# lfunc_search downloaded from the LMFDB on 24 June 2026.
# Search link: https://www.lmfdb.org/L/1/4729/4729.242/r0-0
# Query "{'degree': 1, 'conductor': 4729, 'spectral_label': 'r0-0'}" returned 2363 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.



"1-4729-4729.10-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.10"	[[0, 0.0]]	[]	0	true	true	false	false	-0.48472629785305515	0	0.169462469006	["Character/Dirichlet/4729/10"]
"1-4729-4729.100-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.100"	[[0, 0.0]]	[]	0	true	true	false	false	0.009767995548908576	0	0.246081382123	["Character/Dirichlet/4729/100"]
"1-4729-4729.1000-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1000"	[[0, 0.0]]	[]	0	true	true	false	false	-0.19211835547070266	0	0.639523426322	["Character/Dirichlet/4729/1000"]
"1-4729-4729.1004-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1004"	[[0, 0.0]]	[]	0	true	true	false	false	0.1144654821037574	0	0.527869111036	["Character/Dirichlet/4729/1004"]
"1-4729-4729.1005-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1005"	[[0, 0.0]]	[]	0	true	true	false	false	-0.006641892640483859	0	1.67750545363	["Character/Dirichlet/4729/1005"]
"1-4729-4729.1008-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1008"	[[0, 0.0]]	[]	0	true	true	false	false	0.4045362730517092	0	1.15592681727	["Character/Dirichlet/4729/1008"]
"1-4729-4729.1012-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1012"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3829065722726271	0	0.153305900616	["Character/Dirichlet/4729/1012"]
"1-4729-4729.1013-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1013"	[[0, 0.0]]	[]	0	true	true	false	false	-0.17353501626888948	0	0.758826509301	["Character/Dirichlet/4729/1013"]
"1-4729-4729.1014-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1014"	[[0, 0.0]]	[]	0	true	true	false	false	-0.21293406457975217	0	0.305157323634	["Character/Dirichlet/4729/1014"]
"1-4729-4729.1016-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1016"	[[0, 0.0]]	[]	0	true	true	false	false	0.04798135050636305	0	0.673757002499	["Character/Dirichlet/4729/1016"]
"1-4729-4729.1018-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1018"	[[0, 0.0]]	[]	0	true	true	false	false	-0.1769522268474346	0	0.455207449158	["Character/Dirichlet/4729/1018"]
"1-4729-4729.1022-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1022"	[[0, 0.0]]	[]	0	true	true	false	false	-0.27379655746176007	0	0.794491595191	["Character/Dirichlet/4729/1022"]
"1-4729-4729.1023-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1023"	[[0, 0.0]]	[]	0	true	true	false	false	0.3188941672111971	0	1.36311849775	["Character/Dirichlet/4729/1023"]
"1-4729-4729.1024-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1024"	[[0, 0.0]]	[]	0	true	true	false	false	0.27484917030408973	0	1.05284185931	["Character/Dirichlet/4729/1024"]
"1-4729-4729.1026-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1026"	[[0, 0.0]]	[]	0	true	true	false	false	-0.24465124212395992	0	0.0480943944287	["Character/Dirichlet/4729/1026"]
"1-4729-4729.1029-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1029"	[[0, 0.0]]	[]	0	true	true	false	false	-0.21049769326340026	0	0.372349411484	["Character/Dirichlet/4729/1029"]
"1-4729-4729.1031-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1031"	[[0, 0.0]]	[]	0	true	true	false	false	0.09471395411259632	0	1.00305968251	["Character/Dirichlet/4729/1031"]
"1-4729-4729.1034-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1034"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4494126968335384	0	1.28533620848	["Character/Dirichlet/4729/1034"]
"1-4729-4729.1036-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1036"	[[0, 0.0]]	[]	0	true	true	false	false	-0.34093310462093984	0	1.52518846006	["Character/Dirichlet/4729/1036"]
"1-4729-4729.1037-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1037"	[[0, 0.0]]	[]	0	true	true	false	false	-0.08921027012455118	0	1.0251980174	["Character/Dirichlet/4729/1037"]
"1-4729-4729.104-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.104"	[[0, 0.0]]	[]	0	true	true	false	false	0.06506611162640241	0	1.08906887143	["Character/Dirichlet/4729/104"]
