# lfunc_search downloaded from the LMFDB on 14 June 2026.
# Search link: https://www.lmfdb.org/L/2/21/21.20
# Query "{'degree': 2, 'conductor': 21}" returned 10171 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.



"2-21-1.1-c1-0-0"	0.40949462053205155	0.16768584424468888	2	21	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	4.13559084050773741974089362056	["EllipticCurve/Q/21/a", "ModularForm/GL2/Q/holomorphic/21/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/21/2/a/a"]
"2-21-21.17-c1-0-0"	0.40949462053205155	0.16768584424468888	2	21	"21.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.010086476526973061	0	4.87678738065384911829484710659	["ModularForm/GL2/Q/holomorphic/21/2/g/a/17/1"]
"2-21-21.5-c1-0-0"	0.40949462053205155	0.16768584424468888	2	21	"21.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.010086476526973061	0	4.46412573016302803392609904678	["ModularForm/GL2/Q/holomorphic/21/2/g/a/5/1"]
"2-21-7.2-c1-0-0"	0.40949462053205155	0.16768584424468888	2	21	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14649371361272054	0	3.25829500936861479080741427724	["ModularForm/GL2/Q/holomorphic/21/2/e/a/16/1"]
"2-21-7.4-c1-0-0"	0.40949462053205155	0.16768584424468888	2	21	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14649371361272054	0	5.69031773444673653876183633556	["ModularForm/GL2/Q/holomorphic/21/2/e/a/4/1"]
"2-21-21.11-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.2071070932069753	0	2.02864785952710459585102132239	["ModularForm/GL2/Q/holomorphic/21/3/h/b/11/1"]
"2-21-21.11-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.03290095069846803	0	3.95940027798559613597211892625	["ModularForm/GL2/Q/holomorphic/21/3/h/a/11/1"]
"2-21-21.11-c2-0-2"	0.7564446808304118	0.5722085551566236	2	21	"21.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.06061337959425473	0	4.77249049080509833323078401705	["ModularForm/GL2/Q/holomorphic/21/3/h/b/11/2"]
"2-21-21.2-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.03290095069846803	0	3.36578704399175758378478904088	["ModularForm/GL2/Q/holomorphic/21/3/h/a/2/1"]
"2-21-21.2-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.06061337959425473	0	3.94542690833329969833691836104	["ModularForm/GL2/Q/holomorphic/21/3/h/b/2/2"]
"2-21-21.2-c2-0-2"	0.7564446808304118	0.5722085551566236	2	21	"21.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.2071070932069753	0	4.41658652359578847710526598533	["ModularForm/GL2/Q/holomorphic/21/3/h/b/2/1"]
"2-21-3.2-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.14606114396497957	0	2.72394693730293101925304774111	["ModularForm/GL2/Q/holomorphic/21/3/b/a/8/3"]
"2-21-3.2-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.20577827889749228	0	2.99522710624722766591794103112	["ModularForm/GL2/Q/holomorphic/21/3/b/a/8/4"]
"2-21-3.2-c2-0-2"	0.7564446808304118	0.5722085551566236	2	21	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.14606114396497957	0	4.76189951599963393312237247570	["ModularForm/GL2/Q/holomorphic/21/3/b/a/8/2"]
"2-21-3.2-c2-0-3"	0.7564446808304118	0.5722085551566236	2	21	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.20577827889749228	0	5.21070435478957792899505003238	["ModularForm/GL2/Q/holomorphic/21/3/b/a/8/1"]
"2-21-7.3-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"7.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.10350628638727948	0	3.43300208294605646978948795414	["ModularForm/GL2/Q/holomorphic/21/3/f/c/10/1"]
"2-21-7.3-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"7.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.06316038027938721	0	3.80768540697067178356210859712	["ModularForm/GL2/Q/holomorphic/21/3/f/b/10/1"]
"2-21-7.3-c2-0-2"	0.7564446808304118	0.5722085551566236	2	21	"7.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.2903459061078923	0	5.44308394060920118053063766177	["ModularForm/GL2/Q/holomorphic/21/3/f/a/10/1"]
"2-21-7.5-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"7.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.2903459061078923	0	1.87891378443402310107045640709	["ModularForm/GL2/Q/holomorphic/21/3/f/a/19/1"]
"2-21-7.5-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"7.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.06316038027938721	0	3.17648036595521212323167511635	["ModularForm/GL2/Q/holomorphic/21/3/f/b/19/1"]
"2-21-7.5-c2-0-2"	0.7564446808304118	0.5722085551566236	2	21	"7.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.10350628638727948	0	4.61368157550132321146797149827	["ModularForm/GL2/Q/holomorphic/21/3/f/c/19/1"]
"2-21-7.6-c2-0-0"	0.7564446808304118	0.5722085551566236	2	21	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.02281447417149497	0	3.71728929475028991669282655984	["ModularForm/GL2/Q/holomorphic/21/3/d/a/13/2"]
"2-21-7.6-c2-0-1"	0.7564446808304118	0.5722085551566236	2	21	"7.6"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.02281447417149497	0	4.31814990297491096689755315369	["ModularForm/GL2/Q/holomorphic/21/3/d/a/13/1"]
"2-21-1.1-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.97603540068635921626971898304	["ModularForm/GL2/Q/holomorphic/21/4/a/c/1/1"]
"2-21-1.1-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	3.52383899701414130274827904738	["ModularForm/GL2/Q/holomorphic/21/4/a/c/1/2"]
