LMFDB - L-function 8-3e4-1.1-c41e4-0-0

Dirichlet series

L(s) = 1

− 6.98e4·2^-s + 1.39e10·3^-s − 1.71e12·4^-s + 1.18e14·5^-s − 9.73e14·6^-s + 1.50e17·7^-s + 2.26e18·8^-s + 1.21e20·9^-s − 8.27e18·10^-s + 7.24e20·11^-s − 2.39e22·12^-s − 8.84e21·13^-s − 1.04e22·14^-s + 1.65e24·15^-s − 3.10e24·16^-s − 3.82e25·17^-s − 8.48e24·18^-s + 2.61e26·19^-s − 2.03e26·20^-s + 2.09e27·21^-s − 5.06e25·22^-s − 1.53e28·23^-s + 3.15e28·24^-s − 2.52e28·25^-s + 6.17e26·26^-s + 8.47e29·27^-s − 2.58e29·28^-s + ⋯

L(s) = 1

− 0.0470·2^-s + 2.30·3^-s − 0.781·4^-s + 0.555·5^-s − 0.108·6^-s + 0.711·7^-s + 0.694·8^-s + 10/3·9^-s − 0.0261·10^-s + 0.324·11^-s − 1.80·12^-s − 0.129·13^-s − 0.0335·14^-s + 1.28·15^-s − 0.642·16^-s − 2.28·17^-s − 0.156·18^-s + 1.59·19^-s − 0.434·20^-s + 1.64·21^-s − 0.0152·22^-s − 1.87·23^-s + 1.60·24^-s − 0.555·25^-s + 0.00607·26^-s + 3.84·27^-s − 0.556·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+41/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$81$ = $3^{4}$
Sign:	$1$
Analytic conductor:	$1.04092\times 10^{6}$
Root analytic conductor:	$5.65168$
Motivic weight:	$41$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 81,\ (\ :41/2, 41/2, 41/2, 41/2),\ 1)$

