# lfunc_search downloaded from the LMFDB on 10 June 2026.
# Search link: https://www.lmfdb.org/L/rational?origin=BMF
# Query "{'rational': True, 'is_instance_BMF': 'true'}" returned 90839 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, $\epsilon$, $r$, First zero, Origin]
# For more details, see the definitions at the bottom of the file.



"4-657-1.1-c1e2-0-0"	0.4524076040259261	0.04189088963844789	4	657	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	3.56521670106926636818196449033214	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/73.1/a", "EllipticCurve/2.0.3.1/73.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/73.2/a", "EllipticCurve/2.0.3.1/73.2/a"]
"4-675-1.1-c1e2-0-0"	0.45547494986935055	0.043038585244980714	4	675	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	3.63412818236731287070311550142582	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/75.1/a", "EllipticCurve/2.0.3.1/75.1/a"]
"4-1040-1.1-c1e2-0-0"	0.5074542432183703	0.0663113017107851	4	1040	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	3.19803865538527220934460800981605	["ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/65.2/a", "EllipticCurve/2.0.4.1/65.2/a", "EllipticCurve/2.0.4.1/65.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/65.3/a"]
"4-33e2-1.1-c1e2-0-0"	0.5133286731424862	0.06943558419523554	4	1089	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	3.09000916592887094247460325416532	["EllipticCurve/2.0.3.1/121.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/121.1/a"]
"4-1116-1.1-c1e2-0-0"	0.5164813003769269	0.07115712760503477	4	1116	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.92009081916655404042712155697172	["EllipticCurve/2.0.3.1/124.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/124.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/124.1/a", "EllipticCurve/2.0.3.1/124.2/a"]
"4-1152-1.1-c1e2-0-0"	0.5205970145886866	0.07345251881810042	4	1152	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	3.14826068230388029805668446658192	["Genus2Curve/Q/1152/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/72.1/a", "EllipticCurve/2.0.4.1/72.1/a"]
"4-1323-1.1-c1e2-0-0"	0.5389252285260752	0.0843556270801622	4	1323	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	3.05422074105458389777226041970596	["EllipticCurve/2.0.3.1/147.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/147.2/a"]
"4-11e3-1.1-c1e2-0-0"	0.539738089400763	0.084865714016399	4	1331	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.63044898935838963010520851421528	["EllipticCurve/2.0.11.1/11.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/11.1/a"]
"4-1372-1.1-c1e2-0-0"	0.5438474309689918	0.08747990956461264	4	1372	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.25140513775369021989127014661327	["ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/28.2/a", "EllipticCurve/2.0.7.1/28.2/a"]
"4-1539-1.1-c1e2-0-0"	0.5596908519090441	0.09812797435855602	4	1539	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.78666419482573957815444693786785	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/171.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/171.2/a", "EllipticCurve/2.0.3.1/171.1/a", "EllipticCurve/2.0.3.1/171.2/a"]
"4-1568-1.1-c1e2-0-1"	0.5623090433190886	0.09997703950241446	4	1568	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.48514181668809031419660323365926	["ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/98.1/a", "EllipticCurve/2.0.4.1/98.1/a"]
"4-40e2-1.1-c1e2-0-1"	0.5651562687302705	0.10201738724736169	4	1600	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.76929890617261215013507568310634	["Genus2Curve/Q/1600/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/100.2/a", "EllipticCurve/2.0.4.1/100.2/a"]
"4-1696-1.1-c1e2-0-1"	0.5734492848788719	0.1081384304822034	4	1696	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	3.22192610779849948278652519681582	["Genus2Curve/Q/1696/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/106.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/106.2/a", "EllipticCurve/2.0.4.1/106.2/a", "EllipticCurve/2.0.4.1/106.1/a"]
"4-12e3-1.1-c1e2-0-0"	0.5761353034952639	0.11017877822715062	4	1728	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.51659494204614894722398853037417	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/192.1/a", "EllipticCurve/2.0.3.1/192.1/a"]
"4-42e2-1.1-c1e2-0-0"	0.5791128460752514	0.11247416944021627	4	1764	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.39101862347891857041372548490334	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/196.2/a", "EllipticCurve/2.0.3.1/196.2/a"]
"4-44e2-1.1-c1e2-0-0"	0.5927408952431357	0.12344103856930765	4	1936	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.01168702564605014700712887669433	["EllipticCurve/2.0.4.1/121.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/121.1/a"]