"2-21-1.1-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.0	0	3.78050195514546319844079133815	["ModularForm/GL2/Q/holomorphic/21/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/21/4/a/b"]
"2-21-1.1-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	4.59505455463720251261035786388	["ModularForm/GL2/Q/holomorphic/21/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/21/4/a/a"]
"2-21-21.17-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.3285005550304627	0	1.07429463353818485527786137358	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/3"]
"2-21-21.17-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.1532680955725822	0	1.99943855337412098492483906890	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/2"]
"2-21-21.17-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.01371680826274806	0	3.34255824695526558205209022964	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/5"]
"2-21-21.17-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.3075065674963869	0	3.45078836605789536516177156316	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/1"]
"2-21-21.17-c3-0-4"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.10494455037469853	0	4.06378845584135650286032186020	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/4"]
"2-21-21.17-c3-0-5"	1.113121785844008	1.2390401101205535	2	21	"21.17"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.11611958363104985	0	4.48343967413805275217402009652	["ModularForm/GL2/Q/holomorphic/21/4/g/a/17/6"]
"2-21-21.20-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.34955743478878587	0	1.58894507498030896984846429630	["ModularForm/GL2/Q/holomorphic/21/4/c/b/20/3"]
"2-21-21.20-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.09077828874275753	0	2.39575046885036427936358363073	["ModularForm/GL2/Q/holomorphic/21/4/c/a/20/2"]
"2-21-21.20-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.2261815775743609	0	2.50041469302151453672513236121	["ModularForm/GL2/Q/holomorphic/21/4/c/b/20/4"]
"2-21-21.20-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.09077828874275753	0	3.48417398160779976635838678149	["ModularForm/GL2/Q/holomorphic/21/4/c/a/20/1"]
"2-21-21.20-c3-0-4"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.2261815775743609	0	4.21846996750222375127956771415	["ModularForm/GL2/Q/holomorphic/21/4/c/b/20/2"]
"2-21-21.20-c3-0-5"	1.113121785844008	1.2390401101205535	2	21	"21.20"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.34955743478878587	0	5.49719705219397201986108773057	["ModularForm/GL2/Q/holomorphic/21/4/c/b/20/1"]
"2-21-21.5-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.3075065674963869	0	0.49993227456804802842988061035	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/1"]
"2-21-21.5-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.10494455037469853	0	2.77228712040942353118855861849	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/4"]
"2-21-21.5-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.11611958363104985	0	3.13568813027488427376538127440	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/6"]
"2-21-21.5-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.1532680955725822	0	3.26619686304507911137097675648	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/2"]
"2-21-21.5-c3-0-4"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.01371680826274806	0	3.79934788836565004940873003757	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/5"]
"2-21-21.5-c3-0-5"	1.113121785844008	1.2390401101205535	2	21	"21.5"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.3285005550304627	0	4.35196236005114589951852683636	["ModularForm/GL2/Q/holomorphic/21/4/g/a/5/3"]
"2-21-7.2-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"7.2"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4309316908981119	0	1.28503207381873994848961598039	["ModularForm/GL2/Q/holomorphic/21/4/e/b/16/1"]
"2-21-7.2-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"7.2"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.07845589545650065	0	2.48372776722578918437887004989	["ModularForm/GL2/Q/holomorphic/21/4/e/b/16/2"]
"2-21-7.2-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"7.2"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.1868396197206128	0	4.37886213680069290113882929938	["ModularForm/GL2/Q/holomorphic/21/4/e/a/16/1"]
"2-21-7.2-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"7.2"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.17605425302127914	0	4.49089376306427944573774469808	["ModularForm/GL2/Q/holomorphic/21/4/e/b/16/3"]
"2-21-7.4-c3-0-0"	1.113121785844008	1.2390401101205535	2	21	"7.4"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.1868396197206128	0	2.45162046213851339139222966109	["ModularForm/GL2/Q/holomorphic/21/4/e/a/4/1"]
"2-21-7.4-c3-0-1"	1.113121785844008	1.2390401101205535	2	21	"7.4"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.17605425302127914	0	3.05831546450298624563908847700	["ModularForm/GL2/Q/holomorphic/21/4/e/b/4/3"]
"2-21-7.4-c3-0-2"	1.113121785844008	1.2390401101205535	2	21	"7.4"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.07845589545650065	0	3.38228381260687071477557465693	["ModularForm/GL2/Q/holomorphic/21/4/e/b/4/2"]
