# lfunc_search downloaded from the LMFDB on 14 April 2026.
# Search link: https://www.lmfdb.org/L/2/84/7.4/c5-0
# Query "{'degree': 2, 'conductor': 84, 'spectral_label': 'c5-0'}" returned 392 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.



"2-84-1.1-c5-0-0"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	0.812487711873018734811142710361	["ModularForm/GL2/Q/holomorphic/84/6/a/c/1/1"]
"2-84-1.1-c5-0-1"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.14117276929874442954233315634	["ModularForm/GL2/Q/holomorphic/84/6/a/c/1/2"]
"2-84-1.1-c5-0-2"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.26693560266933299104565739432	["ModularForm/GL2/Q/holomorphic/84/6/a/d/1/1"]
"2-84-1.1-c5-0-3"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.58768645208602917670190450878	["ModularForm/GL2/Q/holomorphic/84/6/a/d/1/2"]
"2-84-1.1-c5-0-4"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	true	true	0.5	1	2.07201297117303168597525594645	["ModularForm/GL2/Q/holomorphic/84/6/a/a/1/1", "ModularForm/GL2/Q/holomorphic/84/6/a/a"]
"2-84-1.1-c5-0-5"	3.6704551304014568	13.472240864290375	2	84	"1.1"	[]	[[2.5, 0.0]]	5	true	true	true	true	0.5	1	2.40539896687688526884775973025	["ModularForm/GL2/Q/holomorphic/84/6/a/b/1/1", "ModularForm/GL2/Q/holomorphic/84/6/a/b"]
"2-84-12.11-c5-0-0"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.2500610043873398	0	0.00439549853971399165938518163	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/36"]
"2-84-12.11-c5-0-1"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.46432115258079876	0	0.01927614219712423702689412886	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/23"]
"2-84-12.11-c5-0-10"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.1703372856333082	0	0.46978029067709947451973382159	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/13"]
"2-84-12.11-c5-0-11"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.08375475758414422	0	0.47357460866168275201787924115	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/3"]
"2-84-12.11-c5-0-12"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.39273443703397043	0	0.47922271355392630691099685477	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/8"]
"2-84-12.11-c5-0-13"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.0014443244376561182	0	0.65451880779990685693918766210	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/39"]
"2-84-12.11-c5-0-14"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.2307578880946309	0	0.65845355966578367630846450997	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/34"]
"2-84-12.11-c5-0-15"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.2604502052857006	0	0.66459641604577503299552956100	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/20"]
"2-84-12.11-c5-0-16"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.44470858430850374	0	0.70213264584398696476617337277	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/26"]
"2-84-12.11-c5-0-17"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.3812244214284343	0	0.71788238289998368117976407402	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/42"]
"2-84-12.11-c5-0-18"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.30452944580431873	0	0.73079576454377485898724372096	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/50"]
"2-84-12.11-c5-0-19"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.38865565739166685	0	0.74953769158305675442226443545	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/60"]
"2-84-12.11-c5-0-2"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.2629834642857535	0	0.06892514298961838987734927429	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/12"]
"2-84-12.11-c5-0-20"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.27945907983627705	0	0.78714265188360292315683293212	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/6"]
"2-84-12.11-c5-0-21"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.25819941499260946	0	0.835045688503849014219927891417	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/18"]
"2-84-12.11-c5-0-22"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.04638089956910333	0	0.841984211211826634926000210072	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/31"]
"2-84-12.11-c5-0-23"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.03220245893475845	0	1.00694290792320026213412330618	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/9"]
"2-84-12.11-c5-0-24"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.08375475758414422	0	1.03921402538742635738433625371	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/4"]
"2-84-12.11-c5-0-25"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.3934428964258199	0	1.04059366690539622898755549841	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/48"]
"2-84-12.11-c5-0-26"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.2307578880946309	0	1.11627757558231275492151192892	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/33"]
"2-84-12.11-c5-0-27"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.1131762774331341	0	1.12379708109284480632934083649	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/27"]
"2-84-12.11-c5-0-28"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.25819941499260946	0	1.14781117750863802795734198282	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/17"]
"2-84-12.11-c5-0-29"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.1703372856333082	0	1.16879221729043806243814129826	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/14"]
"2-84-12.11-c5-0-3"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.4861019909211619	0	0.16247482082430409143549530503	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/15"]
"2-84-12.11-c5-0-30"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.07226797765862747	0	1.18139225763229619955402047081	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/56"]
"2-84-12.11-c5-0-31"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.18324157507929142	0	1.38020405284082715733967909294	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/54"]
"2-84-12.11-c5-0-32"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.0014443244376561182	0	1.39381928314240419988068014800	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/40"]
"2-84-12.11-c5-0-33"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.07226797765862747	0	1.40774071684716718336279114762	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/55"]
"2-84-12.11-c5-0-34"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.1131762774331341	0	1.41064011626807765453477615904	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/28"]
"2-84-12.11-c5-0-35"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.13542338753181796	0	1.43553622291637201654937685748	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/57"]
