LMFDB - L-function 8-3e8-1.1-c37e4-0-0

Dirichlet series

L(s) = 1

− 4.37e5·2^-s − 6.09e9·4^-s + 4.09e12·5^-s + 6.60e15·7^-s + 3.81e15·8^-s − 1.79e18·10^-s − 2.09e19·11^-s + 5.18e19·13^-s − 2.89e21·14^-s + 1.05e22·16^-s − 8.11e22·17^-s − 5.44e23·19^-s − 2.50e22·20^-s + 9.16e24·22^-s + 6.10e24·23^-s − 8.94e25·25^-s − 2.26e25·26^-s − 4.02e25·28^-s − 4.17e27·29^-s + 8.99e27·31^-s − 5.09e27·32^-s + 3.54e28·34^-s + 2.70e28·35^-s + 5.55e28·37^-s + 2.38e29·38^-s + 1.56e28·40^-s + 8.69e29·41^-s + ⋯

L(s) = 1

− 1.18·2^-s − 0.0443·4^-s + 0.480·5^-s + 1.53·7^-s + 0.0748·8^-s − 0.567·10^-s − 1.13·11^-s + 0.127·13^-s − 1.80·14^-s + 0.560·16^-s − 1.39·17^-s − 1.19·19^-s − 0.0213·20^-s + 1.34·22^-s + 0.392·23^-s − 1.22·25^-s − 0.150·26^-s − 0.0680·28^-s − 3.68·29^-s + 2.31·31^-s − 0.727·32^-s + 1.65·34^-s + 0.736·35^-s + 0.540·37^-s + 1.41·38^-s + 0.0359·40^-s + 1.26·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+37/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$6561$ = $3^{8}$
Sign:	$1$
Analytic conductor:	$3.70960\times 10^{7}$
Root analytic conductor:	$8.83417$
Motivic weight:	$37$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 6561,\ (\ :37/2, 37/2, 37/2, 37/2),\ 1)$

