LMFDB - L-function 24-76e12-1.1-c8e12-0-0

Dirichlet series

L(s) = 1

− 568·5^-s + 1.73e3·7^-s + 1.62e4·9^-s + 1.97e3·11^-s − 4.74e4·17^-s − 1.13e5·19^-s + 3.78e5·23^-s − 2.43e6·25^-s − 9.84e5·35^-s + 6.23e6·43^-s − 9.23e6·45^-s + 4.90e6·47^-s − 4.14e7·49^-s − 1.12e6·55^-s + 1.66e7·61^-s + 2.81e7·63^-s + 1.11e8·73^-s + 3.42e6·77^-s + 1.04e8·81^-s + 3.40e7·83^-s + 2.69e7·85^-s + 6.42e7·95^-s + 3.21e7·99^-s − 1.67e7·101^-s − 2.15e8·115^-s − 8.22e7·119^-s − 1.38e9·121^-s + ⋯

L(s) = 1

− 0.908·5^-s + 0.722·7^-s + 2.47·9^-s + 0.134·11^-s − 0.568·17^-s − 0.868·19^-s + 1.35·23^-s − 6.23·25^-s − 0.656·35^-s + 1.82·43^-s − 2.25·45^-s + 1.00·47^-s − 7.19·49^-s − 0.122·55^-s + 1.19·61^-s + 1.78·63^-s + 3.91·73^-s + 0.0974·77^-s + 2.43·81^-s + 0.717·83^-s + 0.516·85^-s + 0.789·95^-s + 0.334·99^-s − 0.161·101^-s − 1.23·115^-s − 0.410·119^-s − 6.46·121^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$2^{24} \cdot 19^{12}$
Sign:	$1$
Analytic conductor:	$7.75785\times 10^{17}$
Root analytic conductor:	$5.56424$
Motivic weight:	$8$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 2^{24} \cdot 19^{12} ,\ ( \ : [4]^{12} ),\ 1 )$