"4-2e11-1.1-c1e2-0-0"	0.6011336530241874	0.13058225567662296	4	2048	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.63895104553179739845027310173351	["ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/32.1/a", "EllipticCurve/2.0.8.1/32.1/a", "ModularForm/GL2/TotallyReal/2.2.8.1/holomorphic/2.2.8.1-32.1-a", "EllipticCurve/2.2.8.1/32.1/a"]
"4-2052-1.1-c1e2-0-0"	0.6014269605781617	0.13083729914474138	4	2052	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.75661394944112353474896301836456	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/228.1/a", "EllipticCurve/2.0.3.1/228.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/228.2/a", "EllipticCurve/2.0.3.1/228.1/a"]
"4-2080-1.1-c1e2-0-0"	0.6034681965736073	0.1326226034215702	4	2080	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.19543188028031805089419316236038	["EllipticCurve/2.0.4.1/130.4/a", "EllipticCurve/2.0.4.1/130.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/130.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/130.1/a"]
"4-2156-1.1-c1e2-0-2"	0.6089066857976458	0.13746842931581987	4	2156	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.71835047120874446997595797973636	["EllipticCurve/2.0.7.1/44.3/a", "EllipticCurve/2.0.7.1/44.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/44.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/44.4/a"]
"4-2169-1.1-c1e2-0-0"	0.6098224964393208	0.13829732058720468	4	2169	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.69938163229631833131760768498385	["Genus2Curve/Q/2169/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/241.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/241.2/a", "EllipticCurve/2.0.3.1/241.1/a", "EllipticCurve/2.0.3.1/241.2/a"]
"4-2254-1.1-c1e2-0-0"	0.6157111685323531	0.1437169942847208	4	2254	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.41431212470637847961403026107162	["ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/46.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/46.3/a", "EllipticCurve/2.0.7.1/46.2/a", "EllipticCurve/2.0.7.1/46.3/a"]
"4-2457-1.1-c1e2-0-0"	0.6291291998669454	0.15666045029172979	4	2457	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.31828391835134807113696350943246	["Genus2Curve/Q/2457/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/273.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/273.1/a", "EllipticCurve/2.0.3.1/273.1/a", "EllipticCurve/2.0.3.1/273.4/a"]
"4-2457-1.1-c1e2-0-1"	0.6291291998669454	0.15666045029172979	4	2457	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.54663383021793368454863755149529	["EllipticCurve/2.0.3.1/273.2/a", "EllipticCurve/2.0.3.1/273.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/273.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/273.3/a"]
"4-2547-1.1-c1e2-0-1"	0.634812967676954	0.1623989283243939	4	2547	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	3.52128976331075074185573155387913	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/283.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/283.1/a", "EllipticCurve/2.0.3.1/283.1/a", "EllipticCurve/2.0.3.1/283.2/a"]
"4-2560-1.1-c1e2-0-0"	0.6356214501948267	0.1632278195957787	4	2560	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.35673748189235633398390138134562	["Genus2Curve/Q/2560/a", "EllipticCurve/2.0.4.1/160.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/160.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/160.1/a", "EllipticCurve/2.0.4.1/160.2/a"]
"4-51e2-1.1-c1e2-0-2"	0.6381512754108779	0.16584201514399236	4	2601	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.14503914986953664301144200994796	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/289.1/a", "EllipticCurve/2.0.3.1/289.1/a"]
"4-2624-1.1-c1e2-0-0"	0.6395573747842911	0.16730851508567318	4	2624	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.39797776147134135959499756883476	["EllipticCurve/2.0.4.1/164.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/164.1/a", "EllipticCurve/2.0.4.1/164.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/164.2/a"]
"4-2700-1.1-c1e2-0-2"	0.6441388514264411	0.17215434097992285	4	2700	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.74628747875934588873509462774083	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/300.1/a", "EllipticCurve/2.0.3.1/300.1/a"]
"4-54e2-1.1-c1e2-0-1"	0.6566522450391183	0.18592668825831668	4	2916	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.44850978333263328730546301919062	["Genus2Curve/Q/2916/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/324.1/a", "EllipticCurve/2.0.3.1/324.1/a"]
"4-3087-1.1-c1e2-0-0"	0.666074351898764	0.19682979652037846	4	3087	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	1.40786771277750203137263322761223	["EllipticCurve/2.0.7.1/63.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.7.1/63.1/a"]
