# lfunc_search downloaded from the LMFDB on 16 April 2026.
# Search link: https://www.lmfdb.org/L/2/77/11.5
# Query "{'degree': 2, 'conductor': 77}" returned 15285 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.



"2-77-77.20-c0-0-0"	0.1960306250888906	0.03842800597274119	2	77	"77.20"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.25728286993039323	0	2.35278045266079068339323690453	["ModularForm/GL2/Q/holomorphic/77/1/j/a/20/1"]
"2-77-77.27-c0-0-0"	0.1960306250888906	0.03842800597274119	2	77	"77.27"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.25728286993039323	0	4.11400874670323536123765252069	["ModularForm/GL2/Q/holomorphic/77/1/j/a/27/1"]
"2-77-77.48-c0-0-0"	0.1960306250888906	0.03842800597274119	2	77	"77.48"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.09621626289782827	0	3.35405452155751025316205560791	["ModularForm/GL2/Q/holomorphic/77/1/j/a/48/1"]
"2-77-77.69-c0-0-0"	0.1960306250888906	0.03842800597274119	2	77	"77.69"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.09621626289782827	0	2.48329704472032446623488429724	["ModularForm/GL2/Q/holomorphic/77/1/j/a/69/1"]
"2-77-1.1-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	2.16581206833890556742345562501	["ModularForm/GL2/Q/holomorphic/77/2/a/d/1/1"]
"2-77-1.1-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	2.52489627324288051568453078479	["EllipticCurve/Q/77/b", "ModularForm/GL2/Q/holomorphic/77/2/a/b/1/1", "ModularForm/GL2/Q/holomorphic/77/2/a/b"]
"2-77-1.1-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	3.26888344602981049454952325554	["EllipticCurve/Q/77/c", "ModularForm/GL2/Q/holomorphic/77/2/a/c/1/1", "ModularForm/GL2/Q/holomorphic/77/2/a/c"]
"2-77-1.1-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	3.28451169009027979961022861738	["ModularForm/GL2/Q/holomorphic/77/2/a/d/1/2"]
"2-77-1.1-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.5	1	4.26020787258843561459488122037	["EllipticCurve/Q/77/a", "ModularForm/GL2/Q/holomorphic/77/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/77/2/a/a"]
"2-77-11.3-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3781130052008296	0	1.38230635777839221939653233589	["ModularForm/GL2/Q/holomorphic/77/2/f/b/36/1"]
"2-77-11.3-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.11758125105665418	0	1.66189493881005694540507173399	["ModularForm/GL2/Q/holomorphic/77/2/f/a/36/1"]
"2-77-11.3-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.09481808209226987	0	1.94291250384609553130849150420	["ModularForm/GL2/Q/holomorphic/77/2/f/b/36/3"]
"2-77-11.3-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.02570751786100452	0	2.96459395047780766182849500593	["ModularForm/GL2/Q/holomorphic/77/2/f/b/36/4"]
"2-77-11.3-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3082901764646599	0	3.49036956941392011556591088240	["ModularForm/GL2/Q/holomorphic/77/2/f/b/36/2"]
"2-77-11.3-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"11.3"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17108038388487568	0	3.77869344625928192220878576802	["ModularForm/GL2/Q/holomorphic/77/2/f/a/36/2"]
"2-77-11.4-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3082901764646599	0	0.75956679774041450742393153840	["ModularForm/GL2/Q/holomorphic/77/2/f/b/15/2"]
"2-77-11.4-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.11758125105665418	0	1.83225717415857255392004335705	["ModularForm/GL2/Q/holomorphic/77/2/f/a/15/1"]
"2-77-11.4-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17108038388487568	0	2.31691111158093558122197604300	["ModularForm/GL2/Q/holomorphic/77/2/f/a/15/2"]
"2-77-11.4-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.09481808209226987	0	2.81582577645182690231715345674	["ModularForm/GL2/Q/holomorphic/77/2/f/b/15/3"]
"2-77-11.4-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.02570751786100452	0	3.45841311397960636141667034005	["ModularForm/GL2/Q/holomorphic/77/2/f/b/15/4"]
"2-77-11.4-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"11.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3781130052008296	0	3.61528827076840361165939371500	["ModularForm/GL2/Q/holomorphic/77/2/f/b/15/1"]
"2-77-11.5-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.23643920891201048	0	2.00084791067807242737199077908	["ModularForm/GL2/Q/holomorphic/77/2/f/a/71/2"]
