LMFDB - L-function 40-117e20-1.1-c11e20-0-0

Dirichlet series

L(s) = 1

+ 1.43e4·4^-s − 1.70e6·13^-s + 1.04e8·16^-s + 5.93e8·25^-s + 2.93e9·43^-s + 1.42e10·49^-s − 2.44e10·52^-s + 3.35e9·61^-s + 5.21e11·64^-s − 1.48e11·79^-s + 8.50e12·100^-s + 1.34e11·103^-s + 3.52e12·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s − 3.66e12·169^-s + 4.21e13·172^-s + 173^-s + 179^-s + 181^-s + ⋯

L(s) = 1

+ 7·4^-s − 1.27·13^-s + 24.9·16^-s + 12.1·25^-s + 3.04·43^-s + 7.22·49^-s − 8.93·52^-s + 0.509·61^-s + 60.6·64^-s − 5.44·79^-s + 85.0·100^-s + 1.14·103^-s + 12.3·121^-s − 2.04·169^-s + 21.3·172^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$3^{40} \cdot 13^{20}$
Sign:	$1$
Analytic conductor:	$1.18801\times 10^{39}$
Root analytic conductor:	$9.48135$
Motivic weight:	$11$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [11/2]^{20} ),\ 1 )$

Particular Values

$L(6)$	$\approx$	$40.89751870$
$L(\frac12)$	$\approx$	$40.89751870$
$L(\frac{13}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	13	$ ( 1 + 65714 p T + 1332376701 p^{3} T^{2} + 731596663000 p^{5} T^{3} + 86040315168013232 p^{7} T^{4} + 7411229450981278720 p^{10} T^{5} + 86040315168013232 p^{18} T^{6} + 731596663000 p^{27} T^{7} + 1332376701 p^{36} T^{8} + 65714 p^{45} T^{9} + p^{55} T^{10} )^{2} $
good	2	$ ( 1 - 7 p^{10} T^{2} + 1548069 p^{4} T^{4} - 442784653 p^{7} T^{6} + 413403803 p^{18} T^{8} - 402973669767 p^{19} T^{10} + 413403803 p^{40} T^{12} - 442784653 p^{51} T^{14} + 1548069 p^{70} T^{16} - 7 p^{98} T^{18} + p^{110} T^{20} )^{2} $
	5	$ ( 1 - 59313194 p T^{2} + 339341417359977 p^{3} T^{4} - $$12\!\cdots\!16$$ p^{5} T^{6} + $$36\!\cdots\!26$$ p^{7} T^{8} - $$79\!\cdots\!32$$ p^{9} T^{10} + $$36\!\cdots\!26$$ p^{29} T^{12} - $$12\!\cdots\!16$$ p^{49} T^{14} + 339341417359977 p^{69} T^{16} - 59313194 p^{89} T^{18} + p^{110} T^{20} )^{2} $
	7	$ ( 1 - 7145296162 T^{2} + 31684008358315057533 T^{4} - $$21\!\cdots\!72$$ p^{2} T^{6} + $$54\!\cdots\!18$$ p^{2} T^{8} - $$24\!\cdots\!32$$ p^{4} T^{10} + $$54\!\cdots\!18$$ p^{24} T^{12} - $$21\!\cdots\!72$$ p^{46} T^{14} + 31684008358315057533 p^{66} T^{16} - 7145296162 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	11	$ ( 1 - 1760583852022 T^{2} + $$13\!\cdots\!09$$ T^{4} - $$58\!\cdots\!36$$ T^{6} + $$17\!\cdots\!10$$ T^{8} - $$46\!\cdots\!64$$ T^{10} + $$17\!\cdots\!10$$ p^{22} T^{12} - $$58\!\cdots\!36$$ p^{44} T^{14} + $$13\!\cdots\!09$$ p^{66} T^{16} - 1760583852022 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	17	$ ( 1 + 168616416527594 T^{2} + $$14\!\cdots\!09$$ T^{4} + $$87\!\cdots\!32$$ T^{6} + $$39\!\cdots\!38$$ T^{8} + $$14\!