# lfunc_search downloaded from the LMFDB on 26 May 2026.
# Search link: https://www.lmfdb.org/L/2/2070/1.1
# Query "{'degree': 2, 'conductor': 2070}" returned 696 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.



"2-2070-1.1-c1-0-0"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.43167941363320819501824277025	["ModularForm/GL2/Q/holomorphic/2070/2/a/t/1/1"]
"2-2070-1.1-c1-0-1"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.52559370077179103750714970746	["EllipticCurve/Q/2070/a", "ModularForm/GL2/Q/holomorphic/2070/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/a"]
"2-2070-1.1-c1-0-10"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.980755982806687577424829286357	["ModularForm/GL2/Q/holomorphic/2070/2/a/t/1/2"]
"2-2070-1.1-c1-0-11"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.06854396603220431591782468674	["ModularForm/GL2/Q/holomorphic/2070/2/a/w/1/1"]
"2-2070-1.1-c1-0-12"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.13950792951787350963064151591	["ModularForm/GL2/Q/holomorphic/2070/2/a/x/1/1"]
"2-2070-1.1-c1-0-13"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.15508694969457450243039729594	["EllipticCurve/Q/2070/n", "ModularForm/GL2/Q/holomorphic/2070/2/a/n/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/n"]
"2-2070-1.1-c1-0-14"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.16008839814667021142299015679	["ModularForm/GL2/Q/holomorphic/2070/2/a/y/1/1"]
"2-2070-1.1-c1-0-15"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.17351616216495372278974762865	["EllipticCurve/Q/2070/r", "ModularForm/GL2/Q/holomorphic/2070/2/a/r/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/r"]
"2-2070-1.1-c1-0-16"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.17981764933870209034850387785	["ModularForm/GL2/Q/holomorphic/2070/2/a/w/1/2"]
"2-2070-1.1-c1-0-17"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.18356153605454565909199857789	["EllipticCurve/Q/2070/o", "ModularForm/GL2/Q/holomorphic/2070/2/a/o/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/o"]
"2-2070-1.1-c1-0-18"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.22508210605556772251166314152	["ModularForm/GL2/Q/holomorphic/2070/2/a/x/1/2"]
"2-2070-1.1-c1-0-19"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.29526058308459081231307607058	["ModularForm/GL2/Q/holomorphic/2070/2/a/z/1/2"]
"2-2070-1.1-c1-0-2"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.60212299356919442244351812269	["ModularForm/GL2/Q/holomorphic/2070/2/a/z/1/1"]
"2-2070-1.1-c1-0-20"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.34409171825676144708222230132	["EllipticCurve/Q/2070/s", "ModularForm/GL2/Q/holomorphic/2070/2/a/s/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/s"]
"2-2070-1.1-c1-0-21"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.39043206902009212889585974407	["ModularForm/GL2/Q/holomorphic/2070/2/a/y/1/2"]
"2-2070-1.1-c1-0-22"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.41543415021040645255010595248	["EllipticCurve/Q/2070/b", "ModularForm/GL2/Q/holomorphic/2070/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/b"]
"2-2070-1.1-c1-0-23"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.42118865869046494644552258938	["ModularForm/GL2/Q/holomorphic/2070/2/a/v/1/1"]
"2-2070-1.1-c1-0-24"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.45402989302954842587531760308	["EllipticCurve/Q/2070/d", "ModularForm/GL2/Q/holomorphic/2070/2/a/d/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/d"]
"2-2070-1.1-c1-0-25"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.49181649525044645554898011447	["EllipticCurve/Q/2070/c", "ModularForm/GL2/Q/holomorphic/2070/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/c"]
"2-2070-1.1-c1-0-26"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.53167294603489718963966945293	["EllipticCurve/Q/2070/e", "ModularForm/GL2/Q/holomorphic/2070/2/a/e/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/e"]
"2-2070-1.1-c1-0-27"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.60335186074833469102358224886	["EllipticCurve/Q/2070/g", "ModularForm/GL2/Q/holomorphic/2070/2/a/g/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/g"]
"2-2070-1.1-c1-0-28"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.64120476496527299640973131628	["ModularForm/GL2/Q/holomorphic/2070/2/a/v/1/2"]
"2-2070-1.1-c1-0-29"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.66257305158845747230756766609	["EllipticCurve/Q/2070/i", "ModularForm/GL2/Q/holomorphic/2070/2/a/i/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/i"]
