# lfunc_search downloaded from the LMFDB on 02 May 2026.
# Search link: https://www.lmfdb.org/L/2/1014/1.1
# Query "{'degree': 2, 'conductor': 1014}" returned 637 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.



"2-1014-1.1-c1-0-0"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.37852683374557655394310314207	["ModularForm/GL2/Q/holomorphic/1014/2/a/l/1/1"]
"2-1014-1.1-c1-0-1"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.62725804158590284016844894758	["ModularForm/GL2/Q/holomorphic/1014/2/a/l/1/2"]
"2-1014-1.1-c1-0-10"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.27432969291731687860536210805	["EllipticCurve/Q/1014/e", "ModularForm/GL2/Q/holomorphic/1014/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/e"]
"2-1014-1.1-c1-0-11"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.34470200992116687103532596200	["ModularForm/GL2/Q/holomorphic/1014/2/a/m/1/3"]
"2-1014-1.1-c1-0-12"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.34806012139143938612331236626	["ModularForm/GL2/Q/holomorphic/1014/2/a/n/1/2"]
"2-1014-1.1-c1-0-13"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.44507304698032299892028334665	["ModularForm/GL2/Q/holomorphic/1014/2/a/o/1/3"]
"2-1014-1.1-c1-0-14"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.50614799649470212666837133027	["EllipticCurve/Q/1014/a", "ModularForm/GL2/Q/holomorphic/1014/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/a"]
"2-1014-1.1-c1-0-15"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.50831117536013935148566119100	["EllipticCurve/Q/1014/c", "ModularForm/GL2/Q/holomorphic/1014/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/c"]
"2-1014-1.1-c1-0-16"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.52147856856187922875477354307	["ModularForm/GL2/Q/holomorphic/1014/2/a/k/1/1"]
"2-1014-1.1-c1-0-17"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.60127985088511496131363056257	["ModularForm/GL2/Q/holomorphic/1014/2/a/n/1/3"]
"2-1014-1.1-c1-0-18"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.61111813697039943435414343039	["ModularForm/GL2/Q/holomorphic/1014/2/a/o/1/2"]
"2-1014-1.1-c1-0-19"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.62728933419036740446353668829	["ModularForm/GL2/Q/holomorphic/1014/2/a/k/1/2"]
"2-1014-1.1-c1-0-2"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.925337255659619455615447351878	["ModularForm/GL2/Q/holomorphic/1014/2/a/m/1/2"]
"2-1014-1.1-c1-0-20"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.69460491308532215457543315446	["EllipticCurve/Q/1014/g", "ModularForm/GL2/Q/holomorphic/1014/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/g"]
"2-1014-1.1-c1-0-21"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.76712137722831607625576187625	["EllipticCurve/Q/1014/b", "ModularForm/GL2/Q/holomorphic/1014/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/b"]
"2-1014-1.1-c1-0-22"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.77945664350906640093555396519	["ModularForm/GL2/Q/holomorphic/1014/2/a/i/1/1"]
"2-1014-1.1-c1-0-23"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.95201350757735839940759885672	["ModularForm/GL2/Q/holomorphic/1014/2/a/i/1/2"]
"2-1014-1.1-c1-0-24"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.99104777947687867489605592496	["ModularForm/GL2/Q/holomorphic/1014/2/a/j/1/1"]
"2-1014-1.1-c1-0-25"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	2.19968356715018438714693743240	["ModularForm/GL2/Q/holomorphic/1014/2/a/j/1/2"]
"2-1014-1.1-c1-0-26"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	2.39404286061527545470700387762	["EllipticCurve/Q/1014/f", "ModularForm/GL2/Q/holomorphic/1014/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/f"]
"2-1014-1.1-c1-0-3"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.938609277818426319051328481758	["EllipticCurve/Q/1014/d", "ModularForm/GL2/Q/holomorphic/1014/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/1014/2/a/d"]
"2-1014-1.1-c1-0-4"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.974560627459520518158610227161	["ModularForm/GL2/Q/holomorphic/1014/2/a/h/1/1"]
"2-1014-1.1-c1-0-5"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.988857246942512103753552122256	["ModularForm/GL2/Q/holomorphic/1014/2/a/l/1/3"]