\cdots\!16$$ T^{10} + $$39\!\cdots\!38$$ p^{22} T^{12} + $$87\!\cdots\!32$$ p^{44} T^{14} + $$14\!\cdots\!09$$ p^{66} T^{16} + 168616416527594 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	19	$ ( 1 - 614851675864042 T^{2} + $$19\!\cdots\!01$$ T^{4} - $$40\!\cdots\!24$$ T^{6} + $$65\!\cdots\!98$$ T^{8} - $$84\!\cdots\!48$$ T^{10} + $$65\!\cdots\!98$$ p^{22} T^{12} - $$40\!\cdots\!24$$ p^{44} T^{14} + $$19\!\cdots\!01$$ p^{66} T^{16} - 614851675864042 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	23	$ ( 1 + 3235272994600742 T^{2} + $$55\!\cdots\!49$$ T^{4} + $$69\!\cdots\!08$$ T^{6} + $$75\!\cdots\!26$$ T^{8} + $$75\!\cdots\!44$$ T^{10} + $$75\!\cdots\!26$$ p^{22} T^{12} + $$69\!\cdots\!08$$ p^{44} T^{14} + $$55\!\cdots\!49$$ p^{66} T^{16} + 3235272994600742 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	29	$ ( 1 + 36278838051552914 T^{2} + $$83\!\cdots\!41$$ T^{4} + $$16\!\cdots\!04$$ T^{6} + $$25\!\cdots\!46$$ T^{8} + $$32\!\cdots\!68$$ T^{10} + $$25\!\cdots\!46$$ p^{22} T^{12} + $$16\!\cdots\!04$$ p^{44} T^{14} + $$83\!\cdots\!41$$ p^{66} T^{16} + 36278838051552914 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	31	$ ( 1 - 144298591895255122 T^{2} + $$10\!\cdots\!33$$ T^{4} - $$55\!\cdots\!48$$ T^{6} + $$20\!\cdots\!42$$ T^{8} - $$59\!\cdots\!32$$ T^{10} + $$20\!\cdots\!42$$ p^{22} T^{12} - $$55\!\cdots\!48$$ p^{44} T^{14} + $$10\!\cdots\!33$$ p^{66} T^{16} - 144298591895255122 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	37	$ ( 1 - 802788737026637362 T^{2} + $$27\!\cdots\!13$$ T^{4} - $$49\!\cdots\!08$$ T^{6} + $$46\!\cdots\!42$$ T^{8} - $$34\!\cdots\!32$$ T^{10} + $$46\!\cdots\!42$$ p^{22} T^{12} - $$49\!\cdots\!08$$ p^{44} T^{14} + $$27\!\cdots\!13$$ p^{66} T^{16} - 802788737026637362 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	41	$ ( 1 - 3803956155835832266 T^{2} + $$69\!\cdots\!29$$ T^{4} - $$80\!\cdots\!48$$ T^{6} + $$68\!\cdots\!50$$ T^{8} - $$43\!\cdots\!92$$ T^{10} + $$68\!\cdots\!50$$ p^{22} T^{12} - $$80\!\cdots\!48$$ p^{44} T^{14} + $$69\!\cdots\!29$$ p^{66} T^{16} - 3803956155835832266 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	43	$ ( 1 - 734249008 T + 4510650370950044223 T^{2} - $$25\!\cdots\!44$$ T^{3} + $$82\!\cdots\!66$$ T^{4} - $$35\!\cdots\!32$$ T^{5} + $$82\!\cdots\!66$$ p^{11} T^{6} - $$25\!\cdots\!44$$ p^{22} T^{7} + 4510650370950044223 p^{33} T^{8} - 734249008 p^{44} T^{9} + p^{55} T^{10} )^{4} $