Particular Values

$L(\frac{9}{2})$	$\approx$	$0.6345821044$
$L(\frac12)$	$\approx$	$0.6345821044$
$L(5)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	19	$ 1 + 113178 T - 6501350742 T^{2} + 1805751996714 p^{2} T^{3} + 34778217524877788 p^{3} T^{4} + 9046105570805971968 p^{5} T^{5} + $$21\!\cdots\!76$$ p^{8} T^{6} + 9046105570805971968 p^{13} T^{7} + 34778217524877788 p^{19} T^{8} + 1805751996714 p^{26} T^{9} - 6501350742 p^{32} T^{10} + 113178 p^{40} T^{11} + p^{48} T^{12} $
good	3	$ 1 - 16261 T^{2} + 5911421 p^{3} T^{4} - 28833700220 p^{3} T^{6} + 172529042280911 p^{3} T^{8} - 145502182283431597 p^{5} T^{10} + $$14\!\cdots\!02$$ p^{7} T^{12} - 145502182283431597 p^{21} T^{14} + 172529042280911 p^{35} T^{16} - 28833700220 p^{51} T^{18} + 5911421 p^{67} T^{20} - 16261 p^{80} T^{22} + p^{96} T^{24} $
	5	$ ( 1 + 284 T + 1337867 T^{2} + 91352806 p T^{3} + 1626883067 p^{4} T^{4} + 2511363517414 p^{3} T^{5} + 773417238473762 p^{4} T^{6} + 2511363517414 p^{11} T^{7} + 1626883067 p^{20} T^{8} + 91352806 p^{25} T^{9} + 1337867 p^{32} T^{10} + 284 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	7	$ ( 1 - 867 T + 3122046 p T^{2} - 466999629 p^{2} T^{3} + 723448749840 p^{3} T^{4} - 104098352967927 p^{4} T^{5} + 104107732575560992 p^{5} T^{6} - 104098352967927 p^{12} T^{7} + 723448749840 p^{19} T^{8} - 466999629 p^{26} T^{9} + 3122046 p^{33} T^{10} - 867 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	11	$ ( 1 - 988 T + 694598813 T^{2} - 1349767625944 T^{3} + 274246782689007851 T^{4} - $$61\!\cdots\!60$$ T^{5} + $$69\!\cdots\!42$$ T^{6} - $$61\!\cdots\!60$$ p^{8} T^{7} + 274246782689007851 p^{16} T^{8} - 1349767625944 p^{24} T^{9} + 694598813 p^{32} T^{10} - 988 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	13	$ 1 - 4824512301 T^{2} + 12161211899503641879 T^{4} - $$20\!\cdots\!72$$ T^{6} + $$27\!\cdots\!05$$ T^{8} - $$28\!\cdots\!67$$ T^{10} + $$25\!\cdots\!50$$ T^{12} - $$28\!\cdots\!67$$ p^{16} T^{14} + $$27\!\cdots\!05$$ p^{32} T^{16} - $$20\!\cdots\!72$$ p^{48} T^{18} + 12161211899503641879 p^{64} T^{20} - 4824512301 p^{80} T^{22} + p^{96} T^{24} $
	17	$ ( 1 + 23723 T + 18072854390 T^{2} + 939136295854493 T^{3} + $$20\!\cdots\!52$$ T^{4} + $$69\!\cdots\!39$$ p T^{5} + $$16\!\cdots\!60$$ T^{6} + $$69\!\cdots\!39$$ p^{9} T^{7} + $$20\!\cdots\!52$$ p^{16} T^{8} + 939136295854493 p^{24} T^{9} + 18072854390 p^{32} T^{10} + 23723 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	23	$ ( 1 - 189475 T + 266949298487 T^{2} - 1566646252588700 p T^{3} + $$34\!\cdots\!89$$ T^{4} - $$39\!\cdots\!25$$ T^{5} + $$31\!\cdots\!86$$ T^{6} - $$39\!\cdots\!25$$ p^{8} T^{7} + $$34\!\cdots\!89$$ p^{16} T^{8} - 1566646252588700 p^{25} T^{9} + 266949298487 p^{32} T^{10} - 189475 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	29	$ 1 + 145504773267 T^{2} + $$47\!\cdots\!67$$ T^{4} - $$66\!\cdots\!84$$ p T^{6} + $$88\!\cdots\!73$$ T^{8} - $$83\!\cdots\!91$$ T^{10} + $$30\!\cdots\!18$$ T^{12} - $$83\!\cdots\!91$$ p^{16} T^{14} + $$88\!\cdots\!73$$ p^{32} T^{16} - $$66\!\cdots\!84$$ p^{49} T^{18} + $$47\!\cdots\!67$$ p^{64} T^{20} + 145504773267 p^{80} T^{22} + p^{96} T^{24} $
	31	$ 1 - 881776438872 T^{2} + $$61\!\cdots\!46$$ T^{4} - $$53\!\cdots\!04$$ T^{6} + $$13\!\cdots\!07$$ T^{8} - $$70\!\cdots\!84$$ T^{10} + $$48\!\cdots\!72$$ T^{12} - $$70\!\cdots\!84$$ p^{16} T^{14} + $$13\!\cdots\!07$$ p^{32} T^{16} - $$53\!\cdots\!04$$ p^{48} T^{18} + $$61\!\cdots\!46$$ p^{64} T^{20} - 881776438872 p^{80} T^{22} + p^{96} T^{24} $
