LMFDB - L-function 6-2e3-1.1-c71e3-0-0

Dirichlet series

L(s) = 1

− 1.03e11·2^-s + 2.36e15·3^-s + 7.08e21·4^-s + 4.75e24·5^-s − 2.43e26·6^-s − 6.85e29·7^-s − 4.05e32·8^-s − 1.35e34·9^-s − 4.89e35·10^-s + 3.96e36·11^-s + 1.67e37·12^-s + 2.52e39·13^-s + 7.06e40·14^-s + 1.12e40·15^-s + 2.09e43·16^-s + 3.54e43·17^-s + 1.39e45·18^-s + 2.84e45·19^-s + 3.36e46·20^-s − 1.62e45·21^-s − 4.09e47·22^-s + 3.72e47·23^-s − 9.58e47·24^-s − 1.72e49·25^-s − 2.60e50·26^-s − 3.53e50·27^-s − 4.85e51·28^-s + ⋯

L(s) = 1

− 2.12·2^-s + 0.0272·3^-s + 3·4^-s + 0.730·5^-s − 0.0578·6^-s − 0.684·7^-s − 3.53·8^-s − 1.79·9^-s − 1.54·10^-s + 0.425·11^-s + 0.0817·12^-s + 0.721·13^-s + 1.45·14^-s + 0.0199·15^-s + 15/4·16^-s + 0.739·17^-s + 3.81·18^-s + 1.14·19^-s + 2.19·20^-s − 0.0186·21^-s − 0.903·22^-s + 0.169·23^-s − 0.0963·24^-s − 0.408·25^-s − 1.52·26^-s − 0.543·27^-s − 2.05·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+71/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$6$
Conductor:	$8$ = $2^{3}$
Sign:	$1$
Analytic conductor:	$260295.$
Root analytic conductor:	$7.99057$
Motivic weight:	$71$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(6,\ 8,\ (\ :71/2, 71/2, 71/2),\ 1)$