Particular Values

$L(21)$	$\approx$	$6.008481472$
$L(\frac12)$	$\approx$	$6.008481472$
$L(\frac{43}{2})$		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	$C_1$	$ ( 1 - p^{20} T )^{4} $
good	2	$C_2 \wr S_4$	$ 1 + 34911 p T + 53875321015 p^{5} T^{2} - 247085990567271 p^{13} T^{3} + 171951681772402383 p^{25} T^{4} - 247085990567271 p^{54} T^{5} + 53875321015 p^{87} T^{6} + 34911 p^{124} T^{7} + p^{164} T^{8} $
	5	$C_2 \wr S_4$	$ 1 - 23707392755256 p T + $$31\!\cdots\!68$$ p^{3} T^{2} - $$16\!\cdots\!68$$ p^{7} T^{3} + $$21\!\cdots\!46$$ p^{13} T^{4} - $$16\!\cdots\!68$$ p^{48} T^{5} + $$31\!\cdots\!68$$ p^{85} T^{6} - 23707392755256 p^{124} T^{7} + p^{164} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 150256264888927136 T + $$23\!\cdots\!48$$ p^{2} T^{2} - $$55\!\cdots\!56$$ p^{4} T^{3} + $$17\!\cdots\!90$$ p^{9} T^{4} - $$55\!\cdots\!56$$ p^{45} T^{5} + $$23\!\cdots\!48$$ p^{84} T^{6} - 150256264888927136 p^{123} T^{7} + p^{164} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - 65891877021040062096 p T + $$18\!\cdots\!32$$ p^{2} T^{2} - $$63\!\cdots\!80$$ p^{5} T^{3} + $$16\!\cdots\!46$$ p^{8} T^{4} - $$63\!\cdots\!80$$ p^{46} T^{5} + $$18\!\cdots\!32$$ p^{84} T^{6} - 65891877021040062096 p^{124} T^{7} + p^{164} T^{8} $
	13	$C_2 \wr S_4$	$ 1 + $$88\!\cdots\!08$$ T + $$65\!\cdots\!88$$ p T^{2} - $$38\!\cdots\!72$$ p^{3} T^{3} + $$12\!\cdots\!50$$ p^{5} T^{4} - $$38\!\cdots\!72$$ p^{44} T^{5} + $$65\!\cdots\!88$$ p^{83} T^{6} + $$88\!\cdots\!08$$ p^{123} T^{7} + p^{164} T^{8} $
	17	$C_2 \wr S_4$	$ 1 + $$38\!\cdots\!88$$ T + $$68\!\cdots\!16$$ p T^{2} + $$48\!\cdots\!60$$ p^{3} T^{3} + $$33\!\cdots\!18$$ p^{5} T^{4} + $$48\!\cdots\!60$$ p^{44} T^{5} + $$68\!\cdots\!16$$ p^{83} T^{6} + $$38\!\cdots\!88$$ p^{123} T^{7} + p^{164} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - $$26\!\cdots\!48$$ T + $$39\!\cdots\!48$$ p T^{2} - $$21\!\cdots\!64$$ p^{3} T^{3} + $$10\!\cdots\!66$$ p^{5} T^{4} - $$21\!\cdots\!64$$ p^{44} T^{5} + $$39\!\cdots\!48$$ p^{83} T^{6} - $$26\!\cdots\!48$$ p^{123} T^{7} + p^{164} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + $$15\!\cdots\!32$$ T + $$86\!\cdots\!76$$ p T^{2} + $$28\!\cdots\!52$$ p^{2} T^{3} + $$10\!\cdots\!50$$ p^{3} T^{4} + $$28\!\cdots\!52$$ p^{43} T^{5} + $$86\!\cdots\!76$$ p^{83} T^{6} + $$15\!\cdots\!32$$ p^{123} T^{7} + p^{164} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + $$10\!\cdots\!64$$ T + $$94\!\cdots\!40$$ p T^{2} + $$31\!\cdots\!88$$ p^{2} T^{3} + $$13\!\cdots\!82$$ p^{3} T^{4} + $$31\!\cdots\!88$$ p^{43} T^{5} + $$94\!\cdots\!40$$ p^{83} T^{6} + $$10\!\cdots\!64$$ p^{123} T^{7} + p^{164} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - $$92\!\cdots\!04$$ T + $$25\!\cdots\!88$$ p T^{2} - $$41\!\cdots\!52$$ p^{2} T^{3} + $$59\!\cdots\!54$$ p^{3} T^{4} - $$41\!\cdots\!52$$ p^{43} T^{5} + $$25\!\cdots\!88$$ p^{83} T^{6} - $$92\!\cdots\!04$$ p^{123} T^{7} + p^{164} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$20\!\cdots\!56$$ T + $$14\!\cdots\!16$$ p T^{2} - $$62\!\cdots\!84$$ p^{2} T^{3} + $$31\!\cdots\!90$$ p^{3} T^{4} - $$62\!\cdots\!84$$ p^{43} T^{5} + $$14\!\cdots\!16$$ p^{83} T^{6} - $$20\!\cdots\!56$$ p^{123} T^{7} + p^{164} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$23\!\cdots\!04$$ T + $$12\!\cdots\!88$$ p T^{2} - $$70\!\cdots\!72$$ T^{3} + $$99\!\cdots\!94$$ T^{4} - $$70\!\cdots\!72$$ p^{41} T^{5} + $$12\!\cdots\!88$$ p^{83} T^{6} - $$23\!\cdots\!04$$ p^{123} T^{7} + p^{164} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$39\!\cdots\!60$$ T + $$29\!\cdots\!00$$ T^{2} - $$85\!\cdots\!80$$ T^{3} + $$37\!\cdots\!98$$ T^{4} - $$85\!\cdots\!80$$ p^{41} T^{5} + $$29\!\cdots\!00$$ p^{82} T^{6} - $$39\!\cdots\!60$$ p^{123} T^{7} + p^{164} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$88\!\cdots\!20$$ T + $$10\!\cdots\!20$$ T^{2} + $$42\!\cdots\!40$$ T^{3} + $$48\!\cdots\!18$$ T^{4} + $$42\!\cdots\!40$$ p^{41} T^{5} + $$10\!\cdots\!20$$ p^{82} T^{6} + $$88\!\cdots\!20$$ p^{123} T^{7} + p^{164} T^{8} $
	53	$C_2 \wr S_4$	$ 1 - $$95\!\cdots\!28$$ T + $$13\!\cdots\!88$$ T^{2} - $$10\!\cdots\!72$$ T^{3} + $$85\!\cdots\!10$$ T^{4} - $$10\!\cdots\!72$$ p^{41} T^{5} + $$13\!\cdots\!88$$ p^{82} T^{6} - $$95\!\cdots\!28$$ p^{123} T^{7} + p^{164} T^{8} $
