# lfunc_search downloaded from the LMFDB on 15 June 2026.
# Search link: https://www.lmfdb.org/L/2/735/15.2
# Query "{'degree': 2, 'conductor': 735}" returned 12488 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.



"2-735-105.44-c0-0-0"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4282255948658145	0	0.802506660407218204182158918960	["ModularForm/GL2/Q/holomorphic/735/1/o/d/569/1"]
"2-735-105.44-c0-0-1"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14649371361272054	0	0.966931755699272803534671108655	["ModularForm/GL2/Q/holomorphic/735/1/o/a/569/1"]
"2-735-105.44-c0-0-2"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.051601452080239404	0	1.57770252888109236123167016501	["ModularForm/GL2/Q/holomorphic/735/1/o/d/569/2"]
"2-735-105.44-c0-0-3"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.051601452080239404	0	1.63469389138249823907523463923	["ModularForm/GL2/Q/holomorphic/735/1/o/c/569/1"]
"2-735-105.44-c0-0-4"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14649371361272054	0	1.68298445597528661425097146136	["ModularForm/GL2/Q/holomorphic/735/1/o/b/569/1"]
"2-735-105.44-c0-0-5"	0.6056507114544296	0.36681278428525677	2	735	"105.44"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.4282255948658145	0	2.63487817523132710027597905007	["ModularForm/GL2/Q/holomorphic/735/1/o/c/569/2"]
"2-735-105.74-c0-0-0"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.051601452080239404	0	0.857268764279335774323015240523	["ModularForm/GL2/Q/holomorphic/735/1/o/c/704/1"]
"2-735-105.74-c0-0-1"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4282255948658145	0	1.37810325116112342368004411780	["ModularForm/GL2/Q/holomorphic/735/1/o/c/704/2"]
"2-735-105.74-c0-0-2"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14649371361272054	0	1.91756844726631483147636055121	["ModularForm/GL2/Q/holomorphic/735/1/o/a/704/1"]
"2-735-105.74-c0-0-3"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14649371361272054	0	2.17800736411101080810521990340	["ModularForm/GL2/Q/holomorphic/735/1/o/b/704/1"]
"2-735-105.74-c0-0-4"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.051601452080239404	0	2.17816830613248210525107850738	["ModularForm/GL2/Q/holomorphic/735/1/o/d/704/2"]
"2-735-105.74-c0-0-5"	0.6056507114544296	0.36681278428525677	2	735	"105.74"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.4282255948658145	0	2.67164587120212222649836366818	["ModularForm/GL2/Q/holomorphic/735/1/o/d/704/1"]
"2-735-15.14-c0-0-0"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	true	true	0.0	0	0.946824080689985765932164937186	["ModularForm/GL2/Q/holomorphic/735/1/f/a/344/1", "ModularForm/GL2/Q/holomorphic/735/1/f/a", "ArtinRepresentation/2.735.4t3.a.a", "ArtinRepresentation/2.735.4t3.a"]
"2-735-15.14-c0-0-1"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.11613813786054256982614528162	["ModularForm/GL2/Q/holomorphic/735/1/f/c/344/1", "ArtinRepresentation/2.735.8t6.c.b"]
"2-735-15.14-c0-0-2"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.26452296747627704830958768689	["ModularForm/GL2/Q/holomorphic/735/1/f/d/344/1", "ArtinRepresentation/2.735.8t6.d.b"]
"2-735-15.14-c0-0-3"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	true	true	0.0	0	1.75562792641656088192317090538	["ModularForm/GL2/Q/holomorphic/735/1/f/b/344/1", "ModularForm/GL2/Q/holomorphic/735/1/f/b", "ArtinRepresentation/2.735.4t3.b.a", "ArtinRepresentation/2.735.4t3.b"]
"2-735-15.14-c0-0-4"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.82901295326634746793103000389	["ModularForm/GL2/Q/holomorphic/735/1/f/c/344/2", "ArtinRepresentation/2.735.8t6.c.a"]
"2-735-15.14-c0-0-5"	0.6056507114544296	0.36681278428525677	2	735	"15.14"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	2.27087558023038572895945376674	["ModularForm/GL2/Q/holomorphic/735/1/f/d/344/2", "ArtinRepresentation/2.735.8t6.d.a"]
"2-735-1.1-c1-0-0"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.66746462687447569978881815058	["EllipticCurve/Q/735/a", "ModularForm/GL2/Q/holomorphic/735/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/a"]
"2-735-1.1-c1-0-1"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.69748826024597739300293562557	["ModularForm/GL2/Q/holomorphic/735/2/a/l/1/1"]
