-81.22-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.146348583672659	0	3.47355894831640250403254915817	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/5"]
"2-3e4-81.22-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.22"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3863885824910972	0	4.34300810695175964535428837500	["ModularForm/GL2/Q/holomorphic/81/2/g/a/22/2"]
"2-3e4-81.25-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3565080035226371	0	0.65313545517410132349936418840	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/2"]
"2-3e4-81.25-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.26344929376370124	0	1.36100808130734169910796253745	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/4"]
"2-3e4-81.25-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.12920385874574278	0	1.89925665093901695527591136577	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/5"]
"2-3e4-81.25-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.06650965375774724	0	2.19511514743989928642532476910	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/1"]
"2-3e4-81.25-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.007624531691340872	0	2.60671830061848423751900771951	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/7"]
"2-3e4-81.25-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.20785892248542162	0	2.64310319093005655054388844285	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/3"]
"2-3e4-81.25-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.11210962536339189	0	3.31757211898587386784856436161	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/6"]
"2-3e4-81.25-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.25"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.15764778128667667	0	3.93903585615150691712739038282	["ModularForm/GL2/Q/holomorphic/81/2/g/a/25/8"]
"2-3e4-81.31-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.15605461415957936	0	1.83644456382169672525381528837	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/5"]
"2-3e4-81.31-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.31043787340367474	0	2.13035887544322534160090050264	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/7"]
"2-3e4-81.31-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.16340184894208334	0	2.59302461954323681844956235086	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/2"]
"2-3e4-81.31-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17508008450305904	0	2.74834737170856407790977333577	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/8"]
"2-3e4-81.31-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07673721942142252	0	2.94346953817535767026451925966	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/6"]
"2-3e4-81.31-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.16910684684144717	0	3.05068515019202117302419510258	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/4"]
"2-3e4-81.31-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3958137276847411	0	3.49829000945817873484427920769	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/3"]
"2-3e4-81.31-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.31"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4844747251443687	0	4.42307380961057283888100753633	["ModularForm/GL2/Q/holomorphic/81/2/g/a/31/1"]
"2-3e4-81.34-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3958137276847411	0	0.960633067984492378598829001577	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/3"]
"2-3e4-81.34-c1-0-1"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4844747251443687	0	1.19987333755570937821319047152	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/1"]
"2-3e4-81.34-c1-0-2"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07673721942142252	0	1.90890229750940663502705729261	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/6"]
"2-3e4-81.34-c1-0-3"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16340184894208334	0	2.04419744164932970852929470195	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/2"]
"2-3e4-81.34-c1-0-4"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16910684684144717	0	2.31736530385878353024429298699	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/4"]
"2-3e4-81.34-c1-0-5"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.15605461415957936	0	3.03942772065242751083855974703	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/5"]
"2-3e4-81.34-c1-0-6"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17508008450305904	0	3.25907691176259324551950257583	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/8"]
"2-3e4-81.34-c1-0-7"	0.8042314693994332	0.6467882563723715	2	81	"81.34"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.31043787340367474	0	4.47398292548583394756206948287	["ModularForm/GL2/Q/holomorphic/81/2/g/a/34/7"]
"2-3e4-81.4-c1-0-0"	0.8042314693994332	0.6467882563723715	2	81	"81.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.23818977495016577	0	0.56687341878834723289625189197	["ModularForm/GL2/Q/holomorphic/81/2/g/a/4/1"]
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"2-3e4-1.1-c21-0-35"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.75619615149841944052690252795	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/3"]
"2-3e4-1.1-c21-0-36"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.77234097809500944487211858741	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/4"]
"2-3e4-1.1-c21-0-37"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.789111228285926468307624155508	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/7"]
"2-3e4-1.1-c21-0-38"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.801818681266095791954270353962	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/5"]
"2-3e4-1.1-c21-0-39"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.806591886959574223956699904832	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/14"]
"2-3e4-1.1-c21-0-4"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.15907476412850810161400878131	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/10"]
"2-3e4-1.1-c21-0-40"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.809239229578106746646565662273	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/9"]
"2-3e4-1.1-c21-0-41"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.816113697917234417576000267272	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/10"]
"2-3e4-1.1-c21-0-42"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.836339020300147156954176015966	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/4"]
"2-3e4-1.1-c21-0-43"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.838556802196304584210109346849	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/15"]
"2-3e4-1.1-c21-0-44"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.864974456706238540213095520037	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/14"]
