# lfunc_search downloaded from the LMFDB on 15 June 2026.
# Search link: https://www.lmfdb.org/L/2/2448/204.131
# Query "{'degree': 2, 'conductor': 2448}" returned 894 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.



"2-2448-204.107-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.107"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.19676958476667414	0	0.45283156965622278424304080244	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/719/1"]
"2-2448-204.107-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.107"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.0008563087513705009	0	1.50493360291353787008980613226	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/719/1"]
"2-2448-204.11-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.057691694781291415	0	1.04231755241500774685889751831	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/1439/1"]
"2-2448-204.11-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.13822158123401224	0	1.54031847829346083137339341349	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/1439/1"]
"2-2448-204.131-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.13623294829818533	0	0.876002160616828662597718224536	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/1151/1"]
"2-2448-204.131-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.05968032771711833	0	1.24927244131102003272740372877	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/1151/1"]
"2-2448-204.143-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.143"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.19676958476667414	0	1.17272242585172904155747782334	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/143/1"]
"2-2448-204.143-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.143"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.0008563087513705009	0	1.36325453808845905103828391990	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/143/1"]
"2-2448-204.167-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.167"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.13822158123401224	0	1.04888224243250465047143734121	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/575/1"]
"2-2448-204.167-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.167"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.057691694781291415	0	1.10858255417144799427874760840	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/575/1"]
"2-2448-204.23-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.23"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3808160556892257	0	0.099398250264513849623277329274	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/431/1"]
"2-2448-204.23-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.23"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.18490277967392207	0	0.950898855193738584967688489464	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/431/1"]
"2-2448-204.71-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.71"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3808160556892257	0	1.54334484609267466648541933268	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/1295/1"]
"2-2448-204.71-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.71"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.18490277967392207	0	1.85295189467096260531453549297	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/1295/1"]
"2-2448-204.95-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"204.95"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.05968032771711833	0	1.05349695281099571760059629402	["ModularForm/GL2/Q/holomorphic/2448/1/ed/b/1727/1"]
"2-2448-204.95-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"204.95"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.13623294829818533	0	1.41147370603204981901870197182	["ModularForm/GL2/Q/holomorphic/2448/1/ed/a/1727/1"]
"2-2448-272.203-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"272.203"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4375	0	0.52721435474503905648081308896	["ModularForm/GL2/Q/holomorphic/2448/1/w/a/2107/1"]
"2-2448-272.203-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"272.203"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.06250000000000001	0	0.794844950494261179873371031602	["ModularForm/GL2/Q/holomorphic/2448/1/w/b/2107/1"]
"2-2448-272.67-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"272.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.06250000000000001	0	1.28042631039178782940483404164	["ModularForm/GL2/Q/holomorphic/2448/1/w/b/883/1"]
"2-2448-272.67-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"272.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4375	0	1.57349791437604523611103441944	["ModularForm/GL2/Q/holomorphic/2448/1/w/a/883/1"]
"2-2448-51.38-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"51.38"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.0784618554133095	0	1.16513325775401711313130644870	["ModularForm/GL2/Q/holomorphic/2448/1/bd/a/1313/2"]
"2-2448-51.38-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"51.38"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.11745142060199415	0	1.21325685058925526138037382116	["ModularForm/GL2/Q/holomorphic/2448/1/bd/a/1313/1"]
"2-2448-51.47-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"51.47"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.11745142060199415	0	0.858527810319889120661194996620	["ModularForm/GL2/Q/holomorphic/2448/1/bd/a/1169/1"]
