LMFDB - L-function 8-2e4-1.1-c81e4-0-0

Dirichlet series

L(s) = 1

+ 4.39e12·2^-s − 7.11e18·3^-s + 1.20e25·4^-s + 2.49e28·5^-s − 3.12e31·6^-s + 1.83e34·7^-s + 2.65e37·8^-s − 3.55e38·9^-s + 1.09e41·10^-s + 3.29e42·11^-s − 8.59e43·12^-s + 5.67e44·13^-s + 8.08e46·14^-s − 1.77e47·15^-s + 5.11e49·16^-s + 1.70e49·17^-s − 1.56e51·18^-s + 5.75e51·19^-s + 3.01e53·20^-s − 1.30e53·21^-s + 1.44e55·22^-s + 1.96e55·23^-s − 1.89e56·24^-s − 1.50e56·25^-s + 2.49e57·26^-s − 9.77e57·27^-s + 2.22e59·28^-s + ⋯

L(s) = 1

+ 2.82·2^-s − 0.337·3^-s + 5·4^-s + 1.22·5^-s − 0.955·6^-s + 1.09·7^-s + 7.07·8^-s − 0.800·9^-s + 3.46·10^-s + 2.19·11^-s − 1.68·12^-s + 0.435·13^-s + 3.08·14^-s − 0.413·15^-s + 35/4·16^-s + 0.250·17^-s − 2.26·18^-s + 0.934·19^-s + 6.12·20^-s − 0.368·21^-s + 6.20·22^-s + 1.39·23^-s − 2.38·24^-s − 0.363·25^-s + 1.23·26^-s − 1.04·27^-s + 5.45·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(82-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+81/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$16$ = $2^{4}$
Sign:	$1$
Analytic conductor:	$4.76880\times 10^{7}$
Root analytic conductor:	$9.11593$
Motivic weight:	$81$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 16,\ (\ :81/2, 81/2, 81/2, 81/2),\ 1)$