"4-3087-1.1-c1e2-0-1"	0.666074351898764	0.19682979652037846	4	3087	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.39777753602607747256751042106524	["EllipticCurve/2.0.3.1/343.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/343.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/343.2/a", "EllipticCurve/2.0.3.1/343.3/a"]
"4-3104-1.1-c1e2-0-0"	0.6669894762113242	0.19791373125988168	4	3104	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.00085898254727558063116708810774	["EllipticCurve/2.0.4.1/194.2/a", "EllipticCurve/2.0.4.1/194.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/194.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/194.1/a"]
"4-3104-1.1-c1e2-0-1"	0.6669894762113242	0.19791373125988168	4	3104	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.64765870112626908625429132934603	["EllipticCurve/2.0.4.1/194.1/b", "EllipticCurve/2.0.4.1/194.2/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/194.2/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/194.1/b"]
"4-3200-1.1-c1e2-0-2"	0.6720878558624275	0.20403477449472338	4	3200	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	2.39961978181641342090490014623718	["ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/200.2/a", "EllipticCurve/2.0.4.1/200.2/a"]
"4-57e2-1.1-c1e2-0-1"	0.6746460490666928	0.20715905697917383	4	3249	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	-1	1	3.23857552173567549926694025030601	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/361.2/a", "EllipticCurve/2.0.3.1/361.2/a"]
"4-3264-1.1-c1e2-0-0"	0.6754233819022539	0.20811546998461783	4	3264	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	2.33446241377498533200596611162702	["Genus2Curve/Q/3264/a", "EllipticCurve/2.0.8.1/51.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/51.1/a", "EllipticCurve/2.0.8.1/51.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/51.4/a"]
"4-3267-1.1-c1e2-0-0"	0.6755785268262101	0.20830675258570663	4	3267	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	1.82001999803537272093437549398377	["EllipticCurve/2.0.3.1/363.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/363.1/a"]
"4-3267-1.1-c1e2-0-1"	0.6755785268262101	0.20830675258570663	4	3267	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.12508272083046916933008974996358	["EllipticCurve/2.0.11.1/27.3/a", "EllipticCurve/2.0.11.1/27.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/27.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/27.3/a"]
"4-3411-1.1-c1e2-0-1"	0.6829029414675669	0.2174883174379692	4	3411	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	3.34254052952856063975225663394009	["EllipticCurve/2.0.3.1/379.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/379.2/a", "EllipticCurve/2.0.3.1/379.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/379.1/a"]
"4-3456-1.1-c1e2-0-3"	0.68514420212082	0.22035755645430125	4	3456	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.12449507348446127913724547554631	["EllipticCurve/2.0.8.1/54.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/54.1/a", "EllipticCurve/2.0.8.1/54.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.8.1/54.4/a"]
"4-3591-1.1-c1e2-0-0"	0.6917392278490195	0.2289652735032974	4	3591	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	2.08672222628823078726316203122174	["EllipticCurve/2.0.3.1/399.2/a", "EllipticCurve/2.0.3.1/399.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/399.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/399.2/a"]
"4-60e2-1.1-c1e2-0-0"	0.6921722416622055	0.2295391213065638	4	3600	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	1.81793015252092636076156145980324	["ModularForm/GL2/ImaginaryQuadratic/2.0.3.1/400.1/a", "EllipticCurve/2.0.3.1/400.1/a"]
"4-60e2-1.1-c1e2-0-1"	0.6921722416622055	0.2295391213065638	4	3600	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	false	1	0	1.82346513010869405208007428429817	["EllipticCurve/2.0.4.1/225.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.4.1/225.2/a"]
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"4-6029793-1.1-c1e2-0-2"	4.428066702927825	384.4648296890192	4	6029793	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.63749519739075866573428463050474	["EllipticCurve/2.0.11.1/49833.1/a", "EllipticCurve/2.0.11.1/49833.6/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49833.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49833.6/a"]
"4-6030519-1.1-c1e2-0-0"	4.428199984090494	384.51112007848275	4	6030519	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.827532433327654464190250519337455	["EllipticCurve/2.0.11.1/49839.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49839.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49839.7/a", "EllipticCurve/2.0.11.1/49839.7/a"]
