# lfunc_search downloaded from the LMFDB on 24 June 2026.
# Search link: https://www.lmfdb.org/L/2/48
# Query "{'degree': 2, 'conductor': 48}" returned 2757 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.



"2-48-1.1-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	3.14826068230388029805668446658	["EllipticCurve/Q/48/a", "ModularForm/GL2/Q/holomorphic/48/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/48/2/a/a"]
"2-48-12.11-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08333333333333334	0	2.52494733590377164458054694339	["ModularForm/GL2/Q/holomorphic/48/2/c/a/47/2"]
"2-48-12.11-c1-0-1"	0.6190976737980414	0.38328192970214603	2	48	"12.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08333333333333334	0	3.63537616080026223751227237238	["ModularForm/GL2/Q/holomorphic/48/2/c/a/47/1"]
"2-48-16.13-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"16.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.015716343788058768	0	2.49704459496081741143324106588	["ModularForm/GL2/Q/holomorphic/48/2/j/a/13/1"]
"2-48-16.13-c1-0-1"	0.6190976737980414	0.38328192970214603	2	48	"16.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1119428600551058	0	2.93277313035644301932611142198	["ModularForm/GL2/Q/holomorphic/48/2/j/a/13/4"]
"2-48-16.13-c1-0-2"	0.6190976737980414	0.38328192970214603	2	48	"16.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.10928365621194125	0	3.86179068547023561844984227152	["ModularForm/GL2/Q/holomorphic/48/2/j/a/13/3"]
"2-48-16.13-c1-0-3"	0.6190976737980414	0.38328192970214603	2	48	"16.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.23694286005510581	0	4.17363002802444845124025890090	["ModularForm/GL2/Q/holomorphic/48/2/j/a/13/2"]
"2-48-16.5-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"16.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.23694286005510581	0	2.03485356374126472316235332668	["ModularForm/GL2/Q/holomorphic/48/2/j/a/37/2"]
"2-48-16.5-c1-0-1"	0.6190976737980414	0.38328192970214603	2	48	"16.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.015716343788058768	0	2.58012679175009916972193838344	["ModularForm/GL2/Q/holomorphic/48/2/j/a/37/1"]
"2-48-16.5-c1-0-2"	0.6190976737980414	0.38328192970214603	2	48	"16.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.10928365621194125	0	3.26001583815327479995642235752	["ModularForm/GL2/Q/holomorphic/48/2/j/a/37/3"]
"2-48-16.5-c1-0-3"	0.6190976737980414	0.38328192970214603	2	48	"16.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1119428600551058	0	3.88431800021340996070071180644	["ModularForm/GL2/Q/holomorphic/48/2/j/a/37/4"]
"2-48-48.11-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.22180372310481786	0	2.19759953500604809690517725123	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/4"]
"2-48-48.11-c1-0-1"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1472177837852683	0	2.62921195346481332736716364162	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/2"]
"2-48-48.11-c1-0-2"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14209670881202832	0	3.10481947319434072200526019304	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/1"]
"2-48-48.11-c1-0-3"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08741749331498082	0	3.49043408298866264078051416716	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/5"]
"2-48-48.11-c1-0-4"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.04785889051872223	0	3.96912143353736020292877361809	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/6"]
"2-48-48.11-c1-0-5"	0.6190976737980414	0.38328192970214603	2	48	"48.11"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2166484142443548	0	4.21289449256991863971045032813	["ModularForm/GL2/Q/holomorphic/48/2/k/a/11/3"]
"2-48-48.35-c1-0-0"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14209670881202832	0	1.69578359853725675531896880474	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/1"]
"2-48-48.35-c1-0-1"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2166484142443548	0	2.29677960166614722649074286943	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/3"]
"2-48-48.35-c1-0-2"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08741749331498082	0	3.18625564779949222248919522397	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/5"]
"2-48-48.35-c1-0-3"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04785889051872223	0	3.23986236753145177440224526434	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/6"]
"2-48-48.35-c1-0-4"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1472177837852683	0	3.49868648684200611022182584110	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/2"]
"2-48-48.35-c1-0-5"	0.6190976737980414	0.38328192970214603	2	48	"48.35"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.22180372310481786	0	4.60154615981983080022896872675	["ModularForm/GL2/Q/holomorphic/48/2/k/a/35/4"]