Particular Values

$L(41)$	$\approx$	$137.9621464$
$L(\frac12)$	$\approx$	$137.9621464$
$L(\frac{83}{2})$		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	2	$C_1$	$ ( 1 - p^{40} T )^{4} $
good	3	$C_2 \wr S_4$	$ 1 + 2370579126212319568 p T + $$68\!\cdots\!32$$ p^{10} T^{2} + $$16\!\cdots\!20$$ p^{23} T^{3} + $$15\!\cdots\!66$$ p^{40} T^{4} + $$16\!\cdots\!20$$ p^{104} T^{5} + $$68\!\cdots\!32$$ p^{172} T^{6} + 2370579126212319568 p^{244} T^{7} + p^{324} T^{8} $
	5	$C_2 \wr S_4$	$ 1 - $$49\!\cdots\!56$$ p T + $$24\!\cdots\!68$$ p^{5} T^{2} - $$29\!\cdots\!36$$ p^{14} T^{3} + $$18\!\cdots\!26$$ p^{25} T^{4} - $$29\!\cdots\!36$$ p^{95} T^{5} + $$24\!\cdots\!68$$ p^{167} T^{6} - $$49\!\cdots\!56$$ p^{244} T^{7} + p^{324} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - $$26\!\cdots\!56$$ p T + $$45\!\cdots\!48$$ p^{6} T^{2} - $$13\!\cdots\!00$$ p^{14} T^{3} + $$10\!\cdots\!46$$ p^{24} T^{4} - $$13\!\cdots\!00$$ p^{95} T^{5} + $$45\!\cdots\!48$$ p^{168} T^{6} - $$26\!\cdots\!56$$ p^{244} T^{7} + p^{324} T^{8} $
	11	$C_2 \wr S_4$	$ 1 - $$29\!\cdots\!08$$ p T + $$62\!\cdots\!08$$ p^{3} T^{2} - $$70\!\cdots\!16$$ p^{8} T^{3} + $$77\!\cdots\!70$$ p^{13} T^{4} - $$70\!\cdots\!16$$ p^{89} T^{5} + $$62\!\cdots\!08$$ p^{165} T^{6} - $$29\!\cdots\!08$$ p^{244} T^{7} + p^{324} T^{8} $
	13	$C_2 \wr S_4$	$ 1 - $$56\!\cdots\!16$$ T + $$26\!\cdots\!92$$ p^{2} T^{2} - $$39\!\cdots\!20$$ p^{6} T^{3} + $$71\!\cdots\!34$$ p^{10} T^{4} - $$39\!\cdots\!20$$ p^{87} T^{5} + $$26\!\cdots\!92$$ p^{164} T^{6} - $$56\!\cdots\!16$$ p^{243} T^{7} + p^{324} T^{8} $
	17	$C_2 \wr S_4$	$ 1 - $$17\!\cdots\!12$$ T + $$48\!\cdots\!16$$ p T^{2} - $$34\!\cdots\!40$$ p^{3} T^{3} + $$14\!\cdots\!62$$ p^{7} T^{4} - $$34\!\cdots\!40$$ p^{84} T^{5} + $$48\!\cdots\!16$$ p^{163} T^{6} - $$17\!\cdots\!12$$ p^{243} T^{7} + p^{324} T^{8} $
	19	$C_2 \wr S_4$	$ 1 - $$57\!\cdots\!20$$ T + $$32\!\cdots\!16$$ p^{2} T^{2} - $$17\!\cdots\!60$$ p^{5} T^{3} + $$17\!\cdots\!54$$ p^{9} T^{4} - $$17\!\cdots\!60$$ p^{86} T^{5} + $$32\!\cdots\!16$$ p^{164} T^{6} - $$57\!\cdots\!20$$ p^{243} T^{7} + p^{324} T^{8} $
	23	$C_2 \wr S_4$	$ 1 - $$37\!\cdots\!44$$ p^{2} T + $$13\!\cdots\!52$$ p^{2} T^{2} - $$65\!\cdots\!20$$ p^{6} T^{3} + $$26\!\cdots\!26$$ p^{8} T^{4} - $$65\!\cdots\!20$$ p^{87} T^{5} + $$13\!\cdots\!52$$ p^{164} T^{6} - $$37\!\cdots\!44$$ p^{245} T^{7} + p^{324} T^{8} $
	29	$C_2 \wr S_4$	$ 1 - $$89\!\cdots\!00$$ T - $$91\!\cdots\!84$$ T^{2} - $$20\!\cdots\!00$$ p^{2} T^{3} + $$22\!\cdots\!66$$ p^{4} T^{4} - $$20\!\cdots\!00$$ p^{83} T^{5} - $$91\!\cdots\!84$$ p^{162} T^{6} - $$89\!\cdots\!00$$ p^{243} T^{7} + p^{324} T^{8} $
	31	$C_2 \wr S_4$	$ 1 - $$44\!\cdots\!08$$ T + $$81\!\cdots\!08$$ p T^{2} - $$24\!\cdots\!36$$ p^{3} T^{3} + $$85\!\cdots\!70$$ p^{5} T^{4} - $$24\!\cdots\!36$$ p^{84} T^{5} + $$81\!\cdots\!08$$ p^{163} T^{6} - $$44\!\cdots\!08$$ p^{243} T^{7} + p^{324} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$28\!\cdots\!92$$ T + $$58\!\cdots\!56$$ p T^{2} + $$45\!\cdots\!40$$ p^{3} T^{3} + $$19\!\cdots\!58$$ p^{5} T^{4} + $$45\!\cdots\!40$$ p^{84} T^{5} + $$58\!\cdots\!56$$ p^{163} T^{6} - $$28\!\cdots\!92$$ p^{243} T^{7} + p^{324} T^{8} $