Particular Values

$L(36)$	$\approx$	$1.693673841$
$L(\frac12)$	$\approx$	$1.693673841$
$L(\frac{73}{2})$		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^{35} T )^{3} $
good	3	$S_4\times C_2$	$ 1 - 262507810300804 p^{2} T + $$22\!\cdots\!77$$ p^{10} T^{2} + $$34\!\cdots\!16$$ p^{25} T^{3} + $$22\!\cdots\!77$$ p^{81} T^{4} - 262507810300804 p^{144} T^{5} + p^{213} T^{6} $
	5	$S_4\times C_2$	$ 1 - $$19\!\cdots\!66$$ p^{2} T + $$51\!\cdots\!51$$ p^{7} T^{2} - $$12\!\cdots\!84$$ p^{15} T^{3} + $$51\!\cdots\!51$$ p^{78} T^{4} - $$19\!\cdots\!66$$ p^{144} T^{5} + p^{213} T^{6} $
	7	$S_4\times C_2$	$ 1 + $$13\!\cdots\!28$$ p^{2} T + $$39\!\cdots\!57$$ p^{8} T^{2} + $$40\!\cdots\!16$$ p^{16} T^{3} + $$39\!\cdots\!57$$ p^{79} T^{4} + $$13\!\cdots\!28$$ p^{144} T^{5} + p^{213} T^{6} $
	11	$S_4\times C_2$	$ 1 - $$36\!\cdots\!96$$ p T + $$62\!\cdots\!95$$ p^{3} T^{2} - $$33\!\cdots\!40$$ p^{8} T^{3} + $$62\!\cdots\!95$$ p^{74} T^{4} - $$36\!\cdots\!96$$ p^{143} T^{5} + p^{213} T^{6} $
	13	$S_4\times C_2$	$ 1 - $$19\!\cdots\!42$$ p T + $$47\!\cdots\!03$$ p^{4} T^{2} - $$41\!\cdots\!64$$ p^{9} T^{3} + $$47\!\cdots\!03$$ p^{75} T^{4} - $$19\!\cdots\!42$$ p^{143} T^{5} + p^{213} T^{6} $
	17	$S_4\times C_2$	$ 1 - $$35\!\cdots\!18$$ T + $$10\!\cdots\!63$$ p^{2} T^{2} - $$54\!\cdots\!72$$ p^{5} T^{3} + $$10\!\cdots\!63$$ p^{73} T^{4} - $$35\!\cdots\!18$$ p^{142} T^{5} + p^{213} T^{6} $
	19	$S_4\times C_2$	$ 1 - $$28\!\cdots\!20$$ T + $$52\!\cdots\!37$$ p^{2} T^{2} - $$14\!\cdots\!40$$ p^{5} T^{3} + $$52\!\cdots\!37$$ p^{73} T^{4} - $$28\!\cdots\!20$$ p^{142} T^{5} + p^{213} T^{6} $
	23	$S_4\times C_2$	$ 1 - $$16\!\cdots\!92$$ p T + $$23\!\cdots\!13$$ p^{4} T^{2} + $$61\!\cdots\!16$$ p^{5} T^{3} + $$23\!\cdots\!13$$ p^{75} T^{4} - $$16\!\cdots\!92$$ p^{143} T^{5} + p^{213} T^{6} $
	29	$S_4\times C_2$	$ 1 - $$28\!\cdots\!70$$ T + $$10\!\cdots\!07$$ p^{2} T^{2} + $$63\!\cdots\!60$$ p^{3} T^{3} + $$10\!\cdots\!07$$ p^{73} T^{4} - $$28\!\cdots\!70$$ p^{142} T^{5} + p^{213} T^{6} $
	31	$S_4\times C_2$	$ 1 + $$19\!\cdots\!44$$ p T + $$77\!\cdots\!75$$ p^{3} T^{2} + $$31\!\cdots\!40$$ p^{5} T^{3} + $$77\!\cdots\!75$$ p^{74} T^{4} + $$19\!\cdots\!44$$ p^{143} T^{5} + p^{213} T^{6} $
	37	$S_4\times C_2$	$ 1 + $$68\!\cdots\!62$$ T + $$20\!\cdots\!51$$ p T^{2} + $$22\!\cdots\!04$$ p^{2} T^{3} + $$20\!\cdots\!51$$ p^{72} T^{4} + $$68\!\cdots\!62$$ p^{142} T^{5} + p^{213} T^{6} $
	41	$S_4\times C_2$	$ 1 - $$25\!\cdots\!86$$ T + $$93\!\cdots\!55$$ T^{2} - $$37\!\cdots\!80$$ p T^{3} + $$93\!\cdots\!55$$ p^{71} T^{4} - $$25\!\cdots\!86$$ p^{142} T^{5} + p^{213} T^{6} $
	43	$S_4\times C_2$	$ 1 - $$18\!\cdots\!36$$ T + $$90\!\cdots\!71$$ p T^{2} - $$20\!\cdots\!68$$ p^{2} T^{3} + $$90\!\cdots\!71$$ p^{72} T^{4} - $$18\!\cdots\!36$$ p^{142} T^{5} + p^{213} T^{6} $
	47	$S_4\times C_2$	$ 1 + $$43\!\cdots\!32$$ T + $$12\!\cdots\!11$$ p T^{2} + $$45\!\cdots\!64$$ p^{2} T^{3} + $$12\!\cdots\!11$$ p^{72} T^{4} + $$43\!\cdots\!32$$ p^{142} T^{5} + p^{213} T^{6} $
	53	$S_4\times C_2$	$ 1 - $$30\!\cdots\!06$$ T + $$14\!\cdots\!51$$ p T^{2} - $$53\!\cdots\!08$$ p^{2} T^{3} + $$14\!\cdots\!51$$ p^{72} T^{4} - $$30\!\cdots\!06$$ p^{142} T^{5} + p^{213} T^{6} $
	59	$S_4\times C_2$	$ 1 - $$46\!\cdots\!40$$ T + $$18\!\cdots\!03$$ p T^{2} - $$37\!\cdots\!20$$ p^{2} T^{3} + $$18\!\cdots\!03$$ p^{72} T^{4} - $$46\!\cdots\!40$$ p^{142} T^{5} + p^{213} T^{6} $