"2-3e4-1.1-c21-0-45"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.880415163710535721454422587404	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/1"]
"2-3e4-1.1-c21-0-46"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.922729950795267712145136384079	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/2"]
"2-3e4-1.1-c21-0-47"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.923613165022022231963105417739	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/11"]
"2-3e4-1.1-c21-0-48"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.961231485411938974384187031404	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/5"]
"2-3e4-1.1-c21-0-49"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.978811966485958104651455242170	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/2"]
"2-3e4-1.1-c21-0-5"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.16696180301161572299353245466	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/12"]
"2-3e4-1.1-c21-0-50"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.987648767078412329837677960975	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/16"]
"2-3e4-1.1-c21-0-51"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	0.993040429298811261745971584393	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/7"]
"2-3e4-1.1-c21-0-52"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.00531330890919917642355584497	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/7"]
"2-3e4-1.1-c21-0-53"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.01713330835769425754066835758	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/3"]
"2-3e4-1.1-c21-0-54"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.02441310318948375277773484037	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/2"]
"2-3e4-1.1-c21-0-55"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.03178882719172204537282408162	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/3"]
"2-3e4-1.1-c21-0-56"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.03843973608077791052317825031	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/5"]
"2-3e4-1.1-c21-0-57"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	1.03935470816728606960182630292	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/19"]
"2-3e4-1.1-c21-0-58"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.04044718785937523000294119943	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/15"]
"2-3e4-1.1-c21-0-59"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.05041060419604603105899433618	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/12"]
"2-3e4-1.1-c21-0-6"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.18726124798912466331991763839	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/1"]
"2-3e4-1.1-c21-0-60"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.09909537047823872291568971789	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/13"]
"2-3e4-1.1-c21-0-61"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	1.13568424933174259957534173189	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/21"]
"2-3e4-1.1-c21-0-62"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.20545986399593910924881034455	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/11"]
"2-3e4-1.1-c21-0-63"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	1.21104786176157053642519267200	["ModularForm/GL2/Q/holomorphic/81/22/a/d/1/20"]
"2-3e4-1.1-c21-0-64"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.23291234323633348924688548544	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/1"]
"2-3e4-1.1-c21-0-65"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.24612075471883602871414657207	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/6"]
"2-3e4-1.1-c21-0-66"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	1.26099657985101031018046554247	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/20"]
"2-3e4-1.1-c21-0-67"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.27345403928610570916439687469	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/2"]
"2-3e4-1.1-c21-0-68"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.28818907891626013404143719570	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/6"]
"2-3e4-1.1-c21-0-69"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.29061854688864251401843040409	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/8"]
"2-3e4-1.1-c21-0-7"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.23576881902663917515866515653	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/5"]
"2-3e4-1.1-c21-0-70"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.32071540762433458785290689702	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/8"]
"2-3e4-1.1-c21-0-71"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.35222817910656458927215295084	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/4"]
"2-3e4-1.1-c21-0-72"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.37038583571279871136816472750	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/9"]
"2-3e4-1.1-c21-0-73"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.37866649329741710683901121490	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/8"]
"2-3e4-1.1-c21-0-74"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.54178961824202162628802405902	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/19"]
"2-3e4-1.1-c21-0-75"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.55680798576073407106540172177	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/18"]
"2-3e4-1.1-c21-0-76"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.56517441648243970620988824663	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/16"]
"2-3e4-1.1-c21-0-77"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.70297394168156231531111327353	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/10"]
"2-3e4-1.1-c21-0-78"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.71646170070627317676808133271	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/17"]
"2-3e4-1.1-c21-0-79"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.74123645291855586074572067577	["ModularForm/GL2/Q/holomorphic/81/22/a/c/1/20"]
"2-3e4-1.1-c21-0-8"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.23580128969284572429547213809	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/13"]
"2-3e4-1.1-c21-0-80"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.74973084081241291117878692638	["ModularForm/GL2/Q/holomorphic/81/22/a/b/1/9"]
"2-3e4-1.1-c21-0-81"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.85595598493669757930082801034	["ModularForm/GL2/Q/holomorphic/81/22/a/a/1/10"]
"2-3e4-1.1-c21-0-9"	15.045818318473751	226.37664887252032	2	81	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.0	0	0.24411759238110008796255282488	["ModularForm/GL2/Q/holomorphic/81/22/a/e/1/18"]


# 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).