	41	$C_2 \wr S_4$	$ 1 - $$24\!\cdots\!88$$ T + $$12\!\cdots\!68$$ T^{2} - $$58\!\cdots\!76$$ p T^{3} + $$40\!\cdots\!70$$ p^{2} T^{4} - $$58\!\cdots\!76$$ p^{82} T^{5} + $$12\!\cdots\!68$$ p^{162} T^{6} - $$24\!\cdots\!88$$ p^{243} T^{7} + p^{324} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$39\!\cdots\!16$$ T + $$11\!\cdots\!68$$ T^{2} - $$50\!\cdots\!40$$ p T^{3} + $$19\!\cdots\!94$$ p^{2} T^{4} - $$50\!\cdots\!40$$ p^{82} T^{5} + $$11\!\cdots\!68$$ p^{162} T^{6} - $$39\!\cdots\!16$$ p^{243} T^{7} + p^{324} T^{8} $
	47	$C_2 \wr S_4$	$ 1 - $$68\!\cdots\!52$$ T + $$26\!\cdots\!16$$ p T^{2} - $$26\!\cdots\!80$$ p^{2} T^{3} + $$52\!\cdots\!22$$ p^{3} T^{4} - $$26\!\cdots\!80$$ p^{83} T^{5} + $$26\!\cdots\!16$$ p^{163} T^{6} - $$68\!\cdots\!52$$ p^{243} T^{7} + p^{324} T^{8} $
	53	$C_2 \wr S_4$	$ 1 + $$12\!\cdots\!24$$ T + $$22\!\cdots\!76$$ p T^{2} + $$26\!\cdots\!20$$ p^{2} T^{3} + $$35\!\cdots\!98$$ p^{3} T^{4} + $$26\!\cdots\!20$$ p^{83} T^{5} + $$22\!\cdots\!76$$ p^{163} T^{6} + $$12\!\cdots\!24$$ p^{243} T^{7} + p^{324} T^{8} $
	59	$C_2 \wr S_4$	$ 1 - $$48\!\cdots\!00$$ T + $$10\!\cdots\!04$$ p T^{2} - $$21\!\cdots\!00$$ p^{2} T^{3} + $$72\!\cdots\!34$$ p^{3} T^{4} - $$21\!\cdots\!00$$ p^{83} T^{5} + $$10\!\cdots\!04$$ p^{163} T^{6} - $$48\!\cdots\!00$$ p^{243} T^{7} + p^{324} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$13\!\cdots\!08$$ T + $$24\!\cdots\!88$$ p T^{2} - $$44\!\cdots\!76$$ p^{2} T^{3} + $$39\!\cdots\!70$$ p^{3} T^{4} - $$44\!\cdots\!76$$ p^{83} T^{5} + $$24\!\cdots\!88$$ p^{163} T^{6} - $$13\!\cdots\!08$$ p^{243} T^{7} + p^{324} T^{8} $
	67	$C_2 \wr S_4$	$ 1 + $$20\!\cdots\!44$$ p T + $$41\!\cdots\!96$$ p T^{2} + $$60\!\cdots\!40$$ p^{2} T^{3} + $$11\!\cdots\!42$$ p^{3} T^{4} + $$60\!\cdots\!40$$ p^{83} T^{5} + $$41\!\cdots\!96$$ p^{163} T^{6} + $$20\!\cdots\!44$$ p^{244} T^{7} + p^{324} T^{8} $
	71	$C_2 \wr S_4$	$ 1 - $$74\!\cdots\!88$$ p T + $$26\!\cdots\!28$$ p^{2} T^{2} + $$13\!\cdots\!24$$ p^{3} T^{3} + $$36\!\cdots\!70$$ p^{4} T^{4} + $$13\!\cdots\!24$$ p^{84} T^{5} + $$26\!\cdots\!28$$ p^{164} T^{6} - $$74\!\cdots\!88$$ p^{244} T^{7} + p^{324} T^{8} $
	73	$C_2 \wr S_4$	$ 1 - $$48\!\cdots\!56$$ T + $$49\!\cdots\!16$$ p T^{2} - $$22\!\cdots\!60$$ p^{2} T^{3} + $$11\!\cdots\!58$$ p^{3} T^{4} - $$22\!\cdots\!60$$ p^{83} T^{5} + $$49\!\cdots\!16$$ p^{163} T^{6} - $$48\!\cdots\!56$$ p^{243} T^{7} + p^{324} T^{8} $
	79	$C_2 \wr S_4$	$ 1 + $$64\!\cdots\!60$$ p T + $$21\!\cdots\!76$$ p^{2} T^{2} + $$10\!\cdots\!20$$ p^{3} T^{3} + $$24\!\cdots\!66$$ p^{4} T^{4} + $$10\!\cdots\!20$$ p^{84} T^{5} + $$21\!\cdots\!76$$ p^{164} T^{6} + $$64\!\cdots\!60$$ p^{244} T^{7} + p^{324} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + $$23\!\cdots\!64$$ T + $$57\!\cdots\!68$$ T^{2} + $$21\!\cdots\!20$$ T^{3} + $$20\!\cdots\!46$$ T^{4} + $$21\!\cdots\!20$$ p^{81} T^{5} + $$57\!\cdots\!68$$ p^{162} T^{6} + $$23\!\cdots\!64$$ p^{243} T^{7} + p^{324} T^{8} $
	89	$C_2 \wr S_4$	$ 1 - $$27\!\cdots\!60$$ T + $$48\!\cdots\!56$$ T^{2} - $$63\!\cdots\!20$$ T^{3} + $$63\!\cdots\!26$$ T^{4} - $$63\!\cdots\!20$$ p^{81} T^{5} + $$48\!\cdots\!56$$ p^{162} T^{6} - $$27\!\cdots\!60$$ p^{243} T^{7} + p^{324} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$15\!\cdots\!08$$ T + $$16\!\cdots\!12$$ T^{2} + $$49\!\cdots\!60$$ T^{3} + $$12\!\cdots\!26$$ T^{4} + $$49\!\cdots\!60$$ p^{81} T^{5} + $$16\!\cdots\!12$$ p^{162} T^{6} + $$15\!\cdots\!08$$ p^{243} T^{7} + p^{324} 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