"2-735-1.1-c1-0-10"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.11133814432095859866628281063	["ModularForm/GL2/Q/holomorphic/735/2/a/m/1/1"]
"2-735-1.1-c1-0-11"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.13876072567237849211933825140	["ModularForm/GL2/Q/holomorphic/735/2/a/k/1/1"]
"2-735-1.1-c1-0-12"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.20900895737299064147839775643	["ModularForm/GL2/Q/holomorphic/735/2/a/o/1/2"]
"2-735-1.1-c1-0-13"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.46386455255413963973625513463	["ModularForm/GL2/Q/holomorphic/735/2/a/i/1/1"]
"2-735-1.1-c1-0-14"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.52422388414270818193100808885	["ModularForm/GL2/Q/holomorphic/735/2/a/j/1/2"]
"2-735-1.1-c1-0-15"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.52483908804904209142058683363	["ModularForm/GL2/Q/holomorphic/735/2/a/l/1/2"]
"2-735-1.1-c1-0-16"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.58410936218206650239601507005	["ModularForm/GL2/Q/holomorphic/735/2/a/n/1/3"]
"2-735-1.1-c1-0-17"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.77352868017345184102614907117	["ModularForm/GL2/Q/holomorphic/735/2/a/h/1/1"]
"2-735-1.1-c1-0-18"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.79182185636844248975787007044	["ModularForm/GL2/Q/holomorphic/735/2/a/o/1/3"]
"2-735-1.1-c1-0-19"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.80163328686849374891334706368	["EllipticCurve/Q/735/d", "ModularForm/GL2/Q/holomorphic/735/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/d"]
"2-735-1.1-c1-0-2"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.887070391376747885592512794311	["ModularForm/GL2/Q/holomorphic/735/2/a/n/1/1"]
"2-735-1.1-c1-0-20"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.84339419921478037654642459241	["EllipticCurve/Q/735/c", "ModularForm/GL2/Q/holomorphic/735/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/c"]
"2-735-1.1-c1-0-21"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.85010008643067655414843348984	["ModularForm/GL2/Q/holomorphic/735/2/a/n/1/4"]
"2-735-1.1-c1-0-22"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.91305429033021291997620793361	["ModularForm/GL2/Q/holomorphic/735/2/a/k/1/2"]
"2-735-1.1-c1-0-23"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.94009500889479360979174607930	["EllipticCurve/Q/735/b", "ModularForm/GL2/Q/holomorphic/735/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/b"]
"2-735-1.1-c1-0-24"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	2.09963858328191631751420322491	["ModularForm/GL2/Q/holomorphic/735/2/a/m/1/2"]
"2-735-1.1-c1-0-25"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	2.12190859142618843532802223967	["ModularForm/GL2/Q/holomorphic/735/2/a/o/1/4"]
"2-735-1.1-c1-0-26"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	2.22463426679336255486529969906	["ModularForm/GL2/Q/holomorphic/735/2/a/i/1/2"]
"2-735-1.1-c1-0-27"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	2.38783044957083797331845206260	["ModularForm/GL2/Q/holomorphic/735/2/a/h/1/2"]
"2-735-1.1-c1-0-3"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.907314652004903767442835188152	["ModularForm/GL2/Q/holomorphic/735/2/a/g/1/1"]
"2-735-1.1-c1-0-4"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.956888202047531280357347023545	["ModularForm/GL2/Q/holomorphic/735/2/a/o/1/1"]
"2-735-1.1-c1-0-5"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.982469269817440515290174005672	["EllipticCurve/Q/735/f", "ModularForm/GL2/Q/holomorphic/735/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/f"]
"2-735-1.1-c1-0-6"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.02739748126678866703037333369	["ModularForm/GL2/Q/holomorphic/735/2/a/g/1/2"]
"2-735-1.1-c1-0-7"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.02887285682465642704345923603	["ModularForm/GL2/Q/holomorphic/735/2/a/n/1/2"]
"2-735-1.1-c1-0-8"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.04599175459547455544259640973	["EllipticCurve/Q/735/e", "ModularForm/GL2/Q/holomorphic/735/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/735/2/a/e"]
"2-735-1.1-c1-0-9"	2.422602845817719	5.869004548564111	2	735	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.10861725756689101543200615962	["ModularForm/GL2/Q/holomorphic/735/2/a/j/1/1"]