"2-84-12.11-c5-0-36"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.39273443703397043	0	1.43631814534852605663930157851	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/7"]
"2-84-12.11-c5-0-37"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.0797429887415555	0	1.43788639396904723906726596464	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/30"]
"2-84-12.11-c5-0-38"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.2604502052857006	0	1.45237804588424988116198555637	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/19"]
"2-84-12.11-c5-0-39"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.03220245893475845	0	1.46491708277735980410321723261	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/10"]
"2-84-12.11-c5-0-4"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.4197613544542498	0	0.20067202421103048229209744347	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/2"]
"2-84-12.11-c5-0-40"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.04638089956910333	0	1.50305223750886855788977346180	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/32"]
"2-84-12.11-c5-0-41"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.04564153217128644	0	1.55323691997535038743505049011	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/45"]
"2-84-12.11-c5-0-42"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.27945907983627705	0	1.69313644624376818327803175585	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/5"]
"2-84-12.11-c5-0-43"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.04564153217128644	0	1.83640442904010640847793057096	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/46"]
"2-84-12.11-c5-0-44"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.4861019909211619	0	2.10405004911612101142781750288	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/16"]
"2-84-12.11-c5-0-45"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.30452944580431873	0	2.12142866171960104137350578286	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/49"]
"2-84-12.11-c5-0-46"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.13542338753181796	0	2.16988905676789030034950090715	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/58"]
"2-84-12.11-c5-0-47"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.4197613544542498	0	2.17798233516576121906816109566	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/1"]
"2-84-12.11-c5-0-48"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.3346889118301095	0	2.37118767704662766822892588090	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/22"]
"2-84-12.11-c5-0-49"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.46432115258079876	0	2.47733073988176928599909242559	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/24"]
"2-84-12.11-c5-0-5"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.0797429887415555	0	0.22019372613187519652971002163	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/29"]
"2-84-12.11-c5-0-50"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.30237501449954274	0	2.50216080002141282375287964474	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/44"]
"2-84-12.11-c5-0-51"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.18324157507929142	0	2.51017195671109730886731615047	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/53"]
"2-84-12.11-c5-0-52"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.44470858430850374	0	2.56965180892596365644820923303	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/25"]
"2-84-12.11-c5-0-53"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.2629834642857535	0	2.63005612143792290846100795786	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/11"]
"2-84-12.11-c5-0-54"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.38865565739166685	0	2.70511662558157729263095340783	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/59"]
"2-84-12.11-c5-0-55"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.4152992402967202	0	2.73276553917137687350129497892	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/51"]
"2-84-12.11-c5-0-56"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.3934428964258199	0	2.74087303708330031576024110564	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/47"]
"2-84-12.11-c5-0-57"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.2500610043873398	0	2.88963713775046128894864088308	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/35"]
"2-84-12.11-c5-0-58"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.33827823786988426	0	3.00654726918856521043454048047	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/37"]
"2-84-12.11-c5-0-59"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.3812244214284343	0	3.19062514448128978645712188869	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/41"]
"2-84-12.11-c5-0-6"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.4152992402967202	0	0.22941840313386989243602766071	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/52"]
"2-84-12.11-c5-0-7"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.33827823786988426	0	0.26542497633481520804244942822	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/38"]
"2-84-12.11-c5-0-8"	3.6704551304014568	13.472240864290375	2	84	"12.11"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.3346889118301095	0	0.34542001794852066956237734740	["ModularForm/GL2/Q/holomorphic/84/6/e/a/71/21"]
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"2-84-84.23-c5-0-71"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.4916564675615988	0	2.77458265945260099622104097197	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/48"]
"2-84-84.23-c5-0-72"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.28912242112330855	0	2.83579728547209729632945008530	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/7"]
"2-84-84.23-c5-0-73"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.4040491947853956	0	3.00738790500934155095295918695	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/62"]
"2-84-84.23-c5-0-74"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.4712687849093544	0	3.11623458808470546720510676118	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/68"]
"2-84-84.23-c5-0-75"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.3244547219361192	0	3.28752469206428075966586863034	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/44"]
"2-84-84.23-c5-0-8"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	0.3367693572351458	0	0.31261897101369439321288609314	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/56"]
"2-84-84.23-c5-0-9"	3.6704551304014568	13.472240864290375	2	84	"84.23"	[]	[[2.5, 0.0]]	5	true	true	false	false	-0.08481059454189672	0	0.34924174451267267609679764561	["ModularForm/GL2/Q/holomorphic/84/6/n/a/23/41"]


# 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).