# lfunc_search downloaded from the LMFDB on 15 June 2026.
# Search link: https://www.lmfdb.org/L/2/105/35.13
# Query "{'degree': 2, 'conductor': 105}" returned 6211 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.



"2-105-1.1-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	1.31435688386558938615415944577	["ModularForm/GL2/Q/holomorphic/105/2/a/b/1/1"]
"2-105-1.1-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"1.1"	[]	[[0.5, 0.0]]	1	true	true	true	true	0.0	0	2.63294756479994443235170715895	["EllipticCurve/Q/105/a", "ModularForm/GL2/Q/holomorphic/105/2/a/a/1/1", "ModularForm/GL2/Q/holomorphic/105/2/a/a"]
"2-105-1.1-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"1.1"	[]	[[0.5, 0.0]]	1	true	true	false	true	0.0	0	2.89131068890819257465193141069	["ModularForm/GL2/Q/holomorphic/105/2/a/b/1/2"]
"2-105-105.104-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2686210124026206	0	0.47956781225255046972740175522	["ModularForm/GL2/Q/holomorphic/105/2/g/a/104/1"]
"2-105-105.104-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.36167490524047163	0	0.66368167937209263439652868992	["ModularForm/GL2/Q/holomorphic/105/2/g/b/104/2"]
"2-105-105.104-c1-0-10"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.05066812082710802	0	3.35833776691349154454458590662	["ModularForm/GL2/Q/holomorphic/105/2/g/c/104/3"]
"2-105-105.104-c1-0-11"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.36167490524047163	0	3.47110627331364208458537828722	["ModularForm/GL2/Q/holomorphic/105/2/g/b/104/1"]
"2-105-105.104-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1452451551881956	0	1.42578641308288919280803743557	["ModularForm/GL2/Q/holomorphic/105/2/g/c/104/2"]
"2-105-105.104-c1-0-3"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13832509475952845	0	1.72878415582342817772241598425	["ModularForm/GL2/Q/holomorphic/105/2/g/b/104/4"]
"2-105-105.104-c1-0-4"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2686210124026206	0	2.13235648433281233133195991540	["ModularForm/GL2/Q/holomorphic/105/2/g/a/104/2"]
"2-105-105.104-c1-0-5"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1452451551881956	0	2.20572488300709862004873935729	["ModularForm/GL2/Q/holomorphic/105/2/g/c/104/1"]
"2-105-105.104-c1-0-6"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07270773638731692	0	2.34515833163741102245773879332	["ModularForm/GL2/Q/holomorphic/105/2/g/a/104/4"]
"2-105-105.104-c1-0-7"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.05066812082710802	0	2.87845056192373894188141371571	["ModularForm/GL2/Q/holomorphic/105/2/g/c/104/4"]
"2-105-105.104-c1-0-8"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13832509475952845	0	2.94173244415565854990122695373	["ModularForm/GL2/Q/holomorphic/105/2/g/b/104/3"]
"2-105-105.104-c1-0-9"	0.9156578079301484	0.8384292212234444	2	105	"105.104"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07270773638731692	0	3.01381048990394622851526182839	["ModularForm/GL2/Q/holomorphic/105/2/g/a/104/3"]
"2-105-105.2-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.37158210339241193	0	1.20366283258916783520851124833	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/10"]
"2-105-105.2-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.40730259974224325	0	1.74408700952844118642915398671	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/12"]
"2-105-105.2-c1-0-10"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3756565167353787	0	3.36232665685655901241319687431	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/3"]
"2-105-105.2-c1-0-11"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4717232330040891	0	4.24656157638019536889397932293	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/1"]
"2-105-105.2-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.03234420349041541	0	1.82756279041259597197516985663	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/4"]
"2-105-105.2-c1-0-3"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.2897385155314049	0	2.13362566073489665501973051282	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/11"]
"2-105-105.2-c1-0-4"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.0352281816623026	0	2.22442289446052811584306463215	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/8"]
"2-105-105.2-c1-0-5"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08101555508463282	0	2.36346477388012276580321747064	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/7"]
"2-105-105.2-c1-0-6"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.24837700888709766	0	2.68273035298435408457039316310	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/2"]
"2-105-105.2-c1-0-7"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.009680605729249489	0	2.77732612026732877221391517755	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/9"]
"2-105-105.2-c1-0-8"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.1033861369132629	0	2.81868405973270212235027282105	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/6"]
"2-105-105.2-c1-0-9"	0.9156578079301484	0.8384292212234444	2	105	"105.2"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.32317359478684593	0	2.96673474953201300034861081821	["ModularForm/GL2/Q/holomorphic/105/2/x/a/2/5"]
"2-105-105.23-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4570315970678113	0	0.65030445642386813680062537777	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/2"]
"2-105-105.23-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3240475366655617	0	0.73766198130993833711533773375	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/7"]
"2-105-105.23-c1-0-10"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4334561014356451	0	3.43927710989046958441187004643	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/5"]
"2-105-105.23-c1-0-11"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.16759483210609072	0	3.44771812668954614786566177576	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/10"]
"2-105-105.23-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.08516654776611865	0	1.62700938985477079762534734678	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/3"]
"2-105-105.23-c1-0-3"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.06409581414098914	0	1.73904343822875846850639853300	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/4"]
"2-105-105.23-c1-0-4"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.13964584466766594	0	1.74454804715826243937081266219	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/6"]
"2-105-105.23-c1-0-5"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.045506301733873274	0	2.18424534059329627715729261420	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/1"]
"2-105-105.23-c1-0-6"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.022071004921324237	0	2.89200156503303298503850693861	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/9"]
"2-105-105.23-c1-0-7"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.07505432498649657	0	2.91561138072949283237632359234	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/8"]
"2-105-105.23-c1-0-8"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.004852878513686204	0	2.92526259499093625769628058426	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/11"]
"2-105-105.23-c1-0-9"	0.9156578079301484	0.8384292212234444	2	105	"105.23"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.10992693499571898	0	3.35825673050670158052135747612	["ModularForm/GL2/Q/holomorphic/105/2/x/a/23/12"]
"2-105-105.32-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.4334561014356451	0	0.33149081114965854031648209193	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/5"]