"2-48-16.11-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.0967138127226401	0	1.64318431323729946503727064285	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/2"]
"2-48-16.11-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.2725712769326917	0	1.74188252212475400268684983584	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/4"]
"2-48-16.11-c2-0-2"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.01117716200874955	0	1.83178116133491254186155661668	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/3"]
"2-48-16.11-c2-0-3"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.08585543703337234	0	2.80832534364437408566934433913	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/7"]
"2-48-16.11-c2-0-4"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.11960387597481355	0	3.19387372975425808843308269309	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/6"]
"2-48-16.11-c2-0-5"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.07132244671409939	0	3.22780090768164610083624752412	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/8"]
"2-48-16.11-c2-0-6"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.4417731855443258	0	3.54736934419977414921133792220	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/1"]
"2-48-16.11-c2-0-7"	1.1436368606027987	1.3079052689294253	2	48	"16.11"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.333618180464215	0	4.10867673311570854249237111966	["ModularForm/GL2/Q/holomorphic/48/3/l/a/43/5"]
"2-48-16.3-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.4417731855443258	0	0.49614870507904859657659736481	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/1"]
"2-48-16.3-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.333618180464215	0	1.38414053555612297403574408004	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/5"]
"2-48-16.3-c2-0-2"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.0967138127226401	0	2.09422229771337631941299255751	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/2"]
"2-48-16.3-c2-0-3"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.01117716200874955	0	2.24667172879860080816651333436	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/3"]
"2-48-16.3-c2-0-4"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.11960387597481355	0	2.55531396308722075764088651644	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/6"]
"2-48-16.3-c2-0-5"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.07132244671409939	0	2.57643030537763233134342356400	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/8"]
"2-48-16.3-c2-0-6"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.08585543703337234	0	3.52608016002707217453223146512	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/7"]
"2-48-16.3-c2-0-7"	1.1436368606027987	1.3079052689294253	2	48	"16.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.2725712769326917	0	3.54826847181597847311203393806	["ModularForm/GL2/Q/holomorphic/48/3/l/a/19/4"]
"2-48-3.2-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.19591327601530364	0	1.51516477393485154652426285439	["ModularForm/GL2/Q/holomorphic/48/3/e/b/17/2"]
"2-48-3.2-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"3.2"	[]	[[1.0, 0.0]]	2	true	true	true	true	0.0	0	2.62671465500727783804393539657	["ModularForm/GL2/Q/holomorphic/48/3/e/a/17/1", "ModularForm/GL2/Q/holomorphic/48/3/e/a"]
"2-48-3.2-c2-0-2"	1.1436368606027987	1.3079052689294253	2	48	"3.2"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.19591327601530364	0	3.29890576054193911342403457106	["ModularForm/GL2/Q/holomorphic/48/3/e/b/17/1"]
"2-48-4.3-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"4.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.08333333333333334	0	2.03879976439589166055243716567	["ModularForm/GL2/Q/holomorphic/48/3/g/a/31/2"]
"2-48-4.3-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"4.3"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.08333333333333334	0	2.76052191293842014090323933196	["ModularForm/GL2/Q/holomorphic/48/3/g/a/31/1"]
"2-48-48.29-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.17847678296325062	0	1.00129983075346991908617434918	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/1"]
"2-48-48.29-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.3980478454623266	0	1.15046671920281075487615447417	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/5"]
"2-48-48.29-c2-0-10"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.05405276137781975	0	3.57237264357135153619771026819	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/10"]
"2-48-48.29-c2-0-11"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.22269152593622502	0	3.60935023540904945344208377328	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/6"]
"2-48-48.29-c2-0-12"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.2524097935535897	0	3.88519139356355624851850737833	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/8"]
