
# lfunc_search downloaded from the LMFDB on 21 April 2026.
# Search link: https://www.lmfdb.org/L/2/1568/56.3
# Query "{'degree': 2, 'conductor': 1568}" returned 946 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.



"2-1568-224.101-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.101"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14550314319363972	0	1.20258403451416158438986234569	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/325/1"]
"2-1568-224.107-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.107"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.09441038027938724	0	1.07289766334608822935180162167	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/1451/1"]
"2-1568-224.11-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.46808961972061286	0	2.18691745071339773104183596590	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/459/1"]
"2-1568-224.117-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.117"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.04199685680636027	0	1.57330090913512116335388911329	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/117/1"]
"2-1568-224.123-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.123"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.2180896197206128	0	1.03495487465391011960551157249	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/1243/1"]
"2-1568-224.157-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.157"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.04199685680636027	0	1.70933040767705908283164863611	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/1501/1"]
"2-1568-224.163-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.163"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.46808961972061286	0	0.41848755519793717508735875497	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/1059/1"]
"2-1568-224.173-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.173"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14550314319363972	0	1.43890372712622404381595271969	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/1293/1"]
"2-1568-224.179-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.179"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.15558961972061278	0	1.28966689237644921432362756086	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/851/1"]
"2-1568-224.213-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.213"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.10449685680636027	0	1.19447181610287746628002711232	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/1109/1"]
"2-1568-224.219-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.219"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.15558961972061278	0	2.05582938551364257748002365164	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/667/1"]
"2-1568-224.45-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.45"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.2919968568063603	0	0.38957572206392821768195466533	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/717/1"]
"2-1568-224.5-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.2919968568063603	0	1.70656024992243671519218408500	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/901/1"]
"2-1568-224.51-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.51"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.2180896197206128	0	1.28974451474655833981326666034	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/275/1"]
"2-1568-224.61-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.61"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.10449685680636027	0	0.981520388703506868513534067892	["ModularForm/GL2/Q/holomorphic/1568/1/bl/a/509/1"]
"2-1568-224.67-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"224.67"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.09441038027938724	0	1.32140934273523698407996108320	["ModularForm/GL2/Q/holomorphic/1568/1/bo/a/67/1"]
"2-1568-28.11-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"28.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.10482704694605388	0	1.26464359384722805519026021915	["ModularForm/GL2/Q/holomorphic/1568/1/r/a/1439/1"]
"2-1568-28.11-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"28.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.14517295305394612	0	1.37932746312004637541757497025	["ModularForm/GL2/Q/holomorphic/1568/1/r/a/1439/2"]
"2-1568-28.23-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"28.23"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.10482704694605388	0	0.909574632935171645474804838652	["ModularForm/GL2/Q/holomorphic/1568/1/r/a/863/1"]