"2-1014-1.1-c1-0-6"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.00170322081288417021166830514	["ModularForm/GL2/Q/holomorphic/1014/2/a/n/1/1"]
"2-1014-1.1-c1-0-7"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.05273906250767299757696084873	["ModularForm/GL2/Q/holomorphic/1014/2/a/h/1/2"]
"2-1014-1.1-c1-0-8"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.08457719255007470071424000931	["ModularForm/GL2/Q/holomorphic/1014/2/a/m/1/1"]
"2-1014-1.1-c1-0-9"	2.845493061836179	8.096830764957835	2	1014	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.18236672390134377279201613725	["ModularForm/GL2/Q/holomorphic/1014/2/a/o/1/1"]
"2-1014-13.10-c1-0-0"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4698380875926084	0	0.01406836951807021542236876520	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/1"]
"2-1014-13.10-c1-0-1"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.30001833630068275	0	0.092556461876884014049368779289	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/3"]
"2-1014-13.10-c1-0-10"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.04064619097038332	0	1.08959055642862441260669015721	["ModularForm/GL2/Q/holomorphic/1014/2/i/c/361/2"]
"2-1014-13.10-c1-0-11"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2582092361795994	0	1.14810231307952175467939165403	["ModularForm/GL2/Q/holomorphic/1014/2/i/d/361/1"]
"2-1014-13.10-c1-0-12"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.16673463452564002	0	1.18188503948486696901808433231	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/1"]
"2-1014-13.10-c1-0-13"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.042687142362950015	0	1.23580470480409717011374851339	["ModularForm/GL2/Q/holomorphic/1014/2/i/e/361/2"]
"2-1014-13.10-c1-0-14"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0967366055307584	0	1.23765240517610202480901393710	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/5"]
"2-1014-13.10-c1-0-15"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.20731285763704999	0	1.30135589303490913090778949103	["ModularForm/GL2/Q/holomorphic/1014/2/i/a/361/1"]
"2-1014-13.10-c1-0-16"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0967366055307584	0	1.42373208103121375824904366364	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/2"]
"2-1014-13.10-c1-0-17"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.30001833630068275	0	1.64917370349581000121612116177	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/6"]
"2-1014-13.10-c1-0-18"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.20731285763704999	0	1.71174666854277450570018856280	["ModularForm/GL2/Q/holomorphic/1014/2/i/f/361/2"]
"2-1014-13.10-c1-0-19"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.16673463452564002	0	1.74002214049556484415371017664	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/4"]
"2-1014-13.10-c1-0-2"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.45731285763705	0	0.17037659392727290547830048564	["ModularForm/GL2/Q/holomorphic/1014/2/i/b/361/1"]
"2-1014-13.10-c1-0-20"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4593538090296167	0	1.98203221996889590288056288733	["ModularForm/GL2/Q/holomorphic/1014/2/i/c/361/1"]
"2-1014-13.10-c1-0-21"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.45731285763705	0	2.19862481020671240957697544092	["ModularForm/GL2/Q/holomorphic/1014/2/i/b/361/2"]
"2-1014-13.10-c1-0-22"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4289578056313994	0	2.31512185939463015109385080253	["ModularForm/GL2/Q/holomorphic/1014/2/i/d/361/2"]
"2-1014-13.10-c1-0-23"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4698380875926084	0	2.51169858205239907591643534146	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/4"]
"2-1014-13.10-c1-0-3"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2419942794792771	0	0.28378706893365250370537829771	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/2"]
"2-1014-13.10-c1-0-4"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.29268714236295	0	0.30343879879023789325653856103	["ModularForm/GL2/Q/holomorphic/1014/2/i/a/361/2"]
"2-1014-13.10-c1-0-5"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.29268714236295	0	0.69846851254721925888064148043	["ModularForm/GL2/Q/holomorphic/1014/2/i/f/361/1"]
