# lfunc_search downloaded from the LMFDB on 16 April 2026.
# Search link: https://www.lmfdb.org/L/2/3800/152.83
# Query "{'degree': 2, 'conductor': 3800}" returned 473 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.



"2-3800-152.11-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.36720997736447175	0	0.68423752526969071470027603493	["ModularForm/GL2/Q/holomorphic/3800/1/bd/a/3051/1", "ArtinRepresentation/2.3800.6t5.d.a"]
"2-3800-152.11-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.033876644031138395	0	0.799358572018411046409269381166	["ModularForm/GL2/Q/holomorphic/3800/1/bd/d/3051/1", "ArtinRepresentation/2.3800.12t18.e.a"]
"2-3800-152.11-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.28387664403113844	0	1.01708103101501137684679887016	["ModularForm/GL2/Q/holomorphic/3800/1/bd/f/3051/2"]
"2-3800-152.11-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.21612335596886162	0	1.06910423054643075536548621834	["ModularForm/GL2/Q/holomorphic/3800/1/bd/f/3051/1"]
"2-3800-152.11-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.033876644031138395	0	1.14076181499277350436845037254	["ModularForm/GL2/Q/holomorphic/3800/1/bd/b/3051/1", "ArtinRepresentation/2.3800.6t5.c.b"]
"2-3800-152.11-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.29945668930219493	0	1.30942933133815676285840403259	["ModularForm/GL2/Q/holomorphic/3800/1/bd/c/3051/1", "ArtinRepresentation/2.3800.12t18.d.a"]
"2-3800-152.11-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.36720997736447175	0	2.72319435301908874115882307727	["ModularForm/GL2/Q/holomorphic/3800/1/bd/e/3051/1", "ArtinRepresentation/2.3800.12t18.f.b"]
"2-3800-152.123-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.28029332098734944	0	0.14961762086333696360935269126	["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/2251/1"]
"2-3800-152.123-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3863733456793173	0	0.63099925734261402372765677228	["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/2251/1"]
"2-3800-152.123-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.25251554320957165	0	0.71757907700913506663827882447	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/2251/1"]
"2-3800-152.123-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.05304001234598393	0	0.75136841781040699102854412938	["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/2251/1"]
"2-3800-152.123-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3863733456793173	0	1.54505486513936447597751823243	["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/2251/1"]
"2-3800-152.123-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.05304001234598393	0	1.56323275383286706708476674064	["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/2251/1"]
"2-3800-152.123-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.24748445679042838	0	1.71637813427509189513980870520	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/2251/2"]
"2-3800-152.131-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.05304001234598393	0	0.73053505432368097083646628390	["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1651/1"]
"2-3800-152.131-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.24748445679042838	0	0.74751150436094506008877153765	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1651/2"]
"2-3800-152.131-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.05304001234598393	0	0.893618540191142883717432693184	["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1651/1"]
"2-3800-152.131-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3863733456793173	0	1.10532613996399056735028032042	["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1651/1"]
"2-3800-152.131-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3863733456793173	0	1.38416345661263996789630470374	["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1651/1"]
"2-3800-152.131-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.28029332098734944	0	1.51538650696186285296774844298	["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1651/1"]
"2-3800-152.131-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.131"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.25251554320957165	0	1.52863128661068222437468926906	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1651/1"]
"2-3800-152.139-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.358699448860543	0	0.000832598882436861477407193692	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1051/1"]
"2-3800-152.139-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4468561066950126	0	0.06633626023977120830009807562	["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1051/1"]