"2-1568-28.23-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"28.23"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.14517295305394612	0	1.86529987310190334708636626818	["ModularForm/GL2/Q/holomorphic/1568/1/r/a/863/2"]
"2-1568-32.11-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"32.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.28125	0	1.07220278636671478710628024423	["ModularForm/GL2/Q/holomorphic/1568/1/x/a/491/1"]
"2-1568-32.19-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"32.19"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.03125000000000001	0	1.36601141306477706755349510952	["ModularForm/GL2/Q/holomorphic/1568/1/x/a/883/1"]
"2-1568-32.27-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"32.27"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.03125000000000001	0	0.830273354711556467065703469392	["ModularForm/GL2/Q/holomorphic/1568/1/x/a/1275/1"]
"2-1568-32.3-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"32.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.28125	0	1.91227291388706616183797107536	["ModularForm/GL2/Q/holomorphic/1568/1/x/a/99/1"]
"2-1568-4.3-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"4.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.375	0	0.58635807685772659355174901311	["ModularForm/GL2/Q/holomorphic/1568/1/d/a/1471/2"]
"2-1568-4.3-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"4.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.125	0	1.42672078864925063800081401682	["ModularForm/GL2/Q/holomorphic/1568/1/d/b/1471/2"]
"2-1568-4.3-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"4.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.125	0	1.52471045353919524105915637814	["ModularForm/GL2/Q/holomorphic/1568/1/d/b/1471/1"]
"2-1568-4.3-c0-0-3"	0.8846094843537162	0.7825339398085478	2	1568	"4.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.375	0	1.87865169974731407308170849971	["ModularForm/GL2/Q/holomorphic/1568/1/d/a/1471/1"]
"2-1568-56.11-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"56.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3849347854135728	0	0.60785572555413721874552331542	["ModularForm/GL2/Q/holomorphic/1568/1/o/b/655/1"]
"2-1568-56.11-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"56.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1868396197206128	0	0.822148109631353808369783193243	["ModularForm/GL2/Q/holomorphic/1568/1/o/a/655/1"]
"2-1568-56.11-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"56.11"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.2384410718008522	0	1.86858885853818821176704884330	["ModularForm/GL2/Q/holomorphic/1568/1/o/b/655/2"]
"2-1568-56.45-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"56.45"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.07324685680636028	0	1.09811358507326840712078854483	["ModularForm/GL2/Q/holomorphic/1568/1/n/a/913/1"]
"2-1568-56.5-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"56.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.07324685680636028	0	1.45717556660701804980797676173	["ModularForm/GL2/Q/holomorphic/1568/1/n/a/1489/1"]
"2-1568-56.51-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"56.51"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.2384410718008522	0	1.10601424514485362009228973435	["ModularForm/GL2/Q/holomorphic/1568/1/o/b/79/2"]
"2-1568-56.51-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"56.51"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3849347854135728	0	1.77387166347442356248440710328	["ModularForm/GL2/Q/holomorphic/1568/1/o/b/79/1"]
"2-1568-56.51-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"56.51"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1868396197206128	0	1.77541335423111088246632853566	["ModularForm/GL2/Q/holomorphic/1568/1/o/a/79/1"]
"2-1568-7.3-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"7.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.1452490541431976	0	0.832843595447161598850104133132	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/129/2"]
"2-1568-7.3-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"7.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.3180829254684844	0	1.62888934129397086818395662768	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/129/1"]
"2-1568-7.3-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"7.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.11014266939733011	0	1.76926603684692060658408155464	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/129/4"]
"2-1568-7.3-c0-0-3"	0.8846094843537162	0.7825339398085478	2	1568	"7.3"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.19685050622343703	0	1.96186963190774911578445444400	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/129/3"]