"1-4729-4729.1040-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1040"	[[0, 0.0]]	[]	0	true	true	false	false	0.4716458425249135	0	0.155317476422	["Character/Dirichlet/4729/1040"]
"1-4729-4729.1042-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1042"	[[0, 0.0]]	[]	0	true	true	false	false	-0.2755567470609277	0	0.961889605245	["Character/Dirichlet/4729/1042"]
"1-4729-4729.1043-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1043"	[[0, 0.0]]	[]	0	true	true	false	false	-0.01359378049612958	0	0.519488928378	["Character/Dirichlet/4729/1043"]
"1-4729-4729.1047-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1047"	[[0, 0.0]]	[]	0	true	true	false	false	0.15005068099299754	0	0.730518034876	["Character/Dirichlet/4729/1047"]
"1-4729-4729.1048-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1048"	[[0, 0.0]]	[]	0	true	true	false	false	-0.1059960704787623	0	0.637779320899	["Character/Dirichlet/4729/1048"]
"1-4729-4729.105-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.105"	[[0, 0.0]]	[]	0	true	true	false	false	0.03128659019571245	0	0.927449654405	["Character/Dirichlet/4729/105"]
"1-4729-4729.1050-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1050"	[[0, 0.0]]	[]	0	true	true	false	false	0.1708966943429039	0	0.783840436522	["Character/Dirichlet/4729/1050"]
"1-4729-4729.1051-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1051"	[[0, 0.0]]	[]	0	true	true	false	false	0.008146438514659104	0	1.81709310395	["Character/Dirichlet/4729/1051"]
"1-4729-4729.1052-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1052"	[[0, 0.0]]	[]	0	true	true	false	false	0.0002674278365504087	0	1.12912647684	["Character/Dirichlet/4729/1052"]
"1-4729-4729.1053-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1053"	[[0, 0.0]]	[]	0	true	true	false	false	0.4578390713349161	0	1.21027557332	["Character/Dirichlet/4729/1053"]
"1-4729-4729.1054-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1054"	[[0, 0.0]]	[]	0	true	true	false	false	-0.27341165200571516	0	0.093557757123	["Character/Dirichlet/4729/1054"]
"1-4729-4729.1055-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1055"	[[0, 0.0]]	[]	0	true	true	false	false	-0.16341647878937332	0	0.0284639844226	["Character/Dirichlet/4729/1055"]
"1-4729-4729.1058-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1058"	[[0, 0.0]]	[]	0	true	true	false	false	0.14385632483929459	0	0.712355846291	["Character/Dirichlet/4729/1058"]
"1-4729-4729.1059-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1059"	[[0, 0.0]]	[]	0	true	true	false	false	0.48989474677159056	0	1.2177537362	["Character/Dirichlet/4729/1059"]
"1-4729-4729.1061-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1061"	[[0, 0.0]]	[]	0	true	true	false	false	-0.34452447995858493	0	1.65364205189	["Character/Dirichlet/4729/1061"]
"1-4729-4729.1062-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1062"	[[0, 0.0]]	[]	0	true	true	false	false	0.17116579632389875	0	0.93364048364	["Character/Dirichlet/4729/1062"]
"1-4729-4729.1064-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1064"	[[0, 0.0]]	[]	0	true	true	false	false	0.015231092850956415	0	0.915654474965	["Character/Dirichlet/4729/1064"]
"1-4729-4729.1065-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1065"	[[0, 0.0]]	[]	0	true	true	false	false	0.42087100953279855	0	0.670548852429	["Character/Dirichlet/4729/1065"]
"1-4729-4729.1072-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1072"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4255383565744797	0	0.81475860459	["Character/Dirichlet/4729/1072"]
"1-4729-4729.1074-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1074"	[[0, 0.0]]	[]	0	true	true	false	false	-0.11106013939171433	0	0.136299600526	["Character/Dirichlet/4729/1074"]
"1-4729-4729.1079-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1079"	[[0, 0.0]]	[]	0	true	true	false	false	-0.05748827446858114	0	0.456305251722	["Character/Dirichlet/4729/1079"]
"1-4729-4729.108-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.108"	[[0, 0.0]]	[]	0	true	true	false	false	0.10829458572824589	0	0.755555455528	["Character/Dirichlet/4729/108"]