"2-3800-152.139-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.11352277336167924	0	0.856843530204907589373333334017	["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1051/1"]
"2-3800-152.139-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.2198105599716541	0	1.16414841980705346392993377780	["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1051/1"]
"2-3800-152.139-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14130055113945703	0	1.60945670089102455026612486123	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1051/2"]
"2-3800-152.139-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.11352277336167924	0	1.77345951636550670047352189157	["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1051/1"]
"2-3800-152.139-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.139"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4468561066950126	0	1.84220947526851715491844553185	["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1051/1"]
"2-3800-152.35-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14130055113945703	0	0.40174486317415070186052478663	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1251/2"]
"2-3800-152.35-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4468561066950126	0	0.58391819050418152457507952929	["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/1251/1"]
"2-3800-152.35-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.11352277336167924	0	0.61065719934470144284918960642	["ModularForm/GL2/Q/holomorphic/3800/1/cv/a/1251/1"]
"2-3800-152.35-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.2198105599716541	0	1.45223696666061835205671100235	["ModularForm/GL2/Q/holomorphic/3800/1/cv/c/1251/1"]
"2-3800-152.35-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.11352277336167924	0	1.50786051014669123445812855798	["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/1251/1"]
"2-3800-152.35-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.358699448860543	0	1.54056080972528817997209847102	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/1251/1"]
"2-3800-152.35-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.35"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4468561066950126	0	1.64588176062350009232667802072	["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/1251/1"]
"2-3800-152.37-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4375	0	0.27668080574447311944771080826	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/2"]
"2-3800-152.37-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3125	0	0.51044433845691587461890953830	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/4"]
"2-3800-152.37-c0-0-10"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	true	true	0.0	0	1.17694312738159186309599882069	["ModularForm/GL2/Q/holomorphic/3800/1/o/b/1101/1", "ModularForm/GL2/Q/holomorphic/3800/1/o/b", "ArtinRepresentation/2.3800.6t3.c.a", "ArtinRepresentation/2.3800.6t3.c"]
"2-3800-152.37-c0-0-11"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.06250000000000001	0	1.21349285112509968177632613427	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/8"]
"2-3800-152.37-c0-0-12"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.22480060681873618762671818129	["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/1", "ArtinRepresentation/2.3800.9t3.a.c"]
"2-3800-152.37-c0-0-13"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.24394717335128879486726389519	["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/2", "ArtinRepresentation/2.3800.9t3.a.b"]
"2-3800-152.37-c0-0-14"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	true	true	0.0	0	1.26627674389081139921670943886	["ModularForm/GL2/Q/holomorphic/3800/1/o/a/1101/1", "ModularForm/GL2/Q/holomorphic/3800/1/o/a", "ArtinRepresentation/2.3800.6t3.b.a", "ArtinRepresentation/2.3800.6t3.b"]
"2-3800-152.37-c0-0-15"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1875	0	1.50711970670807164268213917037	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/5"]
"2-3800-152.37-c0-0-16"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1875	0	1.54463113061535235245841511351	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/6"]
"2-3800-152.37-c0-0-17"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.55059409854462001210166804268	["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/3"]
"2-3800-152.37-c0-0-18"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.68694758213333139295167510214	["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/2"]
"2-3800-152.37-c0-0-19"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.72408719718483107940422132223	["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/3"]
"2-3800-152.37-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.53475195151724166465600046100	["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/2"]