"2-2448-51.47-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"51.47"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.0784618554133095	0	1.42549536807942650921397899794	["ModularForm/GL2/Q/holomorphic/2448/1/bd/a/1169/2"]
"2-2448-51.50-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"51.50"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3479566380076518	0	0.41591831571272995581069633124	["ModularForm/GL2/Q/holomorphic/2448/1/m/a/305/2", "ArtinRepresentation/2.2448.24t22.a.b"]
"2-2448-51.50-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"51.50"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1520433619923482	0	0.939053410406185639029433388456	["ModularForm/GL2/Q/holomorphic/2448/1/m/b/305/2", "ArtinRepresentation/2.2448.24t22.c.a"]
"2-2448-51.50-c0-0-2"	1.10531043192652	1.22171115092559	2	2448	"51.50"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1520433619923482	0	1.59156029747143447406200192966	["ModularForm/GL2/Q/holomorphic/2448/1/m/b/305/1", "ArtinRepresentation/2.2448.24t22.c.b"]
"2-2448-51.50-c0-0-3"	1.10531043192652	1.22171115092559	2	2448	"51.50"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3479566380076518	0	1.72376226840923333116863171027	["ModularForm/GL2/Q/holomorphic/2448/1/m/a/305/1", "ArtinRepresentation/2.2448.24t22.a.a"]
"2-2448-612.115-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.115"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.03338367148323122	0	0.802871719656149056769357132621	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/1951/1"]
"2-2448-612.115-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.115"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.03338367148323122	0	1.37551739855372979944317146701	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/1951/2"]
"2-2448-612.259-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.259"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.25560589370545345	0	1.56667907659407363478024157109	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/2095/2"]
"2-2448-612.259-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.259"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.25560589370545345	0	1.83955429278681031294408746122	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/2095/1"]
"2-2448-612.319-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.319"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.25560589370545345	0	0.30613379476735466629106623673	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/319/1"]
"2-2448-612.319-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.319"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.25560589370545345	0	0.839303937749535795864869331858	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/319/2"]
"2-2448-612.463-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.463"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.03338367148323122	0	1.02285966557474855996019303882	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/463/1"]
"2-2448-612.463-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.463"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.03338367148323122	0	1.40234315621134033404718317464	["ModularForm/GL2/Q/holomorphic/2448/1/de/a/463/2"]
"2-2448-612.475-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.475"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3611111111111111	0	0.05779094705615148349755130440	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1087/2"]
"2-2448-612.475-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.475"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1388888888888889	0	0.64804666363117938097746212262	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1087/1"]
"2-2448-612.475-c0-0-2"	1.10531043192652	1.22171115092559	2	2448	"612.475"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1388888888888889	0	1.11395895719891998197536565771	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1087/4"]
"2-2448-612.475-c0-0-3"	1.10531043192652	1.22171115092559	2	2448	"612.475"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3611111111111111	0	2.16600079847782842015311866959	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1087/3"]
"2-2448-612.67-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"612.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3611111111111111	0	0.76952096587192269131377161077	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1903/3"]
"2-2448-612.67-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"612.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1388888888888889	0	1.38070059276525858755592470640	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1903/1"]
"2-2448-612.67-c0-0-2"	1.10531043192652	1.22171115092559	2	2448	"612.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1388888888888889	0	1.66356359215348786599661323813	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1903/4"]
"2-2448-612.67-c0-0-3"	1.10531043192652	1.22171115092559	2	2448	"612.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3611111111111111	0	2.42965688997818539525440540208	["ModularForm/GL2/Q/holomorphic/2448/1/bw/a/1903/2"]
"2-2448-68.15-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.15"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.13708925704955582	0	1.32161095968943335030478576794	["ModularForm/GL2/Q/holomorphic/2448/1/cf/a/559/1"]
"2-2448-68.19-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.19"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.24259447445521348	0	0.902374834681912290179659613410	["ModularForm/GL2/Q/holomorphic/2448/1/cf/a/1855/1"]
