LMFDB - L-function 24-240e12-1.1-c8e12-0-3

Dirichlet series

L(s) = 1

− 1.31e4·9^-s − 4.16e4·13^-s + 4.40e4·17^-s + 4.68e5·25^-s − 1.52e6·29^-s − 6.65e5·37^-s − 3.35e6·41^-s + 3.24e7·49^-s − 8.85e6·53^-s + 1.16e7·61^-s + 1.19e8·73^-s + 1.00e8·81^-s − 1.62e8·89^-s + 1.25e7·97^-s − 4.84e8·101^-s + 2.38e8·109^-s + 2.82e8·113^-s + 5.46e8·117^-s + 1.07e9·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s − 5.77e8·153^-s + 157^-s + ⋯

L(s) = 1

− 2·9^-s − 1.45·13^-s + 0.527·17^-s + 6/5·25^-s − 2.15·29^-s − 0.355·37^-s − 1.18·41^-s + 5.62·49^-s − 1.12·53^-s + 0.843·61^-s + 4.20·73^-s + 7/3·81^-s − 2.58·89^-s + 0.141·97^-s − 4.65·101^-s + 1.68·109^-s + 1.73·113^-s + 2.91·117^-s + 5.02·121^-s − 1.05·153^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$24$
Conductor:	$2^{48} \cdot 3^{12} \cdot 5^{12}$
Sign:	$1$
Analytic conductor:	$7.62980\times 10^{23}$
Root analytic conductor:	$9.88791$
Motivic weight:	$8$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [4]^{12} ),\ 1 )$