"1-4729-4729.1080-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1080"	[[0, 0.0]]	[]	0	true	true	false	false	0.255637696668638	0	0.111214701215	["Character/Dirichlet/4729/1080"]
"1-4729-4729.1081-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1081"	[[0, 0.0]]	[]	0	true	true	false	false	-0.43002047633063206	0	1.25661881795	["Character/Dirichlet/4729/1081"]
"1-4729-4729.1082-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1082"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3400812980801754	0	1.32236855567	["Character/Dirichlet/4729/1082"]
"1-4729-4729.1083-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1083"	[[0, 0.0]]	[]	0	true	true	false	false	0.3003089612815872	0	0.673938174558	["Character/Dirichlet/4729/1083"]
"1-4729-4729.1084-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1084"	[[0, 0.0]]	[]	0	true	true	false	false	0.4746621246937625	0	0.489154337005	["Character/Dirichlet/4729/1084"]
"1-4729-4729.1087-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1087"	[[0, 0.0]]	[]	0	true	true	false	false	-0.2353196857457703	0	1.6110460453	["Character/Dirichlet/4729/1087"]
"1-4729-4729.1089-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1089"	[[0, 0.0]]	[]	0	true	true	false	false	0.4981637307686415	0	1.08302523823	["Character/Dirichlet/4729/1089"]
"1-4729-4729.1091-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1091"	[[0, 0.0]]	[]	0	true	true	false	false	-0.2764575204335023	0	0.830923436881	["Character/Dirichlet/4729/1091"]
"1-4729-4729.1092-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1092"	[[0, 0.0]]	[]	0	true	true	false	false	0.4740326423649559	0	1.3816135035	["Character/Dirichlet/4729/1092"]
"1-4729-4729.1094-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1094"	[[0, 0.0]]	[]	0	true	true	false	false	0.03216974669090436	0	0.891804865941	["Character/Dirichlet/4729/1094"]
"1-4729-4729.1095-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1095"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4746621246937625	0	0.857174174199	["Character/Dirichlet/4729/1095"]
"1-4729-4729.1097-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1097"	[[0, 0.0]]	[]	0	true	true	false	false	0.26316767768311633	0	1.50192794007	["Character/Dirichlet/4729/1097"]
"1-4729-4729.1099-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1099"	[[0, 0.0]]	[]	0	true	true	false	false	0.42718888966323243	0	0.0481257512213	["Character/Dirichlet/4729/1099"]
"1-4729-4729.1101-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1101"	[[0, 0.0]]	[]	0	true	true	false	false	-0.06292712501723509	0	0.768696190943	["Character/Dirichlet/4729/1101"]
"1-4729-4729.1103-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1103"	[[0, 0.0]]	[]	0	true	true	false	false	0.4981637307686415	0	0.397231566927	["Character/Dirichlet/4729/1103"]
"1-4729-4729.111-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.111"	[[0, 0.0]]	[]	0	true	true	false	false	-0.041226636582892745	0	1.04983558733	["Character/Dirichlet/4729/111"]
"1-4729-4729.1110-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1110"	[[0, 0.0]]	[]	0	true	true	false	false	-0.39697750019784384	0	0.291356845223	["Character/Dirichlet/4729/1110"]
"1-4729-4729.1111-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1111"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3367600457619036	0	0.741516553483	["Character/Dirichlet/4729/1111"]
"1-4729-4729.1114-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1114"	[[0, 0.0]]	[]	0	true	true	false	false	-0.26511139963622077	0	0.645950292563	["Character/Dirichlet/4729/1114"]
"1-4729-4729.1119-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1119"	[[0, 0.0]]	[]	0	true	true	false	false	-0.046063648080305844	0	0.682342859732	["Character/Dirichlet/4729/1119"]
"1-4729-4729.112-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.112"	[[0, 0.0]]	[]	0	true	true	false	false	-0.41652466637394653	0	0.0683611803785	["Character/Dirichlet/4729/112"]
"1-4729-4729.1120-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1120"	[[0, 0.0]]	[]	0	true	true	false	false	-0.44335116585131584	0	1.26919269176	["Character/Dirichlet/4729/1120"]