"4-6031245-1.1-c1e2-0-0"	4.428333253219653	384.5574104679462	4	6031245	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	1.26441591235109421789621462763854	["EllipticCurve/2.0.11.1/49845.6/a", "EllipticCurve/2.0.11.1/49845.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49845.6/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49845.3/a"]
"4-6032092-1.1-c1e2-0-0"	4.428488718664697	384.6114159223203	4	6032092	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.903957850860037810441886036932823	["EllipticCurve/2.0.11.1/49852.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49852.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49852.1/a", "EllipticCurve/2.0.11.1/49852.2/a"]
"4-6032697-1.1-c1e2-0-0"	4.428599755387574	384.6499912468732	4	6032697	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.822824616626915235731029082768916	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49857.2/b", "EllipticCurve/2.0.11.1/49857.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49857.3/a", "EllipticCurve/2.0.11.1/49857.2/b"]
"4-6032697-1.1-c1e2-0-1"	4.428599755387574	384.6499912468732	4	6032697	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	2	0.961443248086103273478345975863868	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49857.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49857.3/b", "EllipticCurve/2.0.11.1/49857.2/a", "EllipticCurve/2.0.11.1/49857.3/b"]
"4-6033907-1.1-c1e2-0-0"	4.4288218037808145	384.727141895979	4	6033907	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.51606683805626619470572762363064	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49867.1/a", "EllipticCurve/2.0.11.1/49867.4/a", "EllipticCurve/2.0.11.1/49867.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49867.4/a"]
"4-6037900-1.1-c1e2-0-0"	4.4295543266147615	384.9817390380282	4	6037900	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.58966529796809253801519822784171	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49900.6/a", "EllipticCurve/2.0.11.1/49900.6/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49900.1/a", "EllipticCurve/2.0.11.1/49900.1/a"]
"4-6038505-1.1-c1e2-0-0"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.71656518685718699238943822665342	["EllipticCurve/2.0.11.1/49905.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.1/a", "EllipticCurve/2.0.11.1/49905.12/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.12/a"]
"4-6038505-1.1-c1e2-0-1"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.941542891212923064343184560755484	["EllipticCurve/2.0.11.1/49905.11/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.11/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.2/a", "EllipticCurve/2.0.11.1/49905.2/a"]
"4-6038505-1.1-c1e2-0-2"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.972329054864684992851575386972002	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.9/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.4/a", "EllipticCurve/2.0.11.1/49905.4/a", "EllipticCurve/2.0.11.1/49905.9/a"]
"4-6038505-1.1-c1e2-0-3"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	1.03249875566838779760737741191314	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.5/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.8/a", "EllipticCurve/2.0.11.1/49905.5/a", "EllipticCurve/2.0.11.1/49905.8/a"]
"4-6038505-1.1-c1e2-0-4"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	2	1.05276925562224896154851867138235	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.7/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.6/a", "EllipticCurve/2.0.11.1/49905.7/a", "EllipticCurve/2.0.11.1/49905.6/a"]
"4-6038505-1.1-c1e2-0-5"	4.429665283225362	385.02031436258113	4	6038505	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	2	1.35089608312135807093404678613093	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.6/b", "EllipticCurve/2.0.11.1/49905.6/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49905.7/b", "EllipticCurve/2.0.11.1/49905.7/b"]
"4-6040683-1.1-c1e2-0-0"	4.430064658001383	385.15918553097157	4	6040683	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.74363083299455655480093194817747	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49923.2/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49923.3/b", "EllipticCurve/2.0.11.1/49923.2/b", "EllipticCurve/2.0.11.1/49923.3/b"]
"4-6040683-1.1-c1e2-0-1"	4.430064658001383	385.15918553097157	4	6040683	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.792578117031574664528198247818047	["EllipticCurve/2.0.11.1/49923.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49923.3/a", "EllipticCurve/2.0.11.1/49923.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49923.2/a"]
"4-6042861-1.1-c1e2-0-0"	4.430463924794496	385.29805669936206	4	6042861	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.62589566318066583824354433566610	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.5/b", "EllipticCurve/2.0.11.1/49941.5/b", "EllipticCurve/2.0.11.1/49941.8/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.8/b"]