Particular Values

$L(19)$	$\approx$	$0.9061870267$
$L(\frac12)$	$\approx$	$0.9061870267$
$L(\frac{39}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	3		$ 1 $
good	2	$C_2 \wr S_4$	$ 1 + 218781 p T + 6173708785 p^{5} T^{2} + 2603177122851 p^{15} T^{3} + 783039221777613 p^{25} T^{4} + 2603177122851 p^{52} T^{5} + 6173708785 p^{79} T^{6} + 218781 p^{112} T^{7} + p^{148} T^{8} $
	5	$C_2 \wr S_4$	$ 1 - 4099829756904 T + $$84\!\cdots\!16$$ p^{3} T^{2} + $$16\!\cdots\!48$$ p^{8} T^{3} + $$34\!\cdots\!82$$ p^{14} T^{4} + $$16\!\cdots\!48$$ p^{45} T^{5} + $$84\!\cdots\!16$$ p^{77} T^{6} - 4099829756904 p^{111} T^{7} + p^{148} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 6605809948153184 T + $$55\!\cdots\!76$$ p T^{2} - $$69\!\cdots\!44$$ p^{4} T^{3} + $$65\!\cdots\!50$$ p^{7} T^{4} - $$69\!\cdots\!44$$ p^{41} T^{5} + $$55\!\cdots\!76$$ p^{75} T^{6} - 6605809948153184 p^{111} T^{7} + p^{148} T^{8} $
	11	$C_2 \wr S_4$	$ 1 + 1904882622926844816 p T + $$76\!\cdots\!52$$ p^{2} T^{2} + $$80\!\cdots\!20$$ p^{5} T^{3} + $$19\!\cdots\!26$$ p^{8} T^{4} + $$80\!\cdots\!20$$ p^{42} T^{5} + $$76\!\cdots\!52$$ p^{76} T^{6} + 1904882622926844816 p^{112} T^{7} + p^{148} T^{8} $
	13	$C_2 \wr S_4$	$ 1 - 51830892788989874168 T + $$47\!\cdots\!04$$ T^{2} - $$13\!\cdots\!52$$ p T^{3} + $$48\!\cdots\!50$$ p^{3} T^{4} - $$13\!\cdots\!52$$ p^{38} T^{5} + $$47\!\cdots\!04$$ p^{74} T^{6} - 51830892788989874168 p^{111} T^{7} + p^{148} T^{8} $
	17	$C_2 \wr S_4$	$ 1 + $$81\!\cdots\!28$$ T + $$59\!\cdots\!56$$ p T^{2} + $$14\!\cdots\!80$$ p^{3} T^{3} + $$31\!\cdots\!18$$ p^{5} T^{4} + $$14\!\cdots\!80$$ p^{40} T^{5} + $$59\!\cdots\!56$$ p^{75} T^{6} + $$81\!\cdots\!28$$ p^{111} T^{7} + p^{148} T^{8} $
	19	$C_2 \wr S_4$	$ 1 + $$54\!\cdots\!32$$ T + $$55\!\cdots\!72$$ T^{2} + $$13\!\cdots\!16$$ p T^{3} + $$44\!\cdots\!34$$ p^{2} T^{4} + $$13\!\cdots\!16$$ p^{38} T^{5} + $$55\!\cdots\!72$$ p^{74} T^{6} + $$54\!\cdots\!32$$ p^{111} T^{7} + p^{148} T^{8} $
	23	$C_2 \wr S_4$	$ 1 - $$61\!\cdots\!88$$ T + $$42\!\cdots\!16$$ p T^{2} - $$32\!\cdots\!16$$ p^{3} T^{3} + $$16\!\cdots\!70$$ p^{3} T^{4} - $$32\!\cdots\!16$$ p^{40} T^{5} + $$42\!\cdots\!16$$ p^{75} T^{6} - $$61\!\cdots\!88$$ p^{111} T^{7} + p^{148} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + $$41\!\cdots\!36$$ T + $$35\!\cdots\!20$$ p T^{2} + $$20\!\cdots\!52$$ p^{2} T^{3} + $$89\!\cdots\!02$$ p^{3} T^{4} + $$20\!\cdots\!52$$ p^{39} T^{5} + $$35\!\cdots\!20$$ p^{75} T^{6} + $$41\!\cdots\!36$$ p^{111} T^{7} + p^{148} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - $$89\!\cdots\!64$$ T + $$23\!\cdots\!28$$ p T^{2} - $$38\!\cdots\!92$$ p^{2} T^{3} + $$57\!\cdots\!94$$ p^{3} T^{4} - $$38\!\cdots\!92$$ p^{39} T^{5} + $$23\!\cdots\!28$$ p^{75} T^{6} - $$89\!\cdots\!64$$ p^{111} T^{7} + p^{148} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$55\!\cdots\!24$$ T + $$22\!\cdots\!72$$ T^{2} - $$20\!\cdots\!44$$ T^{3} + $$28\!\cdots\!70$$ T^{4} - $$20\!\cdots\!44$$ p^{37} T^{5} + $$22\!\cdots\!72$$ p^{74} T^{6} - $$55\!\cdots\!24$$ p^{111} T^{7} + p^{148} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$86\!\cdots\!76$$ T + $$17\!\cdots\!88$$ T^{2} - $$10\!\cdots\!28$$ T^{3} + $$11\!\cdots\!34$$ T^{4} - $$10\!\cdots\!28$$ p^{37} T^{5} + $$17\!\cdots\!88$$ p^{74} T^{6} - $$86\!\cdots\!76$$ p^{111} T^{7} + p^{148} T^{8} $
	43	$C_2 \wr S_4$	$ 1 + $$50\!\cdots\!80$$ T + $$85\!\cdots\!80$$ T^{2} + $$56\!\cdots\!40$$ T^{3} + $$31\!\cdots\!98$$ T^{4} + $$56\!\cdots\!40$$ p^{37} T^{5} + $$85\!\cdots\!80$$ p^{74} T^{6} + $$50\!\cdots\!80$$ p^{111} T^{7} + p^{148} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$42\!\cdots\!20$$ T + $$24\!\cdots\!20$$ T^{2} + $$10\!\cdots\!40$$ T^{3} + $$24\!\cdots\!38$$ T^{4} + $$10\!\cdots\!40$$ p^{37} T^{5} + $$24\!\cdots\!20$$ p^{74} T^{6} + $$42\!\cdots\!20$$ p^{111} T^{7} + p^{148} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - $$12\!\cdots\!88$$ T + $$20\!\cdots\!88$$ T^{2} - $$19\!\cdots\!32$$ T^{3} + $$18\!\cdots\!50$$ T^{4} - $$19\!\cdots\!32$$ p^{37} T^{5} + $$20\!\cdots\!88$$ p^{74} T^{6} - $$12\!\cdots\!88$$ p^{111} T^{7} + p^{148} T^{8} $