"1-4729-4729.1121-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1121"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4191860280356387	0	0.227294239763	["Character/Dirichlet/4729/1121"]
"1-4729-4729.1122-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1122"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4768833821552873	0	1.22638479298	["Character/Dirichlet/4729/1122"]
"1-4729-4729.1125-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1125"	[[0, 0.0]]	[]	0	true	true	false	false	0.22557496312856176	0	0.94186148956	["Character/Dirichlet/4729/1125"]
"1-4729-4729.1126-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1126"	[[0, 0.0]]	[]	0	true	true	false	false	-0.21723520952528863	0	1.53871952939	["Character/Dirichlet/4729/1126"]
"1-4729-4729.1129-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1129"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4077284117677822	0	0.719273280088	["Character/Dirichlet/4729/1129"]
"1-4729-4729.1132-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1132"	[[0, 0.0]]	[]	0	true	true	false	false	0.2478292401273752	0	0.69357399638	["Character/Dirichlet/4729/1132"]
"1-4729-4729.1133-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1133"	[[0, 0.0]]	[]	0	true	true	false	false	-0.07675862018166454	0	1.02869905953	["Character/Dirichlet/4729/1133"]
"1-4729-4729.1134-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1134"	[[0, 0.0]]	[]	0	true	true	false	false	0.4170892740378409	0	0.58048072907	["Character/Dirichlet/4729/1134"]
"1-4729-4729.1135-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1135"	[[0, 0.0]]	[]	0	true	true	false	false	0.4445868715984014	0	0.0402960526285	["Character/Dirichlet/4729/1135"]
"1-4729-4729.1136-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1136"	[[0, 0.0]]	[]	0	true	true	false	false	0.4942693963256676	0	0.524228140056	["Character/Dirichlet/4729/1136"]
"1-4729-4729.1138-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1138"	[[0, 0.0]]	[]	0	true	true	false	false	0.3435633667707114	0	0.294847479243	["Character/Dirichlet/4729/1138"]
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"1-4729-4729.1301-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1301"	[[0, 0.0]]	[]	0	true	true	false	false	0.46578261894576556	0	1.29639885249	["Character/Dirichlet/4729/1301"]
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"1-4729-4729.131-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.131"	[[0, 0.0]]	[]	0	true	true	false	false	0.1907656326287632	0	0.649908731949	["Character/Dirichlet/4729/131"]
"1-4729-4729.1310-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1310"	[[0, 0.0]]	[]	0	true	true	false	false	0.4117708551997955	0	0.330302488095	["Character/Dirichlet/4729/1310"]
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"1-4729-4729.1323-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1323"	[[0, 0.0]]	[]	0	true	true	false	false	0.008026958275359496	0	0.907456274727	["Character/Dirichlet/4729/1323"]
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"1-4729-4729.1328-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.1328"	[[0, 0.0]]	[]	0	true	true	false	false	0.23284768901571273	0	1.17810142978	["Character/Dirichlet/4729/1328"]
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"1-4729-4729.210-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.210"	[[0, 0.0]]	[]	0	true	true	false	false	0.26472611314225014	0	1.22031510227	["Character/Dirichlet/4729/210"]
"1-4729-4729.2100-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2100"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4630458319448499	0	1.45374342872	["Character/Dirichlet/4729/2100"]
"1-4729-4729.2101-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2101"	[[0, 0.0]]	[]	0	true	true	false	false	0.26964700766017446	0	0.44876897487	["Character/Dirichlet/4729/2101"]
"1-4729-4729.2102-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2102"	[[0, 0.0]]	[]	0	true	true	false	false	0.38913998322883486	0	0.153824823668	["Character/Dirichlet/4729/2102"]