"2-2070-1.1-c1-0-3"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.67445417495635941704012673035	["ModularForm/GL2/Q/holomorphic/2070/2/a/u/1/1"]
"2-2070-1.1-c1-0-30"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.85164441327250887995307415846	["EllipticCurve/Q/2070/l", "ModularForm/GL2/Q/holomorphic/2070/2/a/l/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/l"]
"2-2070-1.1-c1-0-31"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.90373541610352756315158154245	["EllipticCurve/Q/2070/m", "ModularForm/GL2/Q/holomorphic/2070/2/a/m/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/m"]
"2-2070-1.1-c1-0-32"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	1.96283676978706544616935979795	["EllipticCurve/Q/2070/p", "ModularForm/GL2/Q/holomorphic/2070/2/a/p/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/p"]
"2-2070-1.1-c1-0-33"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	2.01789084946706752403284220094	["EllipticCurve/Q/2070/q", "ModularForm/GL2/Q/holomorphic/2070/2/a/q/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/q"]
"2-2070-1.1-c1-0-4"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.803817388064460988594638353418	["ModularForm/GL2/Q/holomorphic/2070/2/a/u/1/2"]
"2-2070-1.1-c1-0-5"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.889405566153390861191680356079	["EllipticCurve/Q/2070/f", "ModularForm/GL2/Q/holomorphic/2070/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/f"]
"2-2070-1.1-c1-0-6"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.894778098781870952960807550841	["EllipticCurve/Q/2070/j", "ModularForm/GL2/Q/holomorphic/2070/2/a/j/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/j"]
"2-2070-1.1-c1-0-7"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.917064197038992630100605264992	["EllipticCurve/Q/2070/k", "ModularForm/GL2/Q/holomorphic/2070/2/a/k/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/k"]
"2-2070-1.1-c1-0-8"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	0.935045408651431349885382411049	["EllipticCurve/Q/2070/h", "ModularForm/GL2/Q/holomorphic/2070/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/2070/2/a/h"]
"2-2070-1.1-c1-0-9"	4.065591373761639	16.529033218405047	2	2070	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.946701776765772005968830684738	["ModularForm/GL2/Q/holomorphic/2070/2/a/z/1/3"]
"2-2070-15.2-c1-0-0"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.33434352662085404	0	0.12663989805024893979698020981	["ModularForm/GL2/Q/holomorphic/2070/2/j/j/737/9"]
"2-2070-15.2-c1-0-1"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.26464681792354616	0	0.27823752113009139473362365561	["ModularForm/GL2/Q/holomorphic/2070/2/j/g/737/2"]
"2-2070-15.2-c1-0-10"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08461187771772678	0	0.40350981780533589398634918270	["ModularForm/GL2/Q/holomorphic/2070/2/j/j/737/6"]
"2-2070-15.2-c1-0-11"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13485254242026015	0	0.52350747622087764878506862670	["ModularForm/GL2/Q/holomorphic/2070/2/j/d/737/1"]
"2-2070-15.2-c1-0-12"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3893728247626867	0	0.53144510573403538970634572867	["ModularForm/GL2/Q/holomorphic/2070/2/j/i/737/4"]
"2-2070-15.2-c1-0-13"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.23119467956139372	0	0.59038053212991330761273030176	["ModularForm/GL2/Q/holomorphic/2070/2/j/j/737/7"]
"2-2070-15.2-c1-0-14"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13485254242026015	0	0.60679785617976926785528566512	["ModularForm/GL2/Q/holomorphic/2070/2/j/c/737/2"]
"2-2070-15.2-c1-0-15"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.22399503637581633	0	0.60977528581206708844911088781	["ModularForm/GL2/Q/holomorphic/2070/2/j/i/737/2"]
"2-2070-15.2-c1-0-16"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05003633027231525	0	0.72641794703511211925818299644	["ModularForm/GL2/Q/holomorphic/2070/2/j/h/737/1"]
"2-2070-15.2-c1-0-17"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06106073359504349	0	0.826572020459474788265074205898	["ModularForm/GL2/Q/holomorphic/2070/2/j/c/737/1"]
"2-2070-15.2-c1-0-18"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06106073359504349	0	0.834955379030803557616010833786	["ModularForm/GL2/Q/holomorphic/2070/2/j/e/737/2"]
"2-2070-15.2-c1-0-19"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.22364858136582624	0	0.974021941116558616487650723813	["ModularForm/GL2/Q/holomorphic/2070/2/j/j/737/1"]
"2-2070-15.2-c1-0-2"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.211443042800263	0	0.27892894177034214238214480799	["ModularForm/GL2/Q/holomorphic/2070/2/j/h/737/7"]