"2-3800-152.37-c0-0-20"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4375	0	1.79224391649443448737962314976	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/1"]
"2-3800-152.37-c0-0-21"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.83520021970847513060548334122	["ModularForm/GL2/Q/holomorphic/3800/1/o/f/1101/3", "ArtinRepresentation/2.3800.9t3.a.a"]
"2-3800-152.37-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.790026702321987899437065594714	["ModularForm/GL2/Q/holomorphic/3800/1/o/c/1101/1"]
"2-3800-152.37-c0-0-4"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.821521948736724844879885488022	["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/1"]
"2-3800-152.37-c0-0-5"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.861330360567961029172910553528	["ModularForm/GL2/Q/holomorphic/3800/1/o/e/1101/1"]
"2-3800-152.37-c0-0-6"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3125	0	0.869058986565000773675567928839	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/3"]
"2-3800-152.37-c0-0-7"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.888642821641465817593507004954	["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/2"]
"2-3800-152.37-c0-0-8"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.06250000000000001	0	1.05240540401779602092582256845	["ModularForm/GL2/Q/holomorphic/3800/1/o/g/1101/7"]
"2-3800-152.37-c0-0-9"	1.3771154809983834	1.8964470480054092	2	3800	"152.37"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.16998028326487072405259761940	["ModularForm/GL2/Q/holomorphic/3800/1/o/d/1101/3"]
"2-3800-152.43-c0-0-0"	1.3771154809983834	1.8964470480054092	2	3800	"152.43"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.20064719165680855	0	0.29971038553498067229088233058	["ModularForm/GL2/Q/holomorphic/3800/1/cv/b/651/1"]
"2-3800-152.43-c0-0-1"	1.3771154809983834	1.8964470480054092	2	3800	"152.43"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.20064719165680855	0	0.956308949984380190610881682794	["ModularForm/GL2/Q/holomorphic/3800/1/cv/d/651/1"]
"2-3800-152.43-c0-0-2"	1.3771154809983834	1.8964470480054092	2	3800	"152.43"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.22842496943458634	0	1.02515826655293376180756107969	["ModularForm/GL2/Q/holomorphic/3800/1/cv/f/651/2"]
"2-3800-152.43-c0-0-3"	1.3771154809983834	1.8964470480054092	2	3800	"152.43"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1326861416765248	0	1.36726342816701515364594699614	["ModularForm/GL2/Q/holomorphic/3800/1/cv/e/651/1"]
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"2-3800-5.4-c1-0-47"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17620819117478337	0	0.958357412680870950081031708088	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/11"]
"2-3800-5.4-c1-0-48"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07379180882521663	0	0.958907072719369109721223486395	["ModularForm/GL2/Q/holomorphic/3800/2/d/j/3649/4"]
"2-3800-5.4-c1-0-49"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07379180882521663	0	1.02663657397353437532918763041	["ModularForm/GL2/Q/holomorphic/3800/2/d/b/3649/2"]
"2-3800-5.4-c1-0-5"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32379180882521663	0	0.23519039623746656479886125353	["ModularForm/GL2/Q/holomorphic/3800/2/d/m/3649/6"]
"2-3800-5.4-c1-0-50"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07379180882521663	0	1.02942518999629051448245855101	["ModularForm/GL2/Q/holomorphic/3800/2/d/h/3649/4"]
"2-3800-5.4-c1-0-51"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07379180882521663	0	1.04791710551625612844086626317	["ModularForm/GL2/Q/holomorphic/3800/2/d/i/3649/3"]
"2-3800-5.4-c1-0-52"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.09840946982056016654984022510	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/8"]
"2-3800-5.4-c1-0-53"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07379180882521663	0	1.11889661567589091920564807781	["ModularForm/GL2/Q/holomorphic/3800/2/d/h/3649/3"]
"2-3800-5.4-c1-0-54"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32379180882521663	0	1.12102964089180988767236663900	["ModularForm/GL2/Q/holomorphic/3800/2/d/o/3649/2"]
"2-3800-5.4-c1-0-55"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.13019732281063635205870847342	["ModularForm/GL2/Q/holomorphic/3800/2/d/q/3649/9"]
"2-3800-5.4-c1-0-56"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32379180882521663	0	1.16512632568380846914502097557	["ModularForm/GL2/Q/holomorphic/3800/2/d/m/3649/2"]
"2-3800-5.4-c1-0-57"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.17584901098911734538775764151	["ModularForm/GL2/Q/holomorphic/3800/2/d/k/3649/1"]