"2-1014-13.10-c1-0-6"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2419942794792771	0	0.70315360204804764395148771796	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/5"]
"2-1014-13.10-c1-0-7"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.184990542618529	0	0.855631390080655018313249176957	["ModularForm/GL2/Q/holomorphic/1014/2/i/g/361/3"]
"2-1014-13.10-c1-0-8"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.042687142362950015	0	1.00442739907000154704838934747	["ModularForm/GL2/Q/holomorphic/1014/2/i/e/361/1"]
"2-1014-13.10-c1-0-9"	2.845493061836179	8.096830764957835	2	1014	"13.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.184990542618529	0	1.06774341750737866279433902259	["ModularForm/GL2/Q/holomorphic/1014/2/i/h/361/6"]
"2-1014-13.12-c1-0-0"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3784310658795788	0	0.41626363760206870126841240013	["ModularForm/GL2/Q/holomorphic/1014/2/b/f/337/3"]
"2-1014-13.12-c1-0-1"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3435835209054994	0	0.50529593574935926507859947205	["ModularForm/GL2/Q/holomorphic/1014/2/b/b/337/2"]
"2-1014-13.12-c1-0-10"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.04472809375551671	0	0.974084463606856009848919241523	["ModularForm/GL2/Q/holomorphic/1014/2/b/e/337/2"]
"2-1014-13.12-c1-0-11"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1564164790945006	0	1.00201846058436211373320105939	["ModularForm/GL2/Q/holomorphic/1014/2/b/c/337/2"]
"2-1014-13.12-c1-0-12"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2451473641045361	0	1.09635942724756829502277685855	["ModularForm/GL2/Q/holomorphic/1014/2/b/f/337/1"]
"2-1014-13.12-c1-0-13"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.03970018086954331	0	1.15204312463816871785117229262	["ModularForm/GL2/Q/holomorphic/1014/2/b/f/337/5"]
"2-1014-13.12-c1-0-14"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.04472809375551671	0	1.15608992486730674946899329763	["ModularForm/GL2/Q/holomorphic/1014/2/b/d/337/3"]
"2-1014-13.12-c1-0-15"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1564164790945006	0	1.30260739923111111729756136444	["ModularForm/GL2/Q/holomorphic/1014/2/b/c/337/1"]
"2-1014-13.12-c1-0-16"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3435835209054994	0	1.32191087055635942460036644758	["ModularForm/GL2/Q/holomorphic/1014/2/b/b/337/1"]
"2-1014-13.12-c1-0-17"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04472809375551671	0	1.34206699551082744999465520542	["ModularForm/GL2/Q/holomorphic/1014/2/b/e/337/3"]
"2-1014-13.12-c1-0-18"	2.845493061836179	8.096830764957835	2	1014	"13.12"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.03970018086954331	0	1.51694862897657176102182033136	["ModularForm/GL2/Q/holomorphic/1014/2/b/g/337/5"]
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"2-1014-1.1-c5-0-92"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.24845888010119562271476414111	["ModularForm/GL2/Q/holomorphic/1014/6/a/z/1/5"]
"2-1014-1.1-c5-0-93"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.25567079050312794113365160993	["ModularForm/GL2/Q/holomorphic/1014/6/a/u/1/4"]
"2-1014-1.1-c5-0-94"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.26153837704083217837464483706	["ModularForm/GL2/Q/holomorphic/1014/6/a/y/1/5"]
"2-1014-1.1-c5-0-95"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.0	0	1.26435160915707410398622184785	["ModularForm/GL2/Q/holomorphic/1014/6/a/bg/1/6"]
"2-1014-1.1-c5-0-96"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.27974221343129835220555716004	["ModularForm/GL2/Q/holomorphic/1014/6/a/y/1/4"]
"2-1014-1.1-c5-0-97"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.28264951120351698316990361926	["ModularForm/GL2/Q/holomorphic/1014/6/a/ba/1/1"]
"2-1014-1.1-c5-0-98"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	true	true	0.5	1	1.28747448415357906906886097374	["ModularForm/GL2/Q/holomorphic/1014/6/a/b/1/1", "ModularForm/GL2/Q/holomorphic/1014/6/a/b"]
"2-1014-1.1-c5-0-99"	12.752615154953999	162.6291932903624	2	1014	"1.1"	[]	[[2.5, 0.0]]	5	true	true	false	true	0.5	1	1.29151337710254804684788849534	["ModularForm/GL2/Q/holomorphic/1014/6/a/q/1/3"]


# 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).