"2-735-105.104-c1-0-0"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.26965238551322857	0	0.04537522114221957613950939292	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/11"]
"2-735-105.104-c1-0-1"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3755959463151488	0	0.087111637597271660231347698180	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/17"]
"2-735-105.104-c1-0-10"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14280167663252147	0	0.39355410547069535898582235007	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/13"]
"2-735-105.104-c1-0-11"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.370635464942928	0	0.50253549698247450359974822406	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/6"]
"2-735-105.104-c1-0-12"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.27501990821889444	0	0.52172713841924378648544325467	["ModularForm/GL2/Q/holomorphic/735/2/g/a/734/8"]
"2-735-105.104-c1-0-13"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.43292034785225986	0	0.57302287658251904550117412076	["ModularForm/GL2/Q/holomorphic/735/2/g/a/734/12"]
"2-735-105.104-c1-0-14"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.035626950084510216	0	0.58413804265747357271270940679	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/5"]
"2-735-105.104-c1-0-15"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.22106842234237734	0	0.61500075022244321586752382916	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/1"]
"2-735-105.104-c1-0-16"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2443787397851435	0	0.61759341009963598026894307441	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/16"]
"2-735-105.104-c1-0-17"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0670796521477402	0	0.62137125781132433395461021834	["ModularForm/GL2/Q/holomorphic/735/2/g/a/734/1"]
"2-735-105.104-c1-0-18"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.28956655164585676	0	0.62879886159756260974299140087	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/24"]
"2-735-105.104-c1-0-19"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.06817991606375039	0	0.64080166616684553540964195544	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/18"]
"2-735-105.104-c1-0-2"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.22498009178110556	0	0.12507660068975628630038972586	["ModularForm/GL2/Q/holomorphic/735/2/g/a/734/3"]
"2-735-105.104-c1-0-20"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3699872024610265	0	0.65282474472271208398855588408	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/16"]
"2-735-105.104-c1-0-21"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06817991606375039	0	0.74795180610306666105467322689	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/17"]
"2-735-105.104-c1-0-22"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.26281247591301526	0	0.76182169767505530256928210376	["ModularForm/GL2/Q/holomorphic/735/2/g/b/734/8"]
"2-735-105.104-c1-0-23"	2.422602845817719	5.869004548564111	2	735	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16039363191635433	0	0.78071212717355042395536443366	["ModularForm/GL2/Q/holomorphic/735/2/g/c/734/22"]
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"2-735-1.1-c5-0-95"	10.857352695871162	117.8821075625408	2	735	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.15451763564934123591370816021	["ModularForm/GL2/Q/holomorphic/735/6/a/k/1/2"]
"2-735-1.1-c5-0-96"	10.857352695871162	117.8821075625408	2	735	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.15911183366406483698873278023	["ModularForm/GL2/Q/holomorphic/735/6/a/z/1/12"]
"2-735-1.1-c5-0-97"	10.857352695871162	117.8821075625408	2	735	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.17353268748097569704128628241	["ModularForm/GL2/Q/holomorphic/735/6/a/f/1/2"]
"2-735-1.1-c5-0-98"	10.857352695871162	117.8821075625408	2	735	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.17457848142274655018342721365	["ModularForm/GL2/Q/holomorphic/735/6/a/y/1/3"]
"2-735-1.1-c5-0-99"	10.857352695871162	117.8821075625408	2	735	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.17905896246035643103877532731	["ModularForm/GL2/Q/holomorphic/735/6/a/d/1/1"]