	61	$S_4\times C_2$	$ 1 - $$27\!\cdots\!06$$ p T + $$12\!\cdots\!35$$ p^{2} T^{2} - $$58\!\cdots\!00$$ p^{3} T^{3} + $$12\!\cdots\!35$$ p^{73} T^{4} - $$27\!\cdots\!06$$ p^{143} T^{5} + p^{213} T^{6} $
	67	$S_4\times C_2$	$ 1 - $$43\!\cdots\!24$$ p T + $$92\!\cdots\!33$$ p^{2} T^{2} - $$11\!\cdots\!68$$ p^{3} T^{3} + $$92\!\cdots\!33$$ p^{73} T^{4} - $$43\!\cdots\!24$$ p^{143} T^{5} + p^{213} T^{6} $
	71	$S_4\times C_2$	$ 1 - $$91\!\cdots\!96$$ T + $$80\!\cdots\!85$$ T^{2} - $$38\!\cdots\!00$$ T^{3} + $$80\!\cdots\!85$$ p^{71} T^{4} - $$91\!\cdots\!96$$ p^{142} T^{5} + p^{213} T^{6} $
	73	$S_4\times C_2$	$ 1 - $$28\!\cdots\!86$$ T + $$59\!\cdots\!63$$ T^{2} - $$84\!\cdots\!72$$ T^{3} + $$59\!\cdots\!63$$ p^{71} T^{4} - $$28\!\cdots\!86$$ p^{142} T^{5} + p^{213} T^{6} $
	79	$S_4\times C_2$	$ 1 - $$35\!\cdots\!40$$ T + $$17\!\cdots\!37$$ T^{2} - $$34\!\cdots\!20$$ T^{3} + $$17\!\cdots\!37$$ p^{71} T^{4} - $$35\!\cdots\!40$$ p^{142} T^{5} + p^{213} T^{6} $
	83	$S_4\times C_2$	$ 1 + $$49\!\cdots\!64$$ T + $$13\!\cdots\!33$$ T^{2} + $$21\!\cdots\!48$$ T^{3} + $$13\!\cdots\!33$$ p^{71} T^{4} + $$49\!\cdots\!64$$ p^{142} T^{5} + p^{213} T^{6} $
	89	$S_4\times C_2$	$ 1 + $$17\!\cdots\!70$$ T + $$72\!\cdots\!67$$ T^{2} + $$90\!\cdots\!60$$ T^{3} + $$72\!\cdots\!67$$ p^{71} T^{4} + $$17\!\cdots\!70$$ p^{142} T^{5} + p^{213} T^{6} $
	97	$S_4\times C_2$	$ 1 - $$41\!\cdots\!58$$ T + $$31\!\cdots\!47$$ T^{2} - $$74\!\cdots\!04$$ T^{3} + $$31\!\cdots\!47$$ p^{71} T^{4} - $$41\!\cdots\!58$$ p^{142} T^{5} + p^{213} T^{6} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−12.02150775375769971640294475246, −11.35044125004675883212595408967, −11.14109875602125567823340233419, −10.73768234587935097249065419450, −9.922385362819168541040472256629, −9.602407719289499622842537816045, −9.390456928277208564879103718363, −8.861465666356790097821213590195, −8.309308116085308896947760536579, −8.076428026555780078305276314092, −7.46402528121473953105970301410, −6.70007701866108697943991357577, −6.63912488618128702132546548529, −5.75985300194031720340061788654, −5.62805751643630772634374995411, −5.29138069349547725323626923255, −3.91079472184107988548123664514, −3.41922911598692545131697169932, −3.17556865280986370615013316643, −2.35575346106003753127284087981, −2.27551970500632059377150467691, −1.66064879833718373124201852285, −0.932287180389301422681322737017, −0.70844749435404216599064018316, −0.41980751051285742381739687448, 0.41980751051285742381739687448, 0.70844749435404216599064018316, 0.932287180389301422681322737017, 1.66064879833718373124201852285, 2.27551970500632059377150467691, 2.35575346106003753127284087981, 3.17556865280986370615013316643, 3.41922911598692545131697169932, 3.91079472184107988548123664514, 5.29138069349547725323626923255, 5.62805751643630772634374995411, 5.75985300194031720340061788654, 6.63912488618128702132546548529, 6.70007701866108697943991357577, 7.46402528121473953105970301410, 8.076428026555780078305276314092, 8.309308116085308896947760536579, 8.861465666356790097821213590195, 9.390456928277208564879103718363, 9.602407719289499622842537816045, 9.922385362819168541040472256629, 10.73768234587935097249065419450, 11.14109875602125567823340233419, 11.35044125004675883212595408967, 12.02150775375769971640294475246

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 $\F_{q}$

\(L(36)\)	\(\approx\)	\(1.693673841\)
\(L(\frac12)\)	\(\approx\)	\(1.693673841\)
\(L(\frac{73}{2})\)		not available
\(L(1)\)		not available

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