"2-77-11.5-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.29128852451415516	0	2.39789737135663244476015465875	["ModularForm/GL2/Q/holomorphic/77/2/f/b/71/4"]
"2-77-11.5-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0879591263929021	0	2.59430838486175112491784265723	["ModularForm/GL2/Q/holomorphic/77/2/f/b/71/2"]
"2-77-11.5-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17537260187944556	0	2.98273854383731678286599464081	["ModularForm/GL2/Q/holomorphic/77/2/f/a/71/1"]
"2-77-11.5-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1542688925315214	0	3.18143608442819947579568218588	["ModularForm/GL2/Q/holomorphic/77/2/f/b/71/3"]
"2-77-11.5-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"11.5"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.49744036158204685	0	4.39491536689549981562605544598	["ModularForm/GL2/Q/holomorphic/77/2/f/b/71/1"]
"2-77-11.9-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.49744036158204685	0	1.01474788977872755794881375848	["ModularForm/GL2/Q/holomorphic/77/2/f/b/64/1"]
"2-77-11.9-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1542688925315214	0	1.46380038616775393221737670558	["ModularForm/GL2/Q/holomorphic/77/2/f/b/64/3"]
"2-77-11.9-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.17537260187944556	0	1.85476106025871693512891186687	["ModularForm/GL2/Q/holomorphic/77/2/f/a/64/1"]
"2-77-11.9-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0879591263929021	0	2.78399977587664358326351861803	["ModularForm/GL2/Q/holomorphic/77/2/f/b/64/2"]
"2-77-11.9-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.23643920891201048	0	3.48409995540354735534011642463	["ModularForm/GL2/Q/holomorphic/77/2/f/a/64/2"]
"2-77-11.9-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"11.9"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.29128852451415516	0	4.11910068654873997734361548081	["ModularForm/GL2/Q/holomorphic/77/2/f/b/64/4"]
"2-77-7.2-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3687159358349428	0	1.13021797604519816026999231821	["ModularForm/GL2/Q/holomorphic/77/2/e/a/23/1"]
"2-77-7.2-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3242577621553154	0	1.82956938649959526708922619529	["ModularForm/GL2/Q/holomorphic/77/2/e/b/23/1"]
"2-77-7.2-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.03538260250160944	0	2.31197962326812811421567525558	["ModularForm/GL2/Q/holomorphic/77/2/e/a/23/2"]
"2-77-7.2-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.03877077197593418	0	2.52994888200094150951199022014	["ModularForm/GL2/Q/holomorphic/77/2/e/b/23/2"]
"2-77-7.2-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.22496813101754287	0	3.96176796408239184367296603891	["ModularForm/GL2/Q/holomorphic/77/2/e/b/23/3"]
"2-77-7.2-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"7.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2979507308317239	0	4.41215307715792759859193151860	["ModularForm/GL2/Q/holomorphic/77/2/e/a/23/3"]
"2-77-7.4-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2979507308317239	0	1.79744230101110718502144436565	["ModularForm/GL2/Q/holomorphic/77/2/e/a/67/3"]
"2-77-7.4-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.03877077197593418	0	1.94715995869985345784970239046	["ModularForm/GL2/Q/holomorphic/77/2/e/b/67/2"]
"2-77-7.4-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.22496813101754287	0	2.65292920569784627899543212349	["ModularForm/GL2/Q/holomorphic/77/2/e/b/67/3"]
"2-77-7.4-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.03538260250160944	0	2.96800190682158495625039596705	["ModularForm/GL2/Q/holomorphic/77/2/e/a/67/2"]
"2-77-7.4-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3687159358349428	0	3.38396963030364195163891180328	["ModularForm/GL2/Q/holomorphic/77/2/e/a/67/1"]
"2-77-7.4-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"7.4"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3242577621553154	0	3.63727125864432032688310576714	["ModularForm/GL2/Q/holomorphic/77/2/e/b/67/1"]
"2-77-77.10-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4350732019481732	0	0.15849098895759835765687707390	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/1"]
"2-77-77.10-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16295242964523304	0	1.77699599686626116964957436387	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/3"]