"2-1568-7.5-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"7.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.3180829254684844	0	0.28258707368594528149196406828	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/705/1"]
"2-1568-7.5-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"7.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.19685050622343703	0	0.870721665807901014066193777472	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/705/3"]
"2-1568-7.5-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"7.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.11014266939733011	0	1.28551354866079674503287280046	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/705/4"]
"2-1568-7.5-c0-0-3"	0.8846094843537162	0.7825339398085478	2	1568	"7.5"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.1452490541431976	0	1.44368949395014367166278966183	["ModularForm/GL2/Q/holomorphic/1568/1/s/a/705/2"]
"2-1568-7.6-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"7.6"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.22498009178110556	0	0.980279724311692558501736472713	["ModularForm/GL2/Q/holomorphic/1568/1/c/a/97/4"]
"2-1568-7.6-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"7.6"	[]	[[0.0, 0.0]]	0	true	true	false	false	-0.0670796521477402	0	1.20561902552548704108946457169	["ModularForm/GL2/Q/holomorphic/1568/1/c/a/97/3"]
"2-1568-7.6-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"7.6"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.0670796521477402	0	1.40712236133890665787240401762	["ModularForm/GL2/Q/holomorphic/1568/1/c/a/97/2"]
"2-1568-7.6-c0-0-3"	0.8846094843537162	0.7825339398085478	2	1568	"7.6"	[]	[[0.0, 0.0]]	0	true	true	false	false	0.22498009178110556	0	1.85759256179043283852857333065	["ModularForm/GL2/Q/holomorphic/1568/1/c/a/97/1"]
"2-1568-8.3-c0-0-0"	0.8846094843537162	0.7825339398085478	2	1568	"8.3"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	0.930532943025316041953595192790	["ModularForm/GL2/Q/holomorphic/1568/1/g/b/687/1", "ArtinRepresentation/2.1568.8t6.b.a"]
"2-1568-8.3-c0-0-1"	0.8846094843537162	0.7825339398085478	2	1568	"8.3"	[]	[[0.0, 0.0]]	0	true	true	true	true	0.0	0	1.24937716619636972835746446093	["ModularForm/GL2/Q/holomorphic/1568/1/g/a/687/1", "ModularForm/GL2/Q/holomorphic/1568/1/g/a", "ArtinRepresentation/2.1568.4t3.a.a", "ArtinRepresentation/2.1568.4t3.a"]
"2-1568-8.3-c0-0-2"	0.8846094843537162	0.7825339398085478	2	1568	"8.3"	[]	[[0.0, 0.0]]	0	true	true	false	true	0.0	0	1.61459670199496867397522118930	["ModularForm/GL2/Q/holomorphic/1568/1/g/b/687/2", "ArtinRepresentation/2.1568.8t6.b.b"]
"2-1568-1.1-c1-0-0"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.27731966818152329412669043226	["ModularForm/GL2/Q/holomorphic/1568/2/a/x/1/1"]
"2-1568-1.1-c1-0-1"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.40506550031819905158841571449	["ModularForm/GL2/Q/holomorphic/1568/2/a/v/1/1"]
"2-1568-1.1-c1-0-10"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.962050876203161734055370874216	["ModularForm/GL2/Q/holomorphic/1568/2/a/x/1/2"]
"2-1568-1.1-c1-0-11"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	0.981611028434021842742737928419	["ModularForm/GL2/Q/holomorphic/1568/2/a/u/1/1"]
"2-1568-1.1-c1-0-12"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.01951597661611130937101806782	["ModularForm/GL2/Q/holomorphic/1568/2/a/u/1/2"]
"2-1568-1.1-c1-0-13"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.12552427816405528035864022651	["ModularForm/GL2/Q/holomorphic/1568/2/a/r/1/2"]
"2-1568-1.1-c1-0-14"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.13590008709403894614989037078	["EllipticCurve/Q/1568/f", "ModularForm/GL2/Q/holomorphic/1568/2/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1568/2/a/f"]
"2-1568-1.1-c1-0-15"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	1.16036401648900510175062615275	["EllipticCurve/Q/1568/h", "ModularForm/GL2/Q/holomorphic/1568/2/a/h/1/1", "ModularForm/GL2/Q/holomorphic/1568/2/a/h"]
"2-1568-1.1-c1-0-16"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.16798654435157965154300828509	["ModularForm/GL2/Q/holomorphic/1568/2/a/j/1/1"]
"2-1568-1.1-c1-0-17"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.5	1	1.18726156001547987362434063658	["ModularForm/GL2/Q/holomorphic/1568/2/a/m/1/1"]
"2-1568-1.1-c1-0-18"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.21319714704167851164463767389	["ModularForm/GL2/Q/holomorphic/1568/2/a/p/1/2"]
"2-1568-1.1-c1-0-19"	3.538437937414866	12.52054303693677	2	1568	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.23629472945942244569028190055	["ModularForm/GL2/Q/holomorphic/1568/2/a/x/1/3"]