Particular Values

$L(\frac{9}{2})$	$\approx$	$1.654802353$
$L(\frac12)$	$\approx$	$1.654802353$
$L(5)$		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 $
	3	$ ( 1 + p^{7} T^{2} )^{6} $
	5	$ ( 1 - p^{7} T^{2} )^{6} $
good	7	$ 1 - 32421084 T^{2} + 554742743803746 T^{4} - 18731690762008285300 p^{3} T^{6} + $$33\!\cdots\!45$$ p^{5} T^{8} - $$34\!\cdots\!76$$ p^{6} T^{10} + $$43\!\cdots\!76$$ p^{8} T^{12} - $$34\!\cdots\!76$$ p^{22} T^{14} + $$33\!\cdots\!45$$ p^{37} T^{16} - 18731690762008285300 p^{51} T^{18} + 554742743803746 p^{64} T^{20} - 32421084 p^{80} T^{22} + p^{96} T^{24} $
	11	$ 1 - 97933140 p T^{2} + 47993422768917606 p T^{4} - $$13\!\cdots\!00$$ T^{6} + $$14\!\cdots\!15$$ T^{8} + $$28\!\cdots\!00$$ T^{10} - $$12\!\cdots\!80$$ T^{12} + $$28\!\cdots\!00$$ p^{16} T^{14} + $$14\!\cdots\!15$$ p^{32} T^{16} - $$13\!\cdots\!00$$ p^{48} T^{18} + 47993422768917606 p^{65} T^{20} - 97933140 p^{81} T^{22} + p^{96} T^{24} $
	13	$ ( 1 + 20808 T + 1677312090 T^{2} + 48484875032104 T^{3} + 160154655924163275 p T^{4} + $$55\!\cdots\!00$$ T^{5} + $$22\!\cdots\!08$$ T^{6} + $$55\!\cdots\!00$$ p^{8} T^{7} + 160154655924163275 p^{17} T^{8} + 48484875032104 p^{24} T^{9} + 1677312090 p^{32} T^{10} + 20808 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	17	$ ( 1 - 22008 T + 6722973690 T^{2} - 1252911918597144 T^{3} + 60707469767046336015 T^{4} - $$40\!\cdots\!20$$ T^{5} + $$10\!\cdots\!28$$ T^{6} - $$40\!\cdots\!20$$ p^{8} T^{7} + 60707469767046336015 p^{16} T^{8} - 1252911918597144 p^{24} T^{9} + 6722973690 p^{32} T^{10} - 22008 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	19	$ 1 - 94927903404 T^{2} + $$42\!\cdots\!26$$ T^{4} - $$12\!\cdots\!00$$ T^{6} + $$31\!\cdots\!15$$ T^{8} - $$69\!\cdots\!24$$ T^{10} + $$12\!\cdots\!56$$ T^{12} - $$69\!\cdots\!24$$ p^{16} T^{14} + $$31\!\cdots\!15$$ p^{32} T^{16} - $$12\!\cdots\!00$$ p^{48} T^{18} + $$42\!\cdots\!26$$ p^{64} T^{20} - 94927903404 p^{80} T^{22} + p^{96} T^{24} $
	23	$ 1 - 426086965260 T^{2} + $$10\!\cdots\!66$$ T^{4} - $$18\!\cdots\!00$$ T^{6} + $$10\!\cdots\!05$$ p T^{8} - $$25\!\cdots\!00$$ T^{10} + $$21\!\cdots\!20$$ T^{12} - $$25\!\cdots\!00$$ p^{16} T^{14} + $$10\!\cdots\!05$$ p^{33} T^{16} - $$18\!\cdots\!00$$ p^{48} T^{18} + $$10\!\cdots\!66$$ p^{64} T^{20} - 426086965260 p^{80} T^{22} + p^{96} T^{24} $
	29	$ ( 1 + 26232 p T + 1065384937626 T^{2} + 658176041636737080 T^{3} + $$73\!\cdots\!95$$ T^{4} + $$40\!\cdots\!08$$ T^{5} + $$42\!\cdots\!44$$ T^{6} + $$40\!\cdots\!08$$ p^{8} T^{7} + $$73\!\cdots\!95$$ p^{16} T^{8} + 658176041636737080 p^{24} T^{9} + 1065384937626 p^{32} T^{10} + 26232 p^{41} T^{11} + p^{48} T^{12} )^{2} $
	31	$ 1 - 6595816306092 T^{2} + $$21\!\cdots\!46$$ T^{4} - $$47\!\cdots\!20$$ T^{6} + $$76\!\cdots\!95$$ T^{8} - $$30\!\cdots\!32$$ p T^{10} + $$93\!\cdots\!44$$ p^{2} T^{12} - $$30\!\cdots\!32$$ p^{17} T^{14} + $$76\!\cdots\!95$$ p^{32} T^{16} - $$47\!\cdots\!20$$ p^{48} T^{18} + $$21\!\cdots\!46$$ p^{64} T^{20} - 6595816306092 p^{80} T^{22} + p^{96} T^{24} $