"2-2448-68.43-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.43"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.24259447445521348	0	1.81020485108707554383956443930	["ModularForm/GL2/Q/holomorphic/2448/1/cf/a/2287/1"]
"2-2448-68.47-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.47"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14449478259434237	0	1.04282149973212412337774223049	["ModularForm/GL2/Q/holomorphic/2448/1/bf/b/1135/1"]
"2-2448-68.47-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"68.47"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.35550521740565766	0	1.69431359740932771988829757327	["ModularForm/GL2/Q/holomorphic/2448/1/bf/a/1135/1"]
"2-2448-68.55-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.55"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.35550521740565766	0	0.43571638407969112738144408526	["ModularForm/GL2/Q/holomorphic/2448/1/bf/a/1279/1"]
"2-2448-68.55-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"68.55"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14449478259434237	0	1.59426416658193714061408882785	["ModularForm/GL2/Q/holomorphic/2448/1/bf/b/1279/1"]
"2-2448-68.59-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.59"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.13708925704955582	0	0.72568606334709032157155190322	["ModularForm/GL2/Q/holomorphic/2448/1/cf/a/127/1"]
"2-2448-68.67-c0-0-0"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4166666666666667	0	0.30702869449969487361823145125	["ModularForm/GL2/Q/holomorphic/2448/1/i/c/271/3"]
"2-2448-68.67-c0-0-1"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.25	0	0.920663045924890297779252955113	["ModularForm/GL2/Q/holomorphic/2448/1/i/b/271/2", "ArtinRepresentation/2.2448.8t11.c.b"]
"2-2448-68.67-c0-0-2"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.04945362005170423500176522936	["ModularForm/GL2/Q/holomorphic/2448/1/i/a/271/1", "ArtinRepresentation/2.2448.8t6.a.b"]
"2-2448-68.67-c0-0-3"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.08333333333333334	0	1.11287951769380706409634312955	["ModularForm/GL2/Q/holomorphic/2448/1/i/c/271/4"]
"2-2448-68.67-c0-0-4"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.21909430831505465994027729418	["ModularForm/GL2/Q/holomorphic/2448/1/i/a/271/2", "ArtinRepresentation/2.2448.8t6.a.a"]
"2-2448-68.67-c0-0-5"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.08333333333333334	0	1.27029157480821217922044949304	["ModularForm/GL2/Q/holomorphic/2448/1/i/c/271/2"]
"2-2448-68.67-c0-0-6"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.25	0	1.69446005790377658379292185613	["ModularForm/GL2/Q/holomorphic/2448/1/i/b/271/1", "ArtinRepresentation/2.2448.8t11.c.a"]
"2-2448-68.67-c0-0-7"	1.10531043192652	1.22171115092559	2	2448	"68.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4166666666666667	0	1.99678997631964091703627365040	["ModularForm/GL2/Q/holomorphic/2448/1/i/c/271/1"]
"2-2448-1.1-c1-0-0"	4.4212417277060805	19.547378414809447	2	2448	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.31854708708437416021754671800	["ModularForm/GL2/Q/holomorphic/2448/2/a/z/1/1"]
"2-2448-1.1-c1-0-1"	4.4212417277060805	19.547378414809447	2	2448	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.36890639540784057992858691479	["ModularForm/GL2/Q/holomorphic/2448/2/a/w/1/1"]
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"2-2448-12.11-c3-0-86"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.4312899713409851	0	1.27566270745380755672018168553	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/23"]
"2-2448-12.11-c3-0-87"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4312899713409851	0	1.37445160897148232241770801869	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/12"]
"2-2448-12.11-c3-0-88"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4020433619923482	0	1.38005387069912695897557457318	["ModularForm/GL2/Q/holomorphic/2448/4/e/a/1871/19"]
"2-2448-12.11-c3-0-89"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4020433619923482	0	1.38609503224422355033562235530	["ModularForm/GL2/Q/holomorphic/2448/4/e/a/1871/9"]
"2-2448-12.11-c3-0-9"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.2353766953256815	0	0.13294745478075571430816566629	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/46"]
"2-2448-12.11-c3-0-90"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4312899713409851	0	1.43442772451680596874291859109	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/45"]
"2-2448-12.11-c3-0-91"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4312899713409851	0	1.48355894913231149752218502478	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/9"]
"2-2448-12.11-c3-0-92"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4020433619923482	0	1.70907039038839079206399318386	["ModularForm/GL2/Q/holomorphic/2448/4/e/a/1871/1"]
"2-2448-12.11-c3-0-93"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.2646233046743185	0	1.77450155675330357646196833898	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/5"]
"2-2448-12.11-c3-0-94"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.2353766953256815	0	1.89410925526926231484159257379	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/19"]
"2-2448-12.11-c3-0-95"	12.018181047648312	144.4366756940531	2	2448	"12.11"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.2646233046743185	0	1.91040334506521335448262227909	["ModularForm/GL2/Q/holomorphic/2448/4/e/b/1871/1"]


# 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).