	47	$ ( 1 - 19918316678871212878 T^{2} + $$18\!\cdots\!49$$ T^{4} - $$10\!\cdots\!84$$ T^{6} + $$43\!\cdots\!22$$ T^{8} - $$12\!\cdots\!16$$ T^{10} + $$43\!\cdots\!22$$ p^{22} T^{12} - $$10\!\cdots\!84$$ p^{44} T^{14} + $$18\!\cdots\!49$$ p^{66} T^{16} - 19918316678871212878 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	53	$ ( 1 + 25179853773402024866 T^{2} + $$39\!\cdots\!09$$ T^{4} + $$32\!\cdots\!28$$ T^{6} + $$18\!\cdots\!98$$ T^{8} + $$76\!\cdots\!44$$ T^{10} + $$18\!\cdots\!98$$ p^{22} T^{12} + $$32\!\cdots\!28$$ p^{44} T^{14} + $$39\!\cdots\!09$$ p^{66} T^{16} + 25179853773402024866 p^{88} T^{18} + p^{110} T^{20} )^{2} $
	59	$ ( 1 - $$11\!\cdots\!90$$ T^{2} + $$63\!\cdots\!21$$ T^{4} - $$26\!\cdots\!36$$ T^{6} + $$99\!\cdots\!62$$ T^{8} - $$32\!\cdots\!20$$ T^{10} + $$99\!\cdots\!62$$ p^{22} T^{12} - $$26\!\cdots\!36$$ p^{44} T^{14} + $$63\!\cdots\!21$$ p^{66} T^{16} - $$11\!\cdots\!90$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	61	$ ( 1 - 839692834 T + $$13\!\cdots\!65$$ T^{2} - $$35\!\cdots\!20$$ T^{3} + $$84\!\cdots\!06$$ T^{4} - $$27\!\cdots\!28$$ T^{5} + $$84\!\cdots\!06$$ p^{11} T^{6} - $$35\!\cdots\!20$$ p^{22} T^{7} + $$13\!\cdots\!65$$ p^{33} T^{8} - 839692834 p^{44} T^{9} + p^{55} T^{10} )^{4} $
	67	$ ( 1 - $$76\!\cdots\!62$$ T^{2} + $$30\!\cdots\!45$$ T^{4} - $$77\!\cdots\!00$$ T^{6} + $$14\!\cdots\!34$$ T^{8} - $$20\!\cdots\!56$$ T^{10} + $$14\!\cdots\!34$$ p^{22} T^{12} - $$77\!\cdots\!00$$ p^{44} T^{14} + $$30\!\cdots\!45$$ p^{66} T^{16} - $$76\!\cdots\!62$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	71	$ ( 1 - $$11\!\cdots\!30$$ T^{2} + $$70\!\cdots\!81$$ T^{4} - $$30\!\cdots\!44$$ T^{6} + $$10\!\cdots\!02$$ T^{8} - $$26\!\cdots\!00$$ T^{10} + $$10\!\cdots\!02$$ p^{22} T^{12} - $$30\!\cdots\!44$$ p^{44} T^{14} + $$70\!\cdots\!81$$ p^{66} T^{16} - $$11\!\cdots\!30$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	73	$ ( 1 - $$18\!\cdots\!70$$ T^{2} + $$17\!\cdots\!13$$ T^{4} - $$10\!\cdots\!80$$ T^{6} + $$51\!\cdots\!82$$ T^{8} - $$18\!\cdots\!20$$ T^{10} + $$51\!\cdots\!82$$ p^{22} T^{12} - $$10\!\cdots\!80$$ p^{44} T^{14} + $$17\!\cdots\!13$$ p^{66} T^{16} - $$18\!\cdots\!70$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	79	$ ( 1 + 37249602404 T + $$18\!\cdots\!83$$ T^{2} + $$85\!\cdots\!84$$ p T^{3} + $$25\!\cdots\!98$$ T^{4} + $$62\!\cdots\!28$$ T^{5} + $$25\!\cdots\!98$$ p^{11} T^{6} + $$85\!\cdots\!84$$ p^{23} T^{7} + $$18\!\cdots\!83$$ p^{33} T^{8} + 37249602404 p^{44} T^{9} + p^{55} T^{10} )^{4} $