	37	$ ( 1 + 332880 T + 10215036288426 T^{2} - 4504320213110921200 T^{3} + $$54\!\cdots\!15$$ T^{4} - $$36\!\cdots\!00$$ T^{5} + $$22\!\cdots\!20$$ T^{6} - $$36\!\cdots\!00$$ p^{8} T^{7} + $$54\!\cdots\!15$$ p^{16} T^{8} - 4504320213110921200 p^{24} T^{9} + 10215036288426 p^{32} T^{10} + 332880 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	41	$ ( 1 + 1676100 T + 34043757958626 T^{2} + 58450899471520456500 T^{3} + $$51\!\cdots\!15$$ T^{4} + $$86\!\cdots\!00$$ T^{5} + $$48\!\cdots\!20$$ T^{6} + $$86\!\cdots\!00$$ p^{8} T^{7} + $$51\!\cdots\!15$$ p^{16} T^{8} + 58450899471520456500 p^{24} T^{9} + 34043757958626 p^{32} T^{10} + 1676100 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	43	$ 1 - 41811474239340 T^{2} + $$73\!\cdots\!06$$ T^{4} - $$10\!\cdots\!00$$ T^{6} + $$19\!\cdots\!15$$ T^{8} - $$26\!\cdots\!00$$ T^{10} + $$30\!\cdots\!20$$ T^{12} - $$26\!\cdots\!00$$ p^{16} T^{14} + $$19\!\cdots\!15$$ p^{32} T^{16} - $$10\!\cdots\!00$$ p^{48} T^{18} + $$73\!\cdots\!06$$ p^{64} T^{20} - 41811474239340 p^{80} T^{22} + p^{96} T^{24} $
	47	$ 1 - 134886087454284 T^{2} + $$85\!\cdots\!66$$ T^{4} - $$35\!\cdots\!00$$ T^{6} + $$11\!\cdots\!15$$ T^{8} - $$34\!\cdots\!24$$ T^{10} + $$90\!\cdots\!76$$ T^{12} - $$34\!\cdots\!24$$ p^{16} T^{14} + $$11\!\cdots\!15$$ p^{32} T^{16} - $$35\!\cdots\!00$$ p^{48} T^{18} + $$85\!\cdots\!66$$ p^{64} T^{20} - 134886087454284 p^{80} T^{22} + p^{96} T^{24} $
	53	$ ( 1 + 4427616 T + 278503271311722 T^{2} + $$87\!\cdots\!92$$ T^{3} + $$34\!\cdots\!23$$ T^{4} + $$79\!\cdots\!72$$ T^{5} + $$26\!\cdots\!68$$ T^{6} + $$79\!\cdots\!72$$ p^{8} T^{7} + $$34\!\cdots\!23$$ p^{16} T^{8} + $$87\!\cdots\!92$$ p^{24} T^{9} + 278503271311722 p^{32} T^{10} + 4427616 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	59	$ 1 - 1106855512903260 T^{2} + $$63\!\cdots\!46$$ T^{4} - $$23\!\cdots\!00$$ T^{6} + $$66\!\cdots\!15$$ T^{8} - $$14\!\cdots\!00$$ T^{10} + $$23\!\cdots\!20$$ T^{12} - $$14\!\cdots\!00$$ p^{16} T^{14} + $$66\!\cdots\!15$$ p^{32} T^{16} - $$23\!\cdots\!00$$ p^{48} T^{18} + $$63\!\cdots\!46$$ p^{64} T^{20} - 1106855512903260 p^{80} T^{22} + p^{96} T^{24} $
	61	$ ( 1 - 5842164 T + 655959614740626 T^{2} - $$37\!\cdots\!00$$ T^{3} + $$23\!\cdots\!15$$ T^{4} - $$11\!\cdots\!24$$ T^{5} + $$53\!\cdots\!96$$ T^{6} - $$11\!\cdots\!24$$ p^{8} T^{7} + $$23\!\cdots\!15$$ p^{16} T^{8} - $$37\!\cdots\!00$$ p^{24} T^{9} + 655959614740626 p^{32} T^{10} - 5842164 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	67	$ 1 - 2013921573672684 T^{2} + $$22\!\cdots\!26$$ T^{4} - $$18\!\cdots\!00$$ T^{6} + $$11\!\cdots\!15$$ T^{8} - $$61\!\cdots\!24$$ T^{10} + $$27\!\cdots\!76$$ T^{12} - $$61\!\cdots\!24$$ p^{16} T^{14} + $$11\!\cdots\!15$$ p^{32} T^{16} - $$18\!\cdots\!00$$ p^{48} T^{18} + $$22\!\cdots\!26$$ p^{64} T^{20} - 2013921573672684 p^{80} T^{22} + p^{96} T^{24} $
	71	$ 1 - 6409446080106060 T^{2} + $$19\!\cdots\!26$$ T^{4} - $$35\!\cdots\!00$$ T^{6} + $$45\!\cdots\!15$$ T^{8} - $$43\!\cdots\!00$$ T^{10} + $$32\!\cdots\!20$$ T^{12} - $$43\!\cdots\!00$$ p^{16} T^{14} + $$45\!\cdots\!15$$ p^{32} T^{16} - $$35\!\cdots\!00$$ p^{48} T^{18} + $$19\!\cdots\!26$$ p^{64} T^{20} - 6409446080106060 p^{80} T^{22} + p^{96} T^{24} $
	73	$ ( 1 - 59728548 T + 4139176759667250 T^{2} - $$16\!\cdots\!84$$ T^{3} + $$68\!\cdots\!35$$ T^{4} - $$20\!\cdots\!20$$ T^{5} + $$67\!\cdots\!88$$ T^{6} - $$20\!\cdots\!20$$ p^{8} T^{7} + $$68\!\cdots\!35$$ p^{16} T^{8} - $$16\!\cdots\!84$$ p^{24} T^{9} + 4139176759667250 p^{32} T^{10} - 59728548 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	79	$ 1 - 13031322915074604 T^{2} + $$10\!\cdots\!54$$ p T^{4} - $$32\!\cdots\!00$$ T^{6} + $$90\!\cdots\!15$$ T^{8} - $$19\!\cdots\!24$$ T^{10} + $$33\!\cdots\!56$$ T^{12} - $$19\!\cdots\!24$$ p^{16} T^{14} + $$90\!\cdots\!15$$ p^{32} T^{16} - $$32\!\cdots\!00$$ p^{48} T^{18} + $$10\!\cdots\!54$$ p^{65} T^{20} - 13031322915074604 p^{80} T^{22} + p^{96} T^{24} $