# Label --
#    Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where

#     * $d$ is the degree of $L$.
#     * $N$ is the conductor of $L$.  When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions.
#     * q.k is the label of the primitive Dirichlet character from which the central character is induced.
#     * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation.
#     * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label.


#$\alpha$ (root_analytic_conductor) --
#    If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$.  It plays a role analogous to the root discriminant for number fields.


#$A$ (analytic_conductor) --
#    The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number
#    \[
#    A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N.
#    \]



#$d$ (degree) --
#    The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation

#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
#    \]

#    The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor.



#$N$ (conductor) --
#    The **conductor** of an L-function is the integer $N$  occurring in its functional equation

#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s).
#    \]


#    The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function
#     associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$.

#    In the literature, the word _level_ is sometimes used instead of _conductor_.


#$\chi$ (central_character) --
#    An L-function has an Euler product of the form
#    $L(s) = \prod_p L_p(p^{-s})^{-1}$
#    where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function.
#    Here, $N$ is the conductor of $L$.


#$\mu$ (mus) --
#    All known analytic L-functions have a **functional equation** that can be written in the form
#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
#    \]
#    where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
#    or half-integer,
#    \[
#    \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
#    \]
#    and $\varepsilon$ is the sign of the functional equation.
#    With those restrictions on the spectral parameters, the
#    data in the functional equation is specified uniquely.  The integer $d = J + 2 K$
#    is the degree of the L-function. The integer $N$ is  the conductor (or level)
#    of the L-function.  The pair $[J,K]$ is the signature of the L-function.  The parameters
#    in the functional equation can be used to make up the 4-tuple called the Selberg data.


#    The axioms of the Selberg class are less restrictive than
#    given above.

#    Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

#    For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
#    called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.



#$\nu$ (nus) --
#    All known analytic L-functions have a **functional equation** that can be written in the form
#    \[
#    \Lambda(s) := N^{s/2}
#    \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k)
#    \cdot L(s) = \varepsilon \overline{\Lambda}(1-s),
#    \]
#    where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer
#    or half-integer,
#    \[
#    \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real},
#    \]
#    and $\varepsilon$ is the sign of the functional equation.
#    With those restrictions on the spectral parameters, the
#    data in the functional equation is specified uniquely.  The integer $d = J + 2 K$
#    is the degree of the L-function. The integer $N$ is  the conductor (or level)
#    of the L-function.  The pair $[J,K]$ is the signature of the L-function.  The parameters
#    in the functional equation can be used to make up the 4-tuple called the Selberg data.


#    The axioms of the Selberg class are less restrictive than
#    given above.

#    Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$.

#    For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$,
#    called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers.



#$w$ (motivic_weight) --
#    The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$.

#    If the L-function arises from a motive, then the weight of the motive has the
#    same parity as the motivic weight of the L-function, but the weight of the motive
#    could be larger.  This apparent discrepancy comes from the fact that a Tate twist
#    increases the weight of the motive.  This corresponds to the change of variables
#    $s \mapsto s + j$ in the L-function of the motive.


#prim (primitive) --
#    An L-function is <b>primitive</b> if it cannot be written as a product of nontrivial L-functions.  The "trivial L-function" is the constant function $1$.


#arith (algebraic) --
#    An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$  is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers.


#$\mathbb{Q}$ (rational) --
#    A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes
#    \[
#    L(s)=\prod_pL_p(p^{-s})^{-1}
#    \]
#    with $L_p\in \Z[T]$.


#self-dual (self_dual) --
#    An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real.


#$\operatorname{Arg}(\epsilon)$ (root_angle) --
#    The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$.


#$r$ (order_of_vanishing) --
#    The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point.

#    When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point).


#First zero (z1) --
#    The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$.

#    Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization).

#    The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number.


#Origin (instance_urls) --
#    L-functions arise from many different sources. Already in degree 2 we have examples of
#    L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions).

#    Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).