	83	$ ( 1 - $$10\!\cdots\!02$$ T^{2} + $$49\!\cdots\!69$$ T^{4} - $$14\!\cdots\!56$$ T^{6} + $$30\!\cdots\!62$$ T^{8} - $$46\!\cdots\!64$$ T^{10} + $$30\!\cdots\!62$$ p^{22} T^{12} - $$14\!\cdots\!56$$ p^{44} T^{14} + $$49\!\cdots\!69$$ p^{66} T^{16} - $$10\!\cdots\!02$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	89	$ ( 1 - $$17\!\cdots\!06$$ T^{2} + $$14\!\cdots\!13$$ T^{4} - $$81\!\cdots\!20$$ T^{6} + $$33\!\cdots\!14$$ T^{8} - $$10\!\cdots\!92$$ T^{10} + $$33\!\cdots\!14$$ p^{22} T^{12} - $$81\!\cdots\!20$$ p^{44} T^{14} + $$14\!\cdots\!13$$ p^{66} T^{16} - $$17\!\cdots\!06$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
	97	$ ( 1 - $$34\!\cdots\!82$$ T^{2} + $$69\!\cdots\!33$$ T^{4} - $$95\!\cdots\!08$$ T^{6} + $$98\!\cdots\!22$$ T^{8} - $$79\!\cdots\!72$$ T^{10} + $$98\!\cdots\!22$$ p^{22} T^{12} - $$95\!\cdots\!08$$ p^{44} T^{14} + $$69\!\cdots\!33$$ p^{66} T^{16} - $$34\!\cdots\!82$$ p^{88} T^{18} + p^{110} T^{20} )^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−1.79093131426332470992856236552, −1.72699200029537545557637554968, −1.65880578401681489236153901001, −1.50728726285768694954926247405, −1.38234835731894926946605276518, −1.35068716112911141448158747089, −1.32307668505625585069540084070, −1.14850040294695335526913071017, −1.11963302287260201685349991397, −1.10420257691164125730264346674, −1.03959169100172101849727884766, −0.978954109584008093552250578664, −0.953646232355366916823013326040, −0.953246846716437254848193601616, −0.845031130732693362681701521144, −0.71787598647818502874776105613, −0.66139090889596746294508724203, −0.64730500143762339438765944343, −0.52564770450814157592494959907, −0.50553458840398305344561717964, −0.44971786850280528192799850654, −0.39758962073824696805278366566, −0.36622718670370748872945230711, −0.16943605283748006083619538667, −0.00554531324467586922044425104, 0.00554531324467586922044425104, 0.16943605283748006083619538667, 0.36622718670370748872945230711, 0.39758962073824696805278366566, 0.44971786850280528192799850654, 0.50553458840398305344561717964, 0.52564770450814157592494959907, 0.64730500143762339438765944343, 0.66139090889596746294508724203, 0.71787598647818502874776105613, 0.845031130732693362681701521144, 0.953246846716437254848193601616, 0.953646232355366916823013326040, 0.978954109584008093552250578664, 1.03959169100172101849727884766, 1.10420257691164125730264346674, 1.11963302287260201685349991397, 1.14850040294695335526913071017, 1.32307668505625585069540084070, 1.35068716112911141448158747089, 1.38234835731894926946605276518, 1.50728726285768694954926247405, 1.65880578401681489236153901001, 1.72699200029537545557637554968, 1.79093131426332470992856236552

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 $\F_{q}$

\(L(6)\)	\(\approx\)	\(40.89751870\)
\(L(\frac12)\)	\(\approx\)	\(40.89751870\)
\(L(\frac{13}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more