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"1-4729-4729.2108-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2108"	[[0, 0.0]]	[]	0	true	true	false	false	-0.1508563364343534	0	0.244601351251	["Character/Dirichlet/4729/2108"]
"1-4729-4729.2109-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2109"	[[0, 0.0]]	[]	0	true	true	false	false	0.3282660806506965	0	0.572527832995	["Character/Dirichlet/4729/2109"]
"1-4729-4729.211-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.211"	[[0, 0.0]]	[]	0	true	true	false	false	0.18525577808350918	0	0.279642680977	["Character/Dirichlet/4729/211"]
"1-4729-4729.2110-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2110"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3282660806506965	0	0.721860321505	["Character/Dirichlet/4729/2110"]
"1-4729-4729.2111-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2111"	[[0, 0.0]]	[]	0	true	true	false	false	-0.24938451927799665	0	0.349561281461	["Character/Dirichlet/4729/2111"]
"1-4729-4729.2116-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2116"	[[0, 0.0]]	[]	0	true	true	false	false	0.20417950670986873	0	0.366111642978	["Character/Dirichlet/4729/2116"]
"1-4729-4729.2118-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2118"	[[0, 0.0]]	[]	0	true	true	false	false	-0.05447376544818764	0	0.855566624417	["Character/Dirichlet/4729/2118"]
"1-4729-4729.2122-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2122"	[[0, 0.0]]	[]	0	true	true	false	false	0.2967334181707189	0	0.791669235948	["Character/Dirichlet/4729/2122"]
"1-4729-4729.2124-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2124"	[[0, 0.0]]	[]	0	true	true	false	false	0.18543585826774708	0	0.733782084728	["Character/Dirichlet/4729/2124"]
"1-4729-4729.2127-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2127"	[[0, 0.0]]	[]	0	true	true	false	false	-0.22557496312856176	0	0.347691708886	["Character/Dirichlet/4729/2127"]
"1-4729-4729.2128-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2128"	[[0, 0.0]]	[]	0	true	true	false	false	0.1540652491745107	0	1.06603491499	["Character/Dirichlet/4729/2128"]
"1-4729-4729.2129-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2129"	[[0, 0.0]]	[]	0	true	true	false	false	0.1608436406022098	0	0.870969330198	["Character/Dirichlet/4729/2129"]
"1-4729-4729.213-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.213"	[[0, 0.0]]	[]	0	true	true	false	false	-0.24088161352866005	0	0.410574370698	["Character/Dirichlet/4729/213"]
"1-4729-4729.2130-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2130"	[[0, 0.0]]	[]	0	true	true	false	false	0.2928520483822901	0	0.886374843402	["Character/Dirichlet/4729/2130"]
"1-4729-4729.2131-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2131"	[[0, 0.0]]	[]	0	true	true	false	false	-0.2852962418610275	0	1.96131273001	["Character/Dirichlet/4729/2131"]
"1-4729-4729.2137-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2137"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4206909806020938	0	1.07242588625	["Character/Dirichlet/4729/2137"]
"1-4729-4729.2139-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2139"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4004233949291773	0	0.897427957391	["Character/Dirichlet/4729/2139"]
"1-4729-4729.2141-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2141"	[[0, 0.0]]	[]	0	true	true	false	false	-0.15948768606677452	0	0.51474680848	["Character/Dirichlet/4729/2141"]
"1-4729-4729.2144-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2144"	[[0, 0.0]]	[]	0	true	true	false	false	0.49607517140075436	0	0.610235551495	["Character/Dirichlet/4729/2144"]
"1-4729-4729.2148-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2148"	[[0, 0.0]]	[]	0	true	true	false	false	0.125204972687988	0	0.0993372410278	["Character/Dirichlet/4729/2148"]
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"1-4729-4729.2158-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2158"	[[0, 0.0]]	[]	0	true	true	false	false	-0.19074743441193776	0	0.735514257453	["Character/Dirichlet/4729/2158"]