	37	$ 1 - 18452595723672 T^{2} + $$17\!\cdots\!46$$ T^{4} - $$11\!\cdots\!56$$ T^{6} + $$62\!\cdots\!59$$ T^{8} - $$27\!\cdots\!44$$ T^{10} + $$10\!\cdots\!92$$ T^{12} - $$27\!\cdots\!44$$ p^{16} T^{14} + $$62\!\cdots\!59$$ p^{32} T^{16} - $$11\!\cdots\!56$$ p^{48} T^{18} + $$17\!\cdots\!46$$ p^{64} T^{20} - 18452595723672 p^{80} T^{22} + p^{96} T^{24} $
	41	$ 1 - 28309655963544 T^{2} + $$47\!\cdots\!34$$ T^{4} - $$59\!\cdots\!48$$ T^{6} + $$59\!\cdots\!75$$ T^{8} - $$51\!\cdots\!68$$ T^{10} + $$41\!\cdots\!00$$ T^{12} - $$51\!\cdots\!68$$ p^{16} T^{14} + $$59\!\cdots\!75$$ p^{32} T^{16} - $$59\!\cdots\!48$$ p^{48} T^{18} + $$47\!\cdots\!34$$ p^{64} T^{20} - 28309655963544 p^{80} T^{22} + p^{96} T^{24} $
	43	$ ( 1 - 3117198 T + 41317029786093 T^{2} - $$13\!\cdots\!48$$ T^{3} + $$70\!\cdots\!67$$ T^{4} - $$24\!\cdots\!78$$ T^{5} + $$83\!\cdots\!78$$ T^{6} - $$24\!\cdots\!78$$ p^{8} T^{7} + $$70\!\cdots\!67$$ p^{16} T^{8} - $$13\!\cdots\!48$$ p^{24} T^{9} + 41317029786093 p^{32} T^{10} - 3117198 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	47	$ ( 1 - 2451112 T + 67737990071693 T^{2} + 17683070692706908784 T^{3} + $$19\!\cdots\!07$$ T^{4} + $$47\!\cdots\!68$$ T^{5} + $$44\!\cdots\!18$$ T^{6} + $$47\!\cdots\!68$$ p^{8} T^{7} + $$19\!\cdots\!07$$ p^{16} T^{8} + 17683070692706908784 p^{24} T^{9} + 67737990071693 p^{32} T^{10} - 2451112 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	53	$ 1 - 358790076570237 T^{2} + $$56\!\cdots\!99$$ T^{4} - $$52\!\cdots\!44$$ T^{6} + $$36\!\cdots\!69$$ T^{8} - $$22\!\cdots\!35$$ T^{10} + $$14\!\cdots\!34$$ T^{12} - $$22\!\cdots\!35$$ p^{16} T^{14} + $$36\!\cdots\!69$$ p^{32} T^{16} - $$52\!\cdots\!44$$ p^{48} T^{18} + $$56\!\cdots\!99$$ p^{64} T^{20} - 358790076570237 p^{80} T^{22} + p^{96} T^{24} $
	59	$ 1 - 370046691017349 T^{2} + $$53\!\cdots\!71$$ T^{4} - $$53\!\cdots\!60$$ T^{6} + $$10\!\cdots\!17$$ T^{8} - $$26\!\cdots\!45$$ p T^{10} + $$18\!\cdots\!90$$ T^{12} - $$26\!\cdots\!45$$ p^{17} T^{14} + $$10\!\cdots\!17$$ p^{32} T^{16} - $$53\!\cdots\!60$$ p^{48} T^{18} + $$53\!\cdots\!71$$ p^{64} T^{20} - 370046691017349 p^{80} T^{22} + p^{96} T^{24} $
	61	$ ( 1 - 8301456 T + 881255758655211 T^{2} - $$85\!\cdots\!26$$ T^{3} + $$34\!\cdots\!91$$ T^{4} - $$34\!\cdots\!38$$ T^{5} + $$83\!\cdots\!82$$ T^{6} - $$34\!\cdots\!38$$ p^{8} T^{7} + $$34\!\cdots\!91$$ p^{16} T^{8} - $$85\!\cdots\!26$$ p^{24} T^{9} + 881255758655211 p^{32} T^{10} - 8301456 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	67	$ 1 - 2797688115026589 T^{2} + $$36\!\cdots\!39$$ T^{4} - $$29\!\cdots\!68$$ T^{6} + $$16\!\cdots\!85$$ T^{8} - $$73\!\cdots\!63$$ T^{10} + $$29\!\cdots\!90$$ T^{12} - $$73\!\cdots\!63$$ p^{16} T^{14} + $$16\!\cdots\!85$$ p^{32} T^{16} - $$29\!\cdots\!68$$ p^{48} T^{18} + $$36\!\cdots\!39$$ p^{64} T^{20} - 2797688115026589 p^{80} T^{22} + p^{96} T^{24} $
	71	$ 1 - 3712748008080624 T^{2} + $$71\!\cdots\!50$$ T^{4} - $$94\!\cdots\!28$$ T^{6} + $$94\!\cdots\!51$$ T^{8} - $$78\!\cdots\!28$$ T^{10} + $$54\!\cdots\!36$$ T^{12} - $$78\!\cdots\!28$$ p^{16} T^{14} + $$94\!\cdots\!51$$ p^{32} T^{16} - $$94\!\cdots\!28$$ p^{48} T^{18} + $$71\!\cdots\!50$$ p^{64} T^{20} - 3712748008080624 p^{80} T^{22} + p^{96} T^{24} $