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"2-1568-1.1-c3-0-47"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.828025477470473814446511761864	["ModularForm/GL2/Q/holomorphic/1568/4/a/w/1/3"]
"2-1568-1.1-c3-0-48"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.834304800763994036081903882222	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/3"]
"2-1568-1.1-c3-0-49"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.853294156079299427498068124865	["ModularForm/GL2/Q/holomorphic/1568/4/a/bk/1/7"]
"2-1568-1.1-c3-0-5"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.21834069681023803387830901617	["ModularForm/GL2/Q/holomorphic/1568/4/a/bd/1/2"]
"2-1568-1.1-c3-0-50"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.877413023322437212416597661952	["ModularForm/GL2/Q/holomorphic/1568/4/a/r/1/2"]
"2-1568-1.1-c3-0-51"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.882279093046275753366702520809	["ModularForm/GL2/Q/holomorphic/1568/4/a/bd/1/1"]
"2-1568-1.1-c3-0-52"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.886405138484999349009863231397	["ModularForm/GL2/Q/holomorphic/1568/4/a/bf/1/2"]
"2-1568-1.1-c3-0-53"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.901907139666690843925338723144	["ModularForm/GL2/Q/holomorphic/1568/4/a/bg/1/5"]
"2-1568-1.1-c3-0-54"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.903991568698421102348966655586	["ModularForm/GL2/Q/holomorphic/1568/4/a/ba/1/1"]
"2-1568-1.1-c3-0-55"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.933137019239030679587369711747	["ModularForm/GL2/Q/holomorphic/1568/4/a/be/1/3"]
"2-1568-1.1-c3-0-56"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	0.934927380007703285112561303651	["ModularForm/GL2/Q/holomorphic/1568/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/a"]
"2-1568-1.1-c3-0-57"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.940513044562046480920128781193	["ModularForm/GL2/Q/holomorphic/1568/4/a/y/1/1"]
"2-1568-1.1-c3-0-58"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.952891280722081018837997741865	["ModularForm/GL2/Q/holomorphic/1568/4/a/p/1/1"]
"2-1568-1.1-c3-0-59"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.955343782571350875802192946857	["ModularForm/GL2/Q/holomorphic/1568/4/a/bh/1/2"]
"2-1568-1.1-c3-0-6"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.23189396388551111853772017273	["ModularForm/GL2/Q/holomorphic/1568/4/a/bm/1/2"]
"2-1568-1.1-c3-0-60"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.963887123839347089016629585943	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/4"]
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"2-1568-1.1-c3-0-62"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.971227261407435027235582027600	["ModularForm/GL2/Q/holomorphic/1568/4/a/w/1/1"]
"2-1568-1.1-c3-0-63"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	0.989270903112510098167582382072	["ModularForm/GL2/Q/holomorphic/1568/4/a/bb/1/1"]
"2-1568-1.1-c3-0-64"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	0.997388644218494631556116533723	["ModularForm/GL2/Q/holomorphic/1568/4/a/g/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/g"]
"2-1568-1.1-c3-0-65"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.02933975470221732298582245215	["ModularForm/GL2/Q/holomorphic/1568/4/a/bh/1/1"]
"2-1568-1.1-c3-0-66"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.03259909770252692973119069164	["ModularForm/GL2/Q/holomorphic/1568/4/a/bi/1/5"]
"2-1568-1.1-c3-0-67"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.03925203153416048973365560738	["ModularForm/GL2/Q/holomorphic/1568/4/a/bi/1/6"]
"2-1568-1.1-c3-0-68"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.04770823258275964386904336209	["ModularForm/GL2/Q/holomorphic/1568/4/a/bk/1/6"]
"2-1568-1.1-c3-0-69"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.04829278546981350208969373176	["ModularForm/GL2/Q/holomorphic/1568/4/a/s/1/1"]
"2-1568-1.1-c3-0-7"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.27222223064049448565115103425	["ModularForm/GL2/Q/holomorphic/1568/4/a/w/1/2"]
"2-1568-1.1-c3-0-70"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.05256949027750556133402064275	["ModularForm/GL2/Q/holomorphic/1568/4/a/x/1/3"]
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"2-1568-1.1-c3-0-72"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.09082979706801042316938096761	["ModularForm/GL2/Q/holomorphic/1568/4/a/bj/1/4"]
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"2-1568-1.1-c3-0-74"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.10199398291801684934330632213	["ModularForm/GL2/Q/holomorphic/1568/4/a/bf/1/4"]