"2-105-105.32-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.06409581414098914	0	1.24324151723839757288615306673	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/4"]
"2-105-105.32-c1-0-10"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.004852878513686204	0	3.26257590060564266961992521243	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/11"]
"2-105-105.32-c1-0-11"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.3240475366655617	0	3.38682358923095611902052775664	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/7"]
"2-105-105.32-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.045506301733873274	0	1.40907799761246195810057222594	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/1"]
"2-105-105.32-c1-0-3"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.16759483210609072	0	1.94251051800903210010206832643	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/10"]
"2-105-105.32-c1-0-4"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08516654776611865	0	2.12400642151089781795677030840	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/3"]
"2-105-105.32-c1-0-5"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.07505432498649657	0	2.50341200074928761404236028459	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/8"]
"2-105-105.32-c1-0-6"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.10992693499571898	0	2.57840070936464811318725115299	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/12"]
"2-105-105.32-c1-0-7"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.13964584466766594	0	2.72083510051971430192810921346	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/6"]
"2-105-105.32-c1-0-8"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4570315970678113	0	2.89270157379613377330903261686	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/2"]
"2-105-105.32-c1-0-9"	0.9156578079301484	0.8384292212234444	2	105	"105.32"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.022071004921324237	0	2.90332607184030009539581897638	["ModularForm/GL2/Q/holomorphic/105/2/x/a/32/9"]
"2-105-105.53-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.32317359478684593	0	1.00705138499228596445757631019	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/5"]
"2-105-105.53-c1-0-1"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.4717232330040891	0	1.04329792440247530466316119725	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/1"]
"2-105-105.53-c1-0-10"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.2897385155314049	0	3.73842716984026107285574706401	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/11"]
"2-105-105.53-c1-0-11"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.40730259974224325	0	4.16591415841522533582424218687	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/12"]
"2-105-105.53-c1-0-2"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.3756565167353787	0	1.38870816652073950795454864894	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/3"]
"2-105-105.53-c1-0-3"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.009680605729249489	0	2.03146238300716129021761020825	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/9"]
"2-105-105.53-c1-0-4"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.24837700888709766	0	2.04160873205741505505109639699	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/2"]
"2-105-105.53-c1-0-5"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.0352281816623026	0	2.04814462909678929295636779610	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/8"]
"2-105-105.53-c1-0-6"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.03234420349041541	0	2.24905959127290116076869027754	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/4"]
"2-105-105.53-c1-0-7"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.1033861369132629	0	2.28310621613916264379906146969	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/6"]
"2-105-105.53-c1-0-8"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.08101555508463282	0	2.56624740763306355575309366941	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/7"]
"2-105-105.53-c1-0-9"	0.9156578079301484	0.8384292212234444	2	105	"105.53"	[]	[[0.5, 0.0]]	1	true	true	false	false	0.37158210339241193	0	3.57077880886964560767576589599	["ModularForm/GL2/Q/holomorphic/105/2/x/a/53/10"]
"2-105-105.59-c1-0-0"	0.9156578079301484	0.8384292212234444	2	105	"105.59"	[]	[[0.5, 0.0]]	1	true	true	false	false	-0.049088016423518684	0	1.06146174643284458334543702437	["ModularForm/GL2/Q/holomorphic/105/2/p/a/59/1"]
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"2-105-7.6-c18-0-81"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.41880489453981595	0	1.13936401990681092188554411676	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/39"]
"2-105-7.6-c18-0-82"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.3688206606177069	0	1.14770756367285955596132368321	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/30"]
"2-105-7.6-c18-0-83"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.1407507524475198	0	1.15574618479047855606431020818	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/94"]
"2-105-7.6-c18-0-84"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.03243574836921702	0	1.15610651207437194804461318464	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/80"]
"2-105-7.6-c18-0-85"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.2343531714310249	0	1.26199828921442560248064034179	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/86"]
"2-105-7.6-c18-0-86"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.42331348525713375	0	1.35426133582834535501422662687	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/28"]
"2-105-7.6-c18-0-87"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.4260611539963325	0	1.38345150283578823891716649728	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/66"]
"2-105-7.6-c18-0-88"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.2825235402869574	0	1.39633617975233281196401476390	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/90"]
"2-105-7.6-c18-0-89"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.18261573451881377	0	1.44976400788574640651685305749	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/35"]
"2-105-7.6-c18-0-9"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.2592822150376462	0	0.16888092493967887196278679151	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/3"]
"2-105-7.6-c18-0-90"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.4425181170447661	0	1.51335602464433680235105399998	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/82"]
"2-105-7.6-c18-0-91"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.3614749755552503	0	1.54826976834572945524508137098	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/88"]
"2-105-7.6-c18-0-92"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.17529409469201618	0	1.58139115065484490724620894931	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/96"]
"2-105-7.6-c18-0-93"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.32674167029987494	0	1.63370783298450761045222994095	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/68"]
"2-105-7.6-c18-0-94"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	0.4722815706171559	0	1.70699866506857323101110338828	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/91"]
"2-105-7.6-c18-0-95"	14.685211765420743	215.6554445952518	2	105	"7.6"	[]	[[9.0, 0.0]]	18	true	true	false	false	-0.28000282826796646	0	2.06933003637677316860126900936	["ModularForm/GL2/Q/holomorphic/105/19/h/a/76/78"]


# 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).