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"1-4729-4729.2160-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2160"	[[0, 0.0]]	[]	0	true	true	false	false	-0.008672974190942396	0	0.887379923882	["Character/Dirichlet/4729/2160"]
"1-4729-4729.2162-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2162"	[[0, 0.0]]	[]	0	true	true	false	false	-0.19114546818174183	0	0.59556995622	["Character/Dirichlet/4729/2162"]
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"1-4729-4729.2168-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2168"	[[0, 0.0]]	[]	0	true	true	false	false	0.10790752036567285	0	1.00718618939	["Character/Dirichlet/4729/2168"]
"1-4729-4729.2169-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2169"	[[0, 0.0]]	[]	0	true	true	false	false	-0.47025628603596864	0	0.450627755807	["Character/Dirichlet/4729/2169"]
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"1-4729-4729.2188-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2188"	[[0, 0.0]]	[]	0	true	true	false	false	0.05957901956078963	0	1.31274278582	["Character/Dirichlet/4729/2188"]
"1-4729-4729.2189-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2189"	[[0, 0.0]]	[]	0	true	true	false	false	-0.11535433851803022	0	0.941044050194	["Character/Dirichlet/4729/2189"]
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"1-4729-4729.2207-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.2207"	[[0, 0.0]]	[]	0	true	true	false	false	-0.21761838049323395	0	0.583049000699	["Character/Dirichlet/4729/2207"]
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"1-4729-4729.4023-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4023"	[[0, 0.0]]	[]	0	true	true	false	false	-0.28920724442694795	0	0.538804750154	["Character/Dirichlet/4729/4023"]
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"1-4729-4729.4029-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4029"	[[0, 0.0]]	[]	0	true	true	false	false	0.11355394109035394	0	0.167514253792	["Character/Dirichlet/4729/4029"]
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"1-4729-4729.4042-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4042"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4684844954507653	0	0.687820391182	["Character/Dirichlet/4729/4042"]
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"1-4729-4729.432-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.432"	[[0, 0.0]]	[]	0	true	true	false	false	0.08737280638808644	0	0.935704418977	["Character/Dirichlet/4729/432"]
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"1-4729-4729.4324-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4324"	[[0, 0.0]]	[]	0	true	true	false	false	-0.12638070664321627	0	1.38213365038	["Character/Dirichlet/4729/4324"]
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"1-4729-4729.4335-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4335"	[[0, 0.0]]	[]	0	true	true	false	false	0.4384997652265531	0	0.0541784314013	["Character/Dirichlet/4729/4335"]
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"1-4729-4729.4337-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.4337"	[[0, 0.0]]	[]	0	true	true	false	false	0.390447539199941	0	1.31555344912	["Character/Dirichlet/4729/4337"]
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"1-4729-4729.953-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.953"	[[0, 0.0]]	[]	0	true	true	false	false	-0.14556816052456148	0	0.356868825189	["Character/Dirichlet/4729/953"]
"1-4729-4729.957-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.957"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3739834583841421	0	0.537587113735	["Character/Dirichlet/4729/957"]
"1-4729-4729.96-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.96"	[[0, 0.0]]	[]	0	true	true	false	false	0.07675862018166454	0	1.12578296138	["Character/Dirichlet/4729/96"]
"1-4729-4729.960-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.960"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3336762154833687	0	0.314953145009	["Character/Dirichlet/4729/960"]
"1-4729-4729.961-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.961"	[[0, 0.0]]	[]	0	true	true	false	false	-0.4488069444520875	0	0.204112533998	["Character/Dirichlet/4729/961"]