"2-77-77.10-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.10314647703760225	0	2.17603590953357954119323265445	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/2"]
"2-77-77.10-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.01780367739784156	0	2.58868588445288596816947225705	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/4"]
"2-77-77.10-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1768271043940308	0	3.66251940651857310654816923249	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/5"]
"2-77-77.10-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"77.10"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13237599032993497	0	3.85392829481407896944005942509	["ModularForm/GL2/Q/holomorphic/77/2/i/a/10/6"]
"2-77-77.13-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1565177159783988	0	1.94007870270729304110028665313	["ModularForm/GL2/Q/holomorphic/77/2/l/a/13/1"]
"2-77-77.13-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.006478625433666002	0	2.06229985283481353134123802498	["ModularForm/GL2/Q/holomorphic/77/2/l/b/13/2"]
"2-77-77.13-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.11370978897148369	0	2.43121340342579429470280569553	["ModularForm/GL2/Q/holomorphic/77/2/l/b/13/4"]
"2-77-77.13-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04951945032195582	0	2.96548008747228776688903917910	["ModularForm/GL2/Q/holomorphic/77/2/l/a/13/2"]
"2-77-77.13-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.44562230137221426	0	3.80629814589997344397009653711	["ModularForm/GL2/Q/holomorphic/77/2/l/b/13/1"]
"2-77-77.13-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"77.13"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2324773824440306	0	3.83629951216362426038913270968	["ModularForm/GL2/Q/holomorphic/77/2/l/b/13/3"]
"2-77-77.16-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.15406118781706402	0	1.26071485104049961258309706927	["ModularForm/GL2/Q/holomorphic/77/2/m/a/16/1"]
"2-77-77.16-c1-0-1"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.14913561712894072	0	2.00683923121159820729475508675	["ModularForm/GL2/Q/holomorphic/77/2/m/b/16/4"]
"2-77-77.16-c1-0-2"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07989825607080986	0	2.96981790234124783255510600087	["ModularForm/GL2/Q/holomorphic/77/2/m/b/16/5"]
"2-77-77.16-c1-0-3"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.40712162096934523	0	3.10425634157088678103313057205	["ModularForm/GL2/Q/holomorphic/77/2/m/b/16/1"]
"2-77-77.16-c1-0-4"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.17065490692148924	0	3.41353729897550240359126588616	["ModularForm/GL2/Q/holomorphic/77/2/m/b/16/3"]
"2-77-77.16-c1-0-5"	0.7841225003555625	0.6148480955638592	2	77	"77.16"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.28515273363109195	0	3.81377054583633951087359429779	["ModularForm/GL2/Q/holomorphic/77/2/m/b/16/2"]
"2-77-77.17-c1-0-0"	0.7841225003555625	0.6148480955638592	2	77	"77.17"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.04665630543366399	0	1.28050395726662745365377349817	["ModularForm/GL2/Q/holomorphic/77/2/n/a/17/1"]
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"2-77-1.1-c21-0-95"	14.669613324864354	215.19755510103784	2	77	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.53871187623613327537280937689	["ModularForm/GL2/Q/holomorphic/77/22/a/a/1/3"]
"2-77-1.1-c21-0-96"	14.669613324864354	215.19755510103784	2	77	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.61432860602803582223779913084	["ModularForm/GL2/Q/holomorphic/77/22/a/b/1/19"]
"2-77-1.1-c21-0-97"	14.669613324864354	215.19755510103784	2	77	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.66271691254609769296854574471	["ModularForm/GL2/Q/holomorphic/77/22/a/b/1/22"]
"2-77-1.1-c21-0-98"	14.669613324864354	215.19755510103784	2	77	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.68133676509114233768054470515	["ModularForm/GL2/Q/holomorphic/77/22/a/b/1/26"]
"2-77-1.1-c21-0-99"	14.669613324864354	215.19755510103784	2	77	"1.1"	[]	[[10.5, 0.0]]	21	true	true	false	true	0.5	1	1.80315284305230792066476920944	["ModularForm/GL2/Q/holomorphic/77/22/a/a/1/24"]


# 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).