"2-2070-15.2-c1-0-20"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.160652572919781	0	0.989390899491985626247016631289	["ModularForm/GL2/Q/holomorphic/2070/2/j/i/737/6"]
"2-2070-15.2-c1-0-21"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04728471178127798	0	1.02585947634109265779837056491	["ModularForm/GL2/Q/holomorphic/2070/2/j/i/737/8"]
"2-2070-15.2-c1-0-22"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06106073359504349	0	1.02986406041960055157200638413	["ModularForm/GL2/Q/holomorphic/2070/2/j/d/737/2"]
"2-2070-15.2-c1-0-23"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13485254242026015	0	1.04714976876206036284720017766	["ModularForm/GL2/Q/holomorphic/2070/2/j/e/737/1"]
"2-2070-15.2-c1-0-24"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0639392664049565	0	1.10643053615157827605125065038	["ModularForm/GL2/Q/holomorphic/2070/2/j/f/737/1"]
"2-2070-15.2-c1-0-25"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.10388659028335903	0	1.14842543064496454712299257126	["ModularForm/GL2/Q/holomorphic/2070/2/j/g/737/3"]
"2-2070-15.2-c1-0-26"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.11514745757973988	0	1.18013406386781756225502901628	["ModularForm/GL2/Q/holomorphic/2070/2/j/b/737/1"]
"2-2070-15.2-c1-0-27"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1351458537796251	0	1.22389297170405236484680033856	["ModularForm/GL2/Q/holomorphic/2070/2/j/j/737/3"]
"2-2070-15.2-c1-0-28"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.14862856423402568	0	1.23355146312957017788571298370	["ModularForm/GL2/Q/holomorphic/2070/2/j/i/737/1"]
"2-2070-15.2-c1-0-29"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1458769457429884	0	1.27235833311238239284524974951	["ModularForm/GL2/Q/holomorphic/2070/2/j/h/737/8"]
"2-2070-15.2-c1-0-3"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4360607335950435	0	0.30095788340841676810677763930	["ModularForm/GL2/Q/holomorphic/2070/2/j/a/737/1"]
"2-2070-15.2-c1-0-30"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3110607335950435	0	1.38077067339678394721222102735	["ModularForm/GL2/Q/holomorphic/2070/2/j/b/737/2"]
"2-2070-15.2-c1-0-31"	4.065591373761639	16.529033218405047	2	2070	"15.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.44020897031298717	0	1.42696148927403047481731374953	["ModularForm/GL2/Q/holomorphic/2070/2/j/h/737/5"]
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"2-2070-69.68-c3-0-86"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.13368914045579466	0	1.19091534134188355019040920246	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/48"]
"2-2070-69.68-c3-0-87"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4046584491858614	0	1.19177568687493651135348639249	["ModularForm/GL2/Q/holomorphic/2070/4/e/a/1241/7"]
"2-2070-69.68-c3-0-88"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.471209188305186	0	1.22031150391639218766348505203	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/2"]
"2-2070-69.68-c3-0-89"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.30848922903230924	0	1.25801530967001109618917116914	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/33"]
"2-2070-69.68-c3-0-9"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.30848922903230924	0	0.15890025932612836729764934534	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/16"]
"2-2070-69.68-c3-0-90"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.4996347683741594	0	1.27128928285846214548264393726	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/8"]
"2-2070-69.68-c3-0-91"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.4713370013339156	0	1.31070354727979226004076480952	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/3"]
"2-2070-69.68-c3-0-92"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	-0.33017241145673737	0	1.32879382185001090569934121058	["ModularForm/GL2/Q/holomorphic/2070/4/e/a/1241/3"]
"2-2070-69.68-c3-0-93"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.4375388550320537	0	1.33077011013100103347215174460	["ModularForm/GL2/Q/holomorphic/2070/4/e/a/1241/29"]
"2-2070-69.68-c3-0-94"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.27850981016833687	0	1.33834403443254050288983177478	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/26"]
"2-2070-69.68-c3-0-95"	11.051423153236108	122.13395371188312	2	2070	"69.68"	[]	[[1.5, 0.0]]	3	true	true	false	false	0.399428274798835	0	1.44938414279826077679084409420	["ModularForm/GL2/Q/holomorphic/2070/4/e/b/1241/31"]


# 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).