"1-4729-4729.962-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.962"	[[0, 0.0]]	[]	0	true	true	false	false	0.05562373447574625	0	0.474041063286	["Character/Dirichlet/4729/962"]
"1-4729-4729.964-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.964"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3767496702595261	0	1.63579978706	["Character/Dirichlet/4729/964"]
"1-4729-4729.965-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.965"	[[0, 0.0]]	[]	0	true	true	false	false	-0.05020509989956357	0	0.00651117137956	["Character/Dirichlet/4729/965"]
"1-4729-4729.968-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.968"	[[0, 0.0]]	[]	0	true	true	false	false	0.29670080107781727	0	1.11184047273	["Character/Dirichlet/4729/968"]
"1-4729-4729.97-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.97"	[[0, 0.0]]	[]	0	true	true	false	false	-0.11773153432284869	0	0.523243014783	["Character/Dirichlet/4729/97"]
"1-4729-4729.970-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.970"	[[0, 0.0]]	[]	0	true	true	false	false	-0.20761048672701607	0	0.145108273259	["Character/Dirichlet/4729/970"]
"1-4729-4729.971-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.971"	[[0, 0.0]]	[]	0	true	true	false	false	0.25600949006206725	0	1.47717074308	["Character/Dirichlet/4729/971"]
"1-4729-4729.972-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.972"	[[0, 0.0]]	[]	0	true	true	false	false	0.007971172429385843	0	0.79832730265	["Character/Dirichlet/4729/972"]
"1-4729-4729.975-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.975"	[[0, 0.0]]	[]	0	true	true	false	false	-0.16818693689471692	0	0.49890685477	["Character/Dirichlet/4729/975"]
"1-4729-4729.977-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.977"	[[0, 0.0]]	[]	0	true	true	false	false	-0.25855997191582786	0	1.01492410584	["Character/Dirichlet/4729/977"]
"1-4729-4729.979-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.979"	[[0, 0.0]]	[]	0	true	true	false	false	-0.10184695265373313	0	0.478971360806	["Character/Dirichlet/4729/979"]
"1-4729-4729.98-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.98"	[[0, 0.0]]	[]	0	true	true	false	false	0.12702099802061934	0	0.327955476845	["Character/Dirichlet/4729/98"]
"1-4729-4729.980-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.980"	[[0, 0.0]]	[]	0	true	true	false	false	-0.26108116748115395	0	0.0466825281795	["Character/Dirichlet/4729/980"]
"1-4729-4729.985-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.985"	[[0, 0.0]]	[]	0	true	true	false	false	-0.07247621284096378	0	0.858236696912	["Character/Dirichlet/4729/985"]
"1-4729-4729.986-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.986"	[[0, 0.0]]	[]	0	true	true	false	false	-0.300707288954212	0	0.900443766143	["Character/Dirichlet/4729/986"]
"1-4729-4729.988-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.988"	[[0, 0.0]]	[]	0	true	true	false	false	0.06537519898822693	0	1.08775462189	["Character/Dirichlet/4729/988"]
"1-4729-4729.989-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.989"	[[0, 0.0]]	[]	0	true	true	false	false	0.1802616617916485	0	0.99160912218	["Character/Dirichlet/4729/989"]
"1-4729-4729.993-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.993"	[[0, 0.0]]	[]	0	true	true	false	false	-0.24073936548521502	0	0.693995518279	["Character/Dirichlet/4729/993"]
"1-4729-4729.994-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.994"	[[0, 0.0]]	[]	0	true	true	false	false	0.3383159951450581	0	0.748776936688	["Character/Dirichlet/4729/994"]
"1-4729-4729.996-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.996"	[[0, 0.0]]	[]	0	true	true	false	false	-0.3903587463103232	0	0.341570749537	["Character/Dirichlet/4729/996"]
"1-4729-4729.997-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.997"	[[0, 0.0]]	[]	0	true	true	false	false	0.04593353160043076	0	1.06535914395	["Character/Dirichlet/4729/997"]
"1-4729-4729.999-r0-0-0"	21.961383672410207	21.961383672410207	1	4729	"4729.999"	[[0, 0.0]]	[]	0	true	true	false	false	-0.028870779013647534	0	0.625663178538	["Character/Dirichlet/4729/999"]


# 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).