"2-48-48.29-c2-0-13"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.49092769549390314	0	4.22827261061241541052792511477	["ModularForm/GL2/Q/holomorphic/48/3/i/a/29/2"]
"2-48-48.29-c2-0-2"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.03930806765910266	0	1.72115708998705891254197700275	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/4"]
"2-48-48.29-c2-0-3"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.13797283303416583	0	1.92480159761492582684069068044	["ModularForm/GL2/Q/holomorphic/48/3/i/a/29/1"]
"2-48-48.29-c2-0-4"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.1512171317469339	0	2.11989836425608620263998405131	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/9"]
"2-48-48.29-c2-0-5"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.13587423933128726	0	2.27144330983749714748486608558	["ModularForm/GL2/Q/holomorphic/48/3/i/a/29/3"]
"2-48-48.29-c2-0-6"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.16414831495582125	0	2.57290149009258222400805814303	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/7"]
"2-48-48.29-c2-0-7"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.03291937687154994	0	2.89884597339543381502222992146	["ModularForm/GL2/Q/holomorphic/48/3/i/a/29/4"]
"2-48-48.29-c2-0-8"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.26605116877882784	0	3.05291248250812855849659129831	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/2"]
"2-48-48.29-c2-0-9"	1.1436368606027987	1.3079052689294253	2	48	"48.29"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.2609928931409727	0	3.10261436760194827133415291207	["ModularForm/GL2/Q/holomorphic/48/3/i/b/29/3"]
"2-48-48.5-c2-0-0"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.49092769549390314	0	0.48565899256073123151709289078	["ModularForm/GL2/Q/holomorphic/48/3/i/a/5/2"]
"2-48-48.5-c2-0-1"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.2609928931409727	0	0.65516185204374758761459669322	["ModularForm/GL2/Q/holomorphic/48/3/i/b/5/3"]
"2-48-48.5-c2-0-10"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	-0.03291937687154994	0	2.98152972252694508629880344077	["ModularForm/GL2/Q/holomorphic/48/3/i/a/5/4"]
"2-48-48.5-c2-0-11"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.1512171317469339	0	3.53246933964722492058647193015	["ModularForm/GL2/Q/holomorphic/48/3/i/b/5/9"]
"2-48-48.5-c2-0-12"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.16414831495582125	0	3.81081678404030566707089731590	["ModularForm/GL2/Q/holomorphic/48/3/i/b/5/7"]
"2-48-48.5-c2-0-13"	1.1436368606027987	1.3079052689294253	2	48	"48.5"	[]	[[1.0, 0.0]]	2	true	true	false	false	0.3980478454623266	0	4.20715559226518889787013539308	["ModularForm/GL2/Q/holomorphic/48/3/i/b/5/5"]
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"2-48-1.1-c27-0-23"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	true	true	0.5	1	1.36518480820407349525786112317	["ModularForm/GL2/Q/holomorphic/48/28/a/b/1/1", "ModularForm/GL2/Q/holomorphic/48/28/a/b"]
"2-48-1.1-c27-0-24"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.5	1	1.40193253736648484697684480207	["ModularForm/GL2/Q/holomorphic/48/28/a/k/1/3"]
"2-48-1.1-c27-0-25"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.5	1	1.41539618019930317393472123540	["ModularForm/GL2/Q/holomorphic/48/28/a/k/1/1"]
"2-48-1.1-c27-0-26"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.5	1	1.56558034686874840389075928156	["ModularForm/GL2/Q/holomorphic/48/28/a/j/1/3"]
"2-48-1.1-c27-0-3"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.37558227910059204604098055112	["ModularForm/GL2/Q/holomorphic/48/28/a/h/1/2"]
"2-48-1.1-c27-0-4"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.46832935245818858778381825713	["ModularForm/GL2/Q/holomorphic/48/28/a/f/1/1"]
"2-48-1.1-c27-0-5"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.49699267214560595525886045122	["ModularForm/GL2/Q/holomorphic/48/28/a/h/1/3"]
"2-48-1.1-c27-0-6"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.52071941934844237508838271620	["ModularForm/GL2/Q/holomorphic/48/28/a/g/1/1"]
"2-48-1.1-c27-0-7"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.5	1	0.70234461555399629318439641453	["ModularForm/GL2/Q/holomorphic/48/28/a/i/1/1"]
"2-48-1.1-c27-0-8"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.70455644196356570856753195785	["ModularForm/GL2/Q/holomorphic/48/28/a/l/1/3"]
"2-48-1.1-c27-0-9"	14.889280571010291	221.69067592226452	2	48	"1.1"	[]	[[13.5, 0.0]]	27	true	true	false	true	0.0	0	0.76507524631649378051301296455	["ModularForm/GL2/Q/holomorphic/48/28/a/l/1/4"]
"2-48-12.11-c27-0-0"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.11676356821325924	0	0.03323912860991535384376574462	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/18"]
"2-48-12.11-c27-0-1"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.16489235637314575	0	0.06030390940865500675046889464	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/11"]