	59	$C_2 \wr S_4$	$ 1 + $$18\!\cdots\!08$$ T + $$66\!\cdots\!52$$ T^{2} + $$24\!\cdots\!16$$ T^{3} + $$18\!\cdots\!74$$ T^{4} + $$24\!\cdots\!16$$ p^{41} T^{5} + $$66\!\cdots\!52$$ p^{82} T^{6} + $$18\!\cdots\!08$$ p^{123} T^{7} + p^{164} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$53\!\cdots\!40$$ T + $$67\!\cdots\!56$$ T^{2} - $$24\!\cdots\!60$$ T^{3} + $$16\!\cdots\!26$$ T^{4} - $$24\!\cdots\!60$$ p^{41} T^{5} + $$67\!\cdots\!56$$ p^{82} T^{6} - $$53\!\cdots\!40$$ p^{123} T^{7} + p^{164} T^{8} $
	67	$C_2 \wr S_4$	$ 1 + $$73\!\cdots\!28$$ T + $$45\!\cdots\!12$$ T^{2} + $$17\!\cdots\!00$$ T^{3} + $$56\!\cdots\!06$$ T^{4} + $$17\!\cdots\!00$$ p^{41} T^{5} + $$45\!\cdots\!12$$ p^{82} T^{6} + $$73\!\cdots\!28$$ p^{123} T^{7} + p^{164} T^{8} $
	71	$C_2 \wr S_4$	$ 1 + $$84\!\cdots\!52$$ T + $$22\!\cdots\!48$$ T^{2} + $$13\!\cdots\!64$$ T^{3} + $$24\!\cdots\!70$$ T^{4} + $$13\!\cdots\!64$$ p^{41} T^{5} + $$22\!\cdots\!48$$ p^{82} T^{6} + $$84\!\cdots\!52$$ p^{123} T^{7} + p^{164} T^{8} $
	73	$C_2 \wr S_4$	$ 1 + $$44\!\cdots\!32$$ T + $$51\!\cdots\!68$$ T^{2} - $$16\!\cdots\!72$$ T^{3} + $$13\!\cdots\!70$$ T^{4} - $$16\!\cdots\!72$$ p^{41} T^{5} + $$51\!\cdots\!68$$ p^{82} T^{6} + $$44\!\cdots\!32$$ p^{123} T^{7} + p^{164} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$14\!\cdots\!80$$ T + $$22\!\cdots\!16$$ T^{2} - $$18\!\cdots\!60$$ T^{3} + $$18\!\cdots\!46$$ T^{4} - $$18\!\cdots\!60$$ p^{41} T^{5} + $$22\!\cdots\!16$$ p^{82} T^{6} - $$14\!\cdots\!80$$ p^{123} T^{7} + p^{164} T^{8} $
	83	$C_2 \wr S_4$	$ 1 - $$15\!\cdots\!04$$ T + $$99\!\cdots\!00$$ T^{2} - $$72\!\cdots\!24$$ T^{3} + $$39\!\cdots\!46$$ T^{4} - $$72\!\cdots\!24$$ p^{41} T^{5} + $$99\!\cdots\!00$$ p^{82} T^{6} - $$15\!\cdots\!04$$ p^{123} T^{7} + p^{164} T^{8} $
	89	$C_2 \wr S_4$	$ 1 + $$39\!\cdots\!72$$ T + $$90\!\cdots\!12$$ T^{2} + $$13\!\cdots\!84$$ T^{3} + $$16\!\cdots\!46$$ p T^{4} + $$13\!\cdots\!84$$ p^{41} T^{5} + $$90\!\cdots\!12$$ p^{82} T^{6} + $$39\!\cdots\!72$$ p^{123} T^{7} + p^{164} T^{8} $
	97	$C_2 \wr S_4$	$ 1 - $$36\!\cdots\!52$$ T + $$23\!\cdots\!52$$ T^{2} + $$31\!\cdots\!80$$ T^{3} - $$59\!\cdots\!94$$ T^{4} + $$31\!\cdots\!80$$ p^{41} T^{5} + $$23\!\cdots\!52$$ p^{82} T^{6} - $$36\!\cdots\!52$$ p^{123} T^{7} + p^{164} T^{8} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.90466941294547704442078424401, −10.08139757531473401389038677824, −9.544162973716166835633178740028, −9.487933126176598126347729864258, −9.340635324934120044071582964261, −8.631396541114538771867213723687, −8.168099141132219016509541559910, −8.119062495603604383653260599442, −7.57692701435250980805046229326, −7.21455050126391732412642137143, −6.64583078675594647141102048634, −6.09121522419422348833758769714, −5.67464253258325347012894904810, −4.71538140726844446009909166281, −4.51451479554662000466001581882, −4.16872280881164195584099576642, −4.16279728815865988996188367555, −3.27823885956949930655935892256, −2.80124302747572001597600018569, −2.49429621028374224008217175926, −1.97480057373067621755313571870, −1.91314548178943544225527390400, −1.24455453507659229966356001716, −0.967755440504959470005255867116, −0.23805643869082347528862747299, 0.23805643869082347528862747299, 0.967755440504959470005255867116, 1.24455453507659229966356001716, 1.91314548178943544225527390400, 1.97480057373067621755313571870, 2.49429621028374224008217175926, 2.80124302747572001597600018569, 3.27823885956949930655935892256, 4.16279728815865988996188367555, 4.16872280881164195584099576642, 4.51451479554662000466001581882, 4.71538140726844446009909166281, 5.67464253258325347012894904810, 6.09121522419422348833758769714, 6.64583078675594647141102048634, 7.21455050126391732412642137143, 7.57692701435250980805046229326, 8.119062495603604383653260599442, 8.168099141132219016509541559910, 8.631396541114538771867213723687, 9.340635324934120044071582964261, 9.487933126176598126347729864258, 9.544162973716166835633178740028, 10.08139757531473401389038677824, 10.90466941294547704442078424401

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 $\F_{q}$

\(L(21)\)	\(\approx\)	\(6.008481472\)
\(L(\frac12)\)	\(\approx\)	\(6.008481472\)
\(L(\frac{43}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

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