"4-6042861-1.1-c1e2-0-1"	4.430463924794496	385.29805669936206	4	6042861	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.78388173114520320852949992390926	["EllipticCurve/2.0.11.1/49941.5/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.8/a", "EllipticCurve/2.0.11.1/49941.8/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.5/a"]
"4-6042861-1.1-c1e2-0-2"	4.430463924794496	385.29805669936206	4	6042861	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.803442610493159895974204550614319	["EllipticCurve/2.0.11.1/49941.10/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.10/a", "EllipticCurve/2.0.11.1/49941.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.3/a"]
"4-6042861-1.1-c1e2-0-3"	4.430463924794496	385.29805669936206	4	6042861	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.977959024826627103660329969928200	["EllipticCurve/2.0.11.1/49941.3/b", "EllipticCurve/2.0.11.1/49941.10/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.3/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49941.10/b"]
"4-6044676-1.1-c1e2-0-0"	4.430796564684279	385.41378267302076	4	6044676	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.48794408283683114268400507630003	["EllipticCurve/2.0.11.1/49956.4/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49956.5/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49956.4/b", "EllipticCurve/2.0.11.1/49956.5/b"]
"4-6044676-1.1-c1e2-0-1"	4.430796564684279	385.41378267302076	4	6044676	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	1.20904311087996518887489198436571	["EllipticCurve/2.0.11.1/49956.4/a", "EllipticCurve/2.0.11.1/49956.5/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49956.4/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49956.5/a"]
"4-6045765-1.1-c1e2-0-0"	4.430996112662927	385.48321825721604	4	6045765	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.869465395625634278937067902006564	["EllipticCurve/2.0.11.1/49965.7/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49965.2/a", "EllipticCurve/2.0.11.1/49965.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49965.7/a"]
"4-6047217-1.1-c1e2-0-0"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.65854960941836650068007357373641	["EllipticCurve/2.0.11.1/49977.5/a", "EllipticCurve/2.0.11.1/49977.6/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.5/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.6/b"]
"4-6047217-1.1-c1e2-0-1"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.73775219848919156173400429998725	["EllipticCurve/2.0.11.1/49977.5/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.5/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.6/a", "EllipticCurve/2.0.11.1/49977.6/a"]
"4-6047217-1.1-c1e2-0-2"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.74438465809665317375530056506403	["EllipticCurve/2.0.11.1/49977.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.3/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.8/a", "EllipticCurve/2.0.11.1/49977.8/a"]
"4-6047217-1.1-c1e2-0-3"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	0.76312278125557549657818461974900	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.1/a", "EllipticCurve/2.0.11.1/49977.1/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.10/a", "EllipticCurve/2.0.11.1/49977.10/a"]
"4-6047217-1.1-c1e2-0-4"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	0	0.78720690573544261163950036516350	["EllipticCurve/2.0.11.1/49977.9/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.2/a", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.9/a", "EllipticCurve/2.0.11.1/49977.2/a"]
"4-6047217-1.1-c1e2-0-5"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	1	2	0.941955393701784902748724754891393	["ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.3/b", "EllipticCurve/2.0.11.1/49977.8/b", "EllipticCurve/2.0.11.1/49977.3/b", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.8/b"]
"4-6047217-1.1-c1e2-0-6"	4.431262134704331	385.575799036143	4	6047217	"1.1"	[]	[[0.5, 0.0], [0.5, 0.0]]	1	true	-1	1	1.10766856890013532640157531281923	["EllipticCurve/2.0.11.1/49977.6/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.6/c", "ModularForm/GL2/ImaginaryQuadratic/2.0.11.1/49977.5/c", "EllipticCurve/2.0.11.1/49977.5/c"]


# 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 $1$.


#$\epsilon$ (root_number) --
#    The **sign** of the functional equation of an analytic L-function, also called the **root number**, is the complex number $\varepsilon$ that appears in the functional equation of $\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$.  The sign appears as the 4th entry in the quadruple
#    known as the Selberg data.


#$r$ (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).