"2-48-12.11-c27-0-10"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.24206116633176084	0	0.23679635887347791981672599320	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/32"]
"2-48-12.11-c27-0-11"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.05631357544106523	0	0.25560385776077862447727876418	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/2"]
"2-48-12.11-c27-0-12"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.11900146194501603	0	0.25864552580061209289581320812	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/5"]
"2-48-12.11-c27-0-13"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.4811531107716359	0	0.32244073678436281345717165708	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/13"]
"2-48-12.11-c27-0-14"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.4358746410070748	0	0.33952370799838848664103017738	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/8"]
"2-48-12.11-c27-0-15"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.4727440204176371	0	0.35381481553694560343153147521	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/16"]
"2-48-12.11-c27-0-16"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.2692079743404081	0	0.40287239929973311458974754564	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/29"]
"2-48-12.11-c27-0-17"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.013493239156232068	0	0.41555531645022896083994651862	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/27"]
"2-48-12.11-c27-0-18"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.380998538054984	0	0.41891355790679518955839329399	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/12"]
"2-48-12.11-c27-0-19"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.049903098453407416	0	0.42012045722464388909915597960	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/20"]
"2-48-12.11-c27-0-2"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.3284102814662485	0	0.085844757359096781305128664405	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/2"]
"2-48-12.11-c27-0-20"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.18015990582289873	0	0.44201199425448154249364780701	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/9"]
"2-48-12.11-c27-0-21"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.26768356614866584	0	0.44604263051421835446768111533	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/10"]
"2-48-12.11-c27-0-22"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.11900146194501603	0	0.47917712623036892784173225695	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/6"]
"2-48-12.11-c27-0-23"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.0772250630445796	0	0.57332726764475953217436685060	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/4"]
"2-48-12.11-c27-0-24"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.13692178385265175	0	0.59317522362486551096033136216	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/34"]
"2-48-12.11-c27-0-25"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.05631357544106523	0	0.60570308860581448069160503249	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/1"]
"2-48-12.11-c27-0-26"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.26768356614866584	0	0.61252290184189446638096391134	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/9"]
"2-48-12.11-c27-0-27"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.16489235637314575	0	0.63759427956669435511435223770	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/12"]
"2-48-12.11-c27-0-28"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.1617436147995818	0	0.67220362816640404658093572295	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/36"]
"2-48-12.11-c27-0-29"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.0772250630445796	0	0.68725798748008464548245604108	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/3"]
"2-48-12.11-c27-0-3"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.3605893129156963	0	0.094504415290347563664419168145	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/22"]
"2-48-12.11-c27-0-30"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.23231643385133421	0	0.68813357595102819532832000035	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/7"]
"2-48-12.11-c27-0-31"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.4436864245589348	0	0.70129878813575861317256186096	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/16"]
"2-48-12.11-c27-0-32"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.3035884505193184	0	0.73797222794624067445892157987	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/4"]
"2-48-12.11-c27-0-33"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.0017743102935209212	0	0.76164461209694672037329506696	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/26"]