"2-3800-5.4-c1-0-58"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.19105542465414532992862668459	["ModularForm/GL2/Q/holomorphic/3800/2/d/l/3649/1"]
"2-3800-5.4-c1-0-59"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.20402078652065491988634112902	["ModularForm/GL2/Q/holomorphic/3800/2/d/q/3649/7"]
"2-3800-5.4-c1-0-6"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17620819117478337	0	0.25451196312797026086849428210	["ModularForm/GL2/Q/holomorphic/3800/2/d/q/3649/4"]
"2-3800-5.4-c1-0-60"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.22105117527398968286102999411	["ModularForm/GL2/Q/holomorphic/3800/2/d/q/3649/5"]
"2-3800-5.4-c1-0-61"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.22353173911671455689332639958	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/6"]
"2-3800-5.4-c1-0-62"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07379180882521663	0	1.23088984517142501884850294330	["ModularForm/GL2/Q/holomorphic/3800/2/d/j/3649/5"]
"2-3800-5.4-c1-0-63"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.28912794278658653083113113004	["ModularForm/GL2/Q/holomorphic/3800/2/d/d/3649/1"]
"2-3800-5.4-c1-0-64"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32379180882521663	0	1.34930806915059113116715711415	["ModularForm/GL2/Q/holomorphic/3800/2/d/m/3649/3"]
"2-3800-5.4-c1-0-65"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.35411373358212646674104987239	["ModularForm/GL2/Q/holomorphic/3800/2/d/q/3649/11"]
"2-3800-5.4-c1-0-66"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.38505100929604384259484377236	["ModularForm/GL2/Q/holomorphic/3800/2/d/n/3649/3"]
"2-3800-5.4-c1-0-67"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	1	1.38591285822656762466850296992	["ModularForm/GL2/Q/holomorphic/3800/2/d/c/3649/1"]
"2-3800-5.4-c1-0-68"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17620819117478337	0	1.39208661024847549561436818670	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/10"]
"2-3800-5.4-c1-0-69"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.42479544875468443842243438656	["ModularForm/GL2/Q/holomorphic/3800/2/d/n/3649/1"]
"2-3800-5.4-c1-0-7"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	0	0.25823291424071232560391514923	["ModularForm/GL2/Q/holomorphic/3800/2/d/n/3649/4"]
"2-3800-5.4-c1-0-70"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.45884125301415452683789816744	["ModularForm/GL2/Q/holomorphic/3800/2/d/e/3649/1"]
"2-3800-5.4-c1-0-71"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32379180882521663	0	1.50601740205259182684811960351	["ModularForm/GL2/Q/holomorphic/3800/2/d/o/3649/3"]
"2-3800-5.4-c1-0-72"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	1	1.55884944833532194930608619431	["ModularForm/GL2/Q/holomorphic/3800/2/d/c/3649/2"]
"2-3800-5.4-c1-0-73"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.56372697768125446551306778271	["ModularForm/GL2/Q/holomorphic/3800/2/d/l/3649/3"]
"2-3800-5.4-c1-0-74"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	0	1.58091320554040330649158872221	["ModularForm/GL2/Q/holomorphic/3800/2/d/a/3649/1"]
"2-3800-5.4-c1-0-75"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	1.58550971072670501194788095336	["ModularForm/GL2/Q/holomorphic/3800/2/d/k/3649/4"]
"2-3800-5.4-c1-0-76"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	0	1.64587361621402961688956936471	["ModularForm/GL2/Q/holomorphic/3800/2/d/l/3649/2"]
"2-3800-5.4-c1-0-77"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	0	1.64797676790851794209185882959	["ModularForm/GL2/Q/holomorphic/3800/2/d/n/3649/2"]
"2-3800-5.4-c1-0-78"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.32379180882521663	0	1.79172157845969179923588481772	["ModularForm/GL2/Q/holomorphic/3800/2/d/o/3649/1"]
"2-3800-5.4-c1-0-79"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.42620819117478337	0	1.82820213190766681649474878966	["ModularForm/GL2/Q/holomorphic/3800/2/d/k/3649/2"]
"2-3800-5.4-c1-0-8"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.42620819117478337	0	0.26018709799994795815547576798	["ModularForm/GL2/Q/holomorphic/3800/2/d/n/3649/5"]
"2-3800-5.4-c1-0-80"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.32379180882521663	0	1.83616106002060038410262931476	["ModularForm/GL2/Q/holomorphic/3800/2/d/m/3649/1"]
"2-3800-5.4-c1-0-81"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17620819117478337	0	2.17044310654170533246466897021	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/1"]
"2-3800-5.4-c1-0-9"	5.5084619239935355	30.34315276808656	2	3800	"5.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17620819117478337	0	0.32731549504051104090588380026	["ModularForm/GL2/Q/holomorphic/3800/2/d/p/3649/3"]


# 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).