	73	$ ( 1 - 55634307 T + 2749262694335910 T^{2} - $$10\!\cdots\!77$$ T^{3} + $$38\!\cdots\!92$$ T^{4} - $$12\!\cdots\!67$$ T^{5} + $$37\!\cdots\!60$$ T^{6} - $$12\!\cdots\!67$$ p^{8} T^{7} + $$38\!\cdots\!92$$ p^{16} T^{8} - $$10\!\cdots\!77$$ p^{24} T^{9} + 2749262694335910 p^{32} T^{10} - 55634307 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	79	$ 1 - 9556064147061720 T^{2} + $$45\!\cdots\!14$$ T^{4} - $$14\!\cdots\!40$$ T^{6} + $$35\!\cdots\!67$$ T^{8} - $$68\!\cdots\!08$$ T^{10} + $$11\!\cdots\!12$$ T^{12} - $$68\!\cdots\!08$$ p^{16} T^{14} + $$35\!\cdots\!67$$ p^{32} T^{16} - $$14\!\cdots\!40$$ p^{48} T^{18} + $$45\!\cdots\!14$$ p^{64} T^{20} - 9556064147061720 p^{80} T^{22} + p^{96} T^{24} $
	83	$ ( 1 - 17015194 T + 7178210233571186 T^{2} - $$11\!\cdots\!26$$ T^{3} + $$24\!\cdots\!59$$ T^{4} - $$34\!\cdots\!08$$ T^{5} + $$57\!\cdots\!48$$ T^{6} - $$34\!\cdots\!08$$ p^{8} T^{7} + $$24\!\cdots\!59$$ p^{16} T^{8} - $$11\!\cdots\!26$$ p^{24} T^{9} + 7178210233571186 p^{32} T^{10} - 17015194 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	89	$ 1 - 31951793479202196 T^{2} + $$48\!\cdots\!54$$ T^{4} - $$47\!\cdots\!32$$ T^{6} + $$33\!\cdots\!95$$ T^{8} - $$18\!\cdots\!32$$ T^{10} + $$78\!\cdots\!80$$ T^{12} - $$18\!\cdots\!32$$ p^{16} T^{14} + $$33\!\cdots\!95$$ p^{32} T^{16} - $$47\!\cdots\!32$$ p^{48} T^{18} + $$48\!\cdots\!54$$ p^{64} T^{20} - 31951793479202196 p^{80} T^{22} + p^{96} T^{24} $
	97	$ 1 - 36241612514299152 T^{2} + $$66\!\cdots\!86$$ T^{4} - $$83\!\cdots\!84$$ T^{6} + $$87\!\cdots\!27$$ T^{8} - $$82\!\cdots\!04$$ T^{10} + $$68\!\cdots\!92$$ T^{12} - $$82\!\cdots\!04$$ p^{16} T^{14} + $$87\!\cdots\!27$$ p^{32} T^{16} - $$83\!\cdots\!84$$ p^{48} T^{18} + $$66\!\cdots\!86$$ p^{64} T^{20} - 36241612514299152 p^{80} T^{22} + p^{96} T^{24} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.73465267578108417279905076556, −3.45565005947076058192331600974, −3.28555886083838821420216535523, −3.20613497413139730532883495371, −3.06073758746280504617160129325, −2.82566371631592555852568466472, −2.63291737418777743335360616609, −2.57869723960626517206353229758, −2.30190677382528636595784684115, −2.20123848114354042416285691034, −2.08729005371112591744247555478, −2.00180225405666464485552342023, −1.81062430972215880891427518217, −1.73396186688088441913820095147, −1.59167292129891819353078148544, −1.57415714823088237811916439616, −1.31796211020353894747642226186, −1.04085052994544774689301799381, −0.998945569695833488951140171439, −0.907373421810304308429125255616, −0.883463954204654734289100362288, −0.40825508420211105941500155337, −0.17111537231394259923913301400, −0.16607627143726588730422098666, −0.12897191659208220293611682366, 0.12897191659208220293611682366, 0.16607627143726588730422098666, 0.17111537231394259923913301400, 0.40825508420211105941500155337, 0.883463954204654734289100362288, 0.907373421810304308429125255616, 0.998945569695833488951140171439, 1.04085052994544774689301799381, 1.31796211020353894747642226186, 1.57415714823088237811916439616, 1.59167292129891819353078148544, 1.73396186688088441913820095147, 1.81062430972215880891427518217, 2.00180225405666464485552342023, 2.08729005371112591744247555478, 2.20123848114354042416285691034, 2.30190677382528636595784684115, 2.57869723960626517206353229758, 2.63291737418777743335360616609, 2.82566371631592555852568466472, 3.06073758746280504617160129325, 3.20613497413139730532883495371, 3.28555886083838821420216535523, 3.45565005947076058192331600974, 3.73465267578108417279905076556

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{9}{2})\)	\(\approx\)	\(0.6345821044\)
\(L(\frac12)\)	\(\approx\)	\(0.6345821044\)
\(L(5)\)		not available
\(L(1)\)		not available

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