"2-48-12.11-c27-0-34"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.25	0	0.77583684589535234758679289282	["ModularForm/GL2/Q/holomorphic/48/28/c/a/47/2"]
"2-48-12.11-c27-0-35"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.25	0	0.809977888439251527698318059684	["ModularForm/GL2/Q/holomorphic/48/28/c/a/47/1"]
"2-48-12.11-c27-0-36"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.013493239156232068	0	0.883264796628554114265595693661	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/28"]
"2-48-12.11-c27-0-37"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.4358746410070748	0	0.884434642356225255567098339405	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/7"]
"2-48-12.11-c27-0-38"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.13692178385265175	0	0.886991782283653248983078599624	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/33"]
"2-48-12.11-c27-0-39"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.0017743102935209212	0	0.890579454618688867066734762714	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/25"]
"2-48-12.11-c27-0-4"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.42277493695542046	0	0.10480373524500471419919210564	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/14"]
"2-48-12.11-c27-0-40"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.18015990582289873	0	0.921244123815846598152844961306	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/10"]
"2-48-12.11-c27-0-41"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.2692079743404081	0	1.08425606368239244538638793573	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/30"]
"2-48-12.11-c27-0-42"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.4727440204176371	0	1.08437039109230890724506934618	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/15"]
"2-48-12.11-c27-0-43"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.4087278329984275	0	1.12161318347293549706156467907	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/5"]
"2-48-12.11-c27-0-44"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.42277493695542046	0	1.17372180098159139132469561332	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/13"]
"2-48-12.11-c27-0-45"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.380998538054984	0	1.17551728425937205278568057034	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/11"]
"2-48-12.11-c27-0-46"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.1617436147995818	0	1.17880410849175126001179295954	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/35"]
"2-48-12.11-c27-0-47"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.4811531107716359	0	1.27111162648703267743186296561	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/14"]
"2-48-12.11-c27-0-48"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.3284102814662485	0	1.35183814957657099470855777060	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/1"]
"2-48-12.11-c27-0-49"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.3521802225616975	0	1.44405992111566391515491352475	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/23"]
"2-48-12.11-c27-0-5"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.049903098453407416	0	0.14050419576941586535557129552	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/19"]
"2-48-12.11-c27-0-50"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.3605893129156963	0	1.45489372743779066782378584304	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/21"]
"2-48-12.11-c27-0-51"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.4436864245589348	0	1.53396423716154620077895303105	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/15"]
"2-48-12.11-c27-0-52"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.11676356821325924	0	1.89124447192415111258808536316	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/17"]
"2-48-12.11-c27-0-53"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.24206116633176084	0	2.22926997149567503954857708674	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/31"]
"2-48-12.11-c27-0-6"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	0.3521802225616975	0	0.15134538833176345625577132282	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/24"]
"2-48-12.11-c27-0-7"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.23231643385133421	0	0.18279897071144187520538255244	["ModularForm/GL2/Q/holomorphic/48/28/c/b/47/8"]
"2-48-12.11-c27-0-8"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.3035884505193184	0	0.21718747191149731121109181203	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/3"]
"2-48-12.11-c27-0-9"	14.889280571010291	221.69067592226452	2	48	"12.11"	[]	[[13.5, 0.0]]	27	true	true	false	false	-0.4087278329984275	0	0.22662016373210007544338770728	["ModularForm/GL2/Q/holomorphic/48/28/c/c/47/6"]


# 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).