"2-1568-1.1-c3-0-75"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.10455693994402575392967009580	["ModularForm/GL2/Q/holomorphic/1568/4/a/bj/1/6"]
"2-1568-1.1-c3-0-76"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	1.14028814389029285615172211207	["ModularForm/GL2/Q/holomorphic/1568/4/a/c/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/c"]
"2-1568-1.1-c3-0-77"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.14261892503697959987059363046	["ModularForm/GL2/Q/holomorphic/1568/4/a/bb/1/2"]
"2-1568-1.1-c3-0-78"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.14817490605406951160898369295	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/2"]
"2-1568-1.1-c3-0-79"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.16691808468352178364152717923	["ModularForm/GL2/Q/holomorphic/1568/4/a/q/1/1"]
"2-1568-1.1-c3-0-8"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.33771534129773036597132321718	["ModularForm/GL2/Q/holomorphic/1568/4/a/bm/1/5"]
"2-1568-1.1-c3-0-80"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.16909968094946301716687230363	["ModularForm/GL2/Q/holomorphic/1568/4/a/bg/1/6"]
"2-1568-1.1-c3-0-81"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.18792302607234896228347156728	["ModularForm/GL2/Q/holomorphic/1568/4/a/bf/1/1"]
"2-1568-1.1-c3-0-82"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	1.19365305107458053492591488706	["ModularForm/GL2/Q/holomorphic/1568/4/a/j/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/j"]
"2-1568-1.1-c3-0-83"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.0	0	1.21931858290260886063232639427	["ModularForm/GL2/Q/holomorphic/1568/4/a/o/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/o"]
"2-1568-1.1-c3-0-84"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.23638133778524648707705871923	["ModularForm/GL2/Q/holomorphic/1568/4/a/u/1/1"]
"2-1568-1.1-c3-0-85"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.25713115351347081478055555684	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/5"]
"2-1568-1.1-c3-0-86"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.26240478908946003454571785716	["ModularForm/GL2/Q/holomorphic/1568/4/a/z/1/2"]
"2-1568-1.1-c3-0-87"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.27975530094756740118881855681	["ModularForm/GL2/Q/holomorphic/1568/4/a/ba/1/2"]
"2-1568-1.1-c3-0-88"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.28400605538516048004771921345	["ModularForm/GL2/Q/holomorphic/1568/4/a/bd/1/4"]
"2-1568-1.1-c3-0-89"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.29942872359821158700221753519	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/1"]
"2-1568-1.1-c3-0-9"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	0.34571322495483059550262907510	["ModularForm/GL2/Q/holomorphic/1568/4/a/bm/1/1"]
"2-1568-1.1-c3-0-90"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.31391802718413985887057152036	["ModularForm/GL2/Q/holomorphic/1568/4/a/bh/1/5"]
"2-1568-1.1-c3-0-91"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.32143946956064500211968104263	["ModularForm/GL2/Q/holomorphic/1568/4/a/be/1/4"]
"2-1568-1.1-c3-0-92"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.32241307755792186807057244039	["ModularForm/GL2/Q/holomorphic/1568/4/a/be/1/2"]
"2-1568-1.1-c3-0-93"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.32838474964324578418192773572	["ModularForm/GL2/Q/holomorphic/1568/4/a/bh/1/3"]
"2-1568-1.1-c3-0-94"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.34408156110582528767643406639	["ModularForm/GL2/Q/holomorphic/1568/4/a/bm/1/10"]
"2-1568-1.1-c3-0-95"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.0	0	1.37931629697241341100412697327	["ModularForm/GL2/Q/holomorphic/1568/4/a/bc/1/4"]
"2-1568-1.1-c3-0-96"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.39898722872989206477408680285	["ModularForm/GL2/Q/holomorphic/1568/4/a/bl/1/6"]
"2-1568-1.1-c3-0-97"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	1.40091296592002474174728119552	["ModularForm/GL2/Q/holomorphic/1568/4/a/k/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/k"]
"2-1568-1.1-c3-0-98"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	true	true	0.5	1	1.41232498102149543240140126004	["ModularForm/GL2/Q/holomorphic/1568/4/a/f/1/1", "ModularForm/GL2/Q/holomorphic/1568/4/a/f"]
"2-1568-1.1-c3-0-99"	9.618471546404935	92.51499488900133	2	1568	"1.1"	[]	[[1.5, 0.0]]	3	true	true	false	true	0.5	1	1.41902385036328021067047024579	["ModularForm/GL2/Q/holomorphic/1568/4/a/q/1/2"]


# 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).