−8.642066739537162167075912311131, −8.064074493307377880060133356257, −7.76401563729633409174038891880, −7.39560665790946365817304013983, −7.06172054261726660448778580391, −6.39903658654996153532488242974, −6.24623737766837934786737635905, −6.13225866704962659377101924309, −5.90454407952346430133305049148, −5.35298506312240518733494336163, −5.08197484228443642121743744335, −4.99223663401325734079075783047, −4.39892408420480836732326441934, −4.09308719670146413002772945969, −3.87859343166051952528204844995, −3.60782561744738757361252162056, −3.10909811154511923734058298154, −2.58042340022306084397919131183, −2.50877781832259357027743113090, −2.30191900944389513159855027303, −1.66027834677990205904022508826, −1.27505859728183029454835287117, −1.13982943665743700207569397739, −1.07905802816476386157795905783, −0.43404206843309624699846110995, 0.43404206843309624699846110995, 1.07905802816476386157795905783, 1.13982943665743700207569397739, 1.27505859728183029454835287117, 1.66027834677990205904022508826, 2.30191900944389513159855027303, 2.50877781832259357027743113090, 2.58042340022306084397919131183, 3.10909811154511923734058298154, 3.60782561744738757361252162056, 3.87859343166051952528204844995, 4.09308719670146413002772945969, 4.39892408420480836732326441934, 4.99223663401325734079075783047, 5.08197484228443642121743744335, 5.35298506312240518733494336163, 5.90454407952346430133305049148, 6.13225866704962659377101924309, 6.24623737766837934786737635905, 6.39903658654996153532488242974, 7.06172054261726660448778580391, 7.39560665790946365817304013983, 7.76401563729633409174038891880, 8.064074493307377880060133356257, 8.642066739537162167075912311131

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 $\F_{q}$

\(L(41)\)	\(\approx\)	\(137.9621464\)
\(L(\frac12)\)	\(\approx\)	\(137.9621464\)
\(L(\frac{83}{2})\)		not available
\(L(1)\)		not available

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L-functions

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