"2-21-7.4-c3-0-3"	1.113121785844008	1.2390401101205535	2	21	"7.4"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.4309316908981119	0	5.16970437169280649720345940778	["ModularForm/GL2/Q/holomorphic/21/4/e/b/4/1"]
"2-21-21.11-c4-0-0"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.4757086988189201	0	0.13488343211258448805478056331	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/4"]
"2-21-21.11-c4-0-1"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.3762664331961642	0	0.916316394075687453818861136345	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/2"]
"2-21-21.11-c4-0-2"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.2855424718160169	0	1.09447029849217569637223041483	["ModularForm/GL2/Q/holomorphic/21/5/h/a/11/1"]
"2-21-21.11-c4-0-3"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	0.23500011531275314	0	2.21356303044645060205519549065	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/1"]
"2-21-21.11-c4-0-4"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	0.0697137595806058	0	2.33484256665381316649402326706	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/3"]
"2-21-21.11-c4-0-5"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.023872907380243717	0	2.52784668260452574388921526710	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/5"]
"2-21-21.11-c4-0-6"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	0.08299874188259491	0	3.52397951808616864513965984695	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/8"]
"2-21-21.11-c4-0-7"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	0.18926564412429467	0	4.00552268290051356058457023530	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/7"]
"2-21-21.11-c4-0-8"	1.4733530550281326	2.1707692247607313	2	21	"21.11"	[]	[[2.0, 0.0]]	4	true	true	false	false	0.13748615406351053	0	4.03224829856238470519313190070	["ModularForm/GL2/Q/holomorphic/21/5/h/b/11/6"]
"2-21-21.2-c4-0-0"	1.4733530550281326	2.1707692247607313	2	21	"21.2"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.23500011531275314	0	0.24651093793199083311280767667	["ModularForm/GL2/Q/holomorphic/21/5/h/b/2/1"]
"2-21-21.2-c4-0-1"	1.4733530550281326	2.1707692247607313	2	21	"21.2"	[]	[[2.0, 0.0]]	4	true	true	false	false	-0.0697137595806058	0	1.48177338138975177424949868852	["ModularForm/GL2/Q/holomorphic/21/5/h/b/2/3"]
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"2-21-7.6-c42-0-46"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.23997926363461142	0	0.986334335374897736696759651635	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/5"]
"2-21-7.6-c42-0-47"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.4554393383960121	0	1.10520322116826612205326680551	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/12"]
"2-21-7.6-c42-0-48"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.39726177254273376	0	1.12526018816901556439017308915	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/30"]
"2-21-7.6-c42-0-49"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.49904595360359344	0	1.23396104588841132618454142033	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/35"]
"2-21-7.6-c42-0-5"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.20670245255670427	0	0.13028860593243207602213878349	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/25"]
"2-21-7.6-c42-0-50"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.25843821235040376	0	1.38244820528711669786730538591	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/55"]
"2-21-7.6-c42-0-51"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.29822650422563446	0	1.40684213079370207788192241058	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/52"]
"2-21-7.6-c42-0-52"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.45777191157958763	0	1.45143382355089834491979032829	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/49"]
"2-21-7.6-c42-0-53"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.05361432179804942	0	1.62881760594702202672727255514	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/9"]
"2-21-7.6-c42-0-54"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.3967774558756698	0	1.73636811405923975383534004133	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/53"]
"2-21-7.6-c42-0-55"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.23508777768159667	0	1.91260379103234102230191823639	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/43"]
"2-21-7.6-c42-0-6"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.49904595360359344	0	0.13387667201220976547263385771	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/36"]
"2-21-7.6-c42-0-7"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	-0.37860966266827134	0	0.14478415492796187110733832744	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/20"]
"2-21-7.6-c42-0-8"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.05361432179804942	0	0.14540596108147811119249861076	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/10"]
"2-21-7.6-c42-0-9"	15.317577649922836	234.62818506141556	2	21	"7.6"	[]	[[21.0, 0.0]]	42	true	true	false	false	0.23508777768159667	0	0.19871998313692615096636349193	["ModularForm/GL2/Q/holomorphic/21/43/d/a/13/44"]


# 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).