	83	$ 1 - 14529347128037100 T^{2} + $$11\!\cdots\!86$$ T^{4} - $$59\!\cdots\!00$$ T^{6} + $$23\!\cdots\!15$$ T^{8} - $$72\!\cdots\!00$$ T^{10} + $$18\!\cdots\!20$$ T^{12} - $$72\!\cdots\!00$$ p^{16} T^{14} + $$23\!\cdots\!15$$ p^{32} T^{16} - $$59\!\cdots\!00$$ p^{48} T^{18} + $$11\!\cdots\!86$$ p^{64} T^{20} - 14529347128037100 p^{80} T^{22} + p^{96} T^{24} $
	89	$ ( 1 + 81221196 T + 19382128149455826 T^{2} + $$11\!\cdots\!00$$ T^{3} + $$15\!\cdots\!15$$ T^{4} + $$76\!\cdots\!76$$ T^{5} + $$77\!\cdots\!56$$ T^{6} + $$76\!\cdots\!76$$ p^{8} T^{7} + $$15\!\cdots\!15$$ p^{16} T^{8} + $$11\!\cdots\!00$$ p^{24} T^{9} + 19382128149455826 p^{32} T^{10} + 81221196 p^{40} T^{11} + p^{48} T^{12} )^{2} $
	97	$ ( 1 - 6254844 T + 30808955255531682 T^{2} - $$10\!\cdots\!28$$ T^{3} + $$41\!\cdots\!63$$ T^{4} - $$21\!\cdots\!68$$ T^{5} + $$37\!\cdots\!08$$ T^{6} - $$21\!\cdots\!68$$ p^{8} T^{7} + $$41\!\cdots\!63$$ p^{16} T^{8} - $$10\!\cdots\!28$$ p^{24} T^{9} + 30808955255531682 p^{32} T^{10} - 6254844 p^{40} T^{11} + p^{48} T^{12} )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−2.58339475351512624186548518392, −2.57605854023446247975252911630, −2.51359518182734026312033713193, −2.44631680627292310561870494848, −2.25885010741083450670831282828, −2.21924371068344999778348246281, −2.19983906916056997933915847544, −2.04369303556354111626394615404, −2.00363135444026466895157838096, −1.74185401923045111668894766724, −1.59473667169619970213383079705, −1.48451676590606331601379135781, −1.45190902073699999355919668662, −1.30864878602342066188437756159, −1.15028814524935139724519485503, −1.03946159033552533764950707606, −0.909224192318237011795478150978, −0.901934337503687234978010112417, −0.68844259587791985882378386815, −0.53119825501001265058058529079, −0.50246314787558435136525981670, −0.44875948414707444248672665872, −0.26490954737372195505519725674, −0.13883380705690155691149184467, −0.079914345368298142970872618503, 0.079914345368298142970872618503, 0.13883380705690155691149184467, 0.26490954737372195505519725674, 0.44875948414707444248672665872, 0.50246314787558435136525981670, 0.53119825501001265058058529079, 0.68844259587791985882378386815, 0.901934337503687234978010112417, 0.909224192318237011795478150978, 1.03946159033552533764950707606, 1.15028814524935139724519485503, 1.30864878602342066188437756159, 1.45190902073699999355919668662, 1.48451676590606331601379135781, 1.59473667169619970213383079705, 1.74185401923045111668894766724, 2.00363135444026466895157838096, 2.04369303556354111626394615404, 2.19983906916056997933915847544, 2.21924371068344999778348246281, 2.25885010741083450670831282828, 2.44631680627292310561870494848, 2.51359518182734026312033713193, 2.57605854023446247975252911630, 2.58339475351512624186548518392

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 $\F_{q}$

\(L(\frac{9}{2})\)	\(\approx\)	\(1.654802353\)
\(L(\frac12)\)	\(\approx\)	\(1.654802353\)
\(L(5)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more