	59	$C_2 \wr S_4$	$ 1 - $$13\!\cdots\!88$$ T + $$62\!\cdots\!12$$ T^{2} + $$19\!\cdots\!84$$ T^{3} - $$29\!\cdots\!66$$ T^{4} + $$19\!\cdots\!84$$ p^{37} T^{5} + $$62\!\cdots\!12$$ p^{74} T^{6} - $$13\!\cdots\!88$$ p^{111} T^{7} + p^{148} T^{8} $
	61	$C_2 \wr S_4$	$ 1 + $$12\!\cdots\!60$$ T + $$43\!\cdots\!16$$ T^{2} + $$36\!\cdots\!40$$ T^{3} + $$72\!\cdots\!46$$ T^{4} + $$36\!\cdots\!40$$ p^{37} T^{5} + $$43\!\cdots\!16$$ p^{74} T^{6} + $$12\!\cdots\!60$$ p^{111} T^{7} + p^{148} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$16\!\cdots\!48$$ T + $$20\!\cdots\!72$$ T^{2} - $$16\!\cdots\!00$$ T^{3} + $$11\!\cdots\!66$$ T^{4} - $$16\!\cdots\!00$$ p^{37} T^{5} + $$20\!\cdots\!72$$ p^{74} T^{6} - $$16\!\cdots\!48$$ p^{111} T^{7} + p^{148} T^{8} $
	71	$C_2 \wr S_4$	$ 1 + $$10\!\cdots\!88$$ T + $$55\!\cdots\!68$$ T^{2} + $$47\!\cdots\!16$$ T^{3} + $$14\!\cdots\!70$$ T^{4} + $$47\!\cdots\!16$$ p^{37} T^{5} + $$55\!\cdots\!68$$ p^{74} T^{6} + $$10\!\cdots\!88$$ p^{111} T^{7} + p^{148} T^{8} $
	73	$C_2 \wr S_4$	$ 1 + $$19\!\cdots\!48$$ T + $$28\!\cdots\!68$$ T^{2} + $$45\!\cdots\!52$$ T^{3} + $$35\!\cdots\!90$$ T^{4} + $$45\!\cdots\!52$$ p^{37} T^{5} + $$28\!\cdots\!68$$ p^{74} T^{6} + $$19\!\cdots\!48$$ p^{111} T^{7} + p^{148} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$42\!\cdots\!20$$ T + $$83\!\cdots\!36$$ T^{2} - $$13\!\cdots\!40$$ T^{3} - $$27\!\cdots\!14$$ T^{4} - $$13\!\cdots\!40$$ p^{37} T^{5} + $$83\!\cdots\!36$$ p^{74} T^{6} - $$42\!\cdots\!20$$ p^{111} T^{7} + p^{148} T^{8} $
	83	$C_2 \wr S_4$	$ 1 - $$46\!\cdots\!24$$ T + $$47\!\cdots\!60$$ T^{2} - $$14\!\cdots\!44$$ T^{3} + $$74\!\cdots\!46$$ T^{4} - $$14\!\cdots\!44$$ p^{37} T^{5} + $$47\!\cdots\!60$$ p^{74} T^{6} - $$46\!\cdots\!24$$ p^{111} T^{7} + p^{148} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - $$31\!\cdots\!52$$ T + $$35\!\cdots\!52$$ T^{2} - $$89\!\cdots\!84$$ T^{3} + $$60\!\cdots\!34$$ T^{4} - $$89\!\cdots\!84$$ p^{37} T^{5} + $$35\!\cdots\!52$$ p^{74} T^{6} - $$31\!\cdots\!52$$ p^{111} T^{7} + p^{148} T^{8} $
	97	$C_2 \wr S_4$	$ 1 - $$44\!\cdots\!48$$ T + $$13\!\cdots\!12$$ T^{2} - $$24\!\cdots\!40$$ T^{3} + $$18\!\cdots\!86$$ T^{4} - $$24\!\cdots\!40$$ p^{37} T^{5} + $$13\!\cdots\!12$$ p^{74} T^{6} - $$44\!\cdots\!48$$ p^{111} T^{7} + p^{148} T^{8} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.757942440246284617315481444702, −8.198214539228176137496746117651, −8.182242227282327811052116258004, −7.87511309095839839874909499433, −7.54720713749742113369378399355, −6.95643697468195119764446931995, −6.81802612383089184368840713825, −6.10880970572664590835727833554, −5.93993614889857618017393582226, −5.59795792675660459153752481295, −5.16107299421062086737074944040, −4.87716626848294671609516296283, −4.61326015245259160371731162947, −3.99770135500324232484230150378, −3.87800239795090749687210611690, −3.57770801743257322787710755178, −2.75345715904718220366226058916, −2.36713612551349998305232084055, −2.36567096958845444449601641358, −1.80289714052451018836049793860, −1.78182685151025830588465770767, −1.20008575568362234212282145998, −0.73282083784995701345709659954, −0.44934044528701019307714488010, −0.21108633911225696834721495038, 0.21108633911225696834721495038, 0.44934044528701019307714488010, 0.73282083784995701345709659954, 1.20008575568362234212282145998, 1.78182685151025830588465770767, 1.80289714052451018836049793860, 2.36567096958845444449601641358, 2.36713612551349998305232084055, 2.75345715904718220366226058916, 3.57770801743257322787710755178, 3.87800239795090749687210611690, 3.99770135500324232484230150378, 4.61326015245259160371731162947, 4.87716626848294671609516296283, 5.16107299421062086737074944040, 5.59795792675660459153752481295, 5.93993614889857618017393582226, 6.10880970572664590835727833554, 6.81802612383089184368840713825, 6.95643697468195119764446931995, 7.54720713749742113369378399355, 7.87511309095839839874909499433, 8.182242227282327811052116258004, 8.198214539228176137496746117651, 8.757942440246284617315481444702

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 $\F_{q}$

\(L(19)\)	\(\approx\)	\(0.9061870267\)
\(L(\frac12)\)	\(\approx\)	\(0.9061870267\)
\(L(\frac{39}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Origins of factors

Downloads

Learn more