LMFDB - $L(s)$, of degree 6, motivic weight 11, conductor $3375$, and trivial character

Dirichlet series

L(s) = 1

− 2^-s + 729·3^-s − 693·4^-s + 9.37e3·5^-s − 729·6^-s − 1.46e4·7^-s − 9.95e3·8^-s + 3.54e5·9^-s − 9.37e3·10^-s + 5.40e5·11^-s − 5.05e5·12^-s + 8.40e5·13^-s + 1.46e4·14^-s + 6.83e6·15^-s − 9.19e5·16^-s + 1.51e7·17^-s − 3.54e5·18^-s + 1.77e7·19^-s − 6.49e6·20^-s − 1.06e7·21^-s − 5.40e5·22^-s − 2.81e7·23^-s − 7.25e6·24^-s + 5.85e7·25^-s − 8.40e5·26^-s + 1.43e8·27^-s + 1.01e7·28^-s + ⋯

L(s) = 1

− 0.0220·2^-s + 1.73·3^-s − 0.338·4^-s + 1.34·5^-s − 0.0382·6^-s − 0.328·7^-s − 0.107·8^-s + 2·9^-s − 0.0296·10^-s + 1.01·11^-s − 0.586·12^-s + 0.628·13^-s + 0.00725·14^-s + 2.32·15^-s − 0.219·16^-s + 2.59·17^-s − 0.0441·18^-s + 1.64·19^-s − 0.453·20^-s − 0.568·21^-s − 0.0223·22^-s − 0.911·23^-s − 0.186·24^-s + 6/5·25^-s − 0.0138·26^-s + 1.92·27^-s + 0.111·28^-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned} \]

\[\begin{aligned} \Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s+11/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

$ d $	=	$6$
$ N $	=	$3375$ = $3^{3} \cdot 5^{3}$
$ \varepsilon $	=	$1$
motivic weight	=	$11$
character	:	induced by $\chi_{15} (1, \cdot )$
primitive	:	no
self-dual	:	yes
analytic rank	=	0
Selberg data	=	$(6,\ 3375,\ (\ :11/2, 11/2, 11/2),\ 1)$
$L(6)$	$\approx$	$10.6184$
$L(\frac12)$	$\approx$	$10.6184$
$L(\frac{13}{2})$		not available
$L(1)$		not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, $F_p$ is a polynomial of degree 6. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 5.

	$p$	$\Gal(F_p)$	$F_p$
bad	3	$C_1$	$ ( 1 - p^{5} T )^{3} $
	5	$C_1$	$ ( 1 - p^{5} T )^{3} $
good	2	$S_4\times C_2$	$ 1 + T + 347 p T^{2} + 709 p^{4} T^{3} + 347 p^{12} T^{4} + p^{22} T^{5} + p^{33} T^{6} $
	7	$S_4\times C_2$	$ 1 + 14608 T + 461534355 p T^{2} + 1634953686752 p^{2} T^{3} + 461534355 p^{12} T^{4} + 14608 p^{22} T^{5} + p^{33} T^{6} $
	11	$S_4\times C_2$	$ 1 - 540620 T + 871024121641 T^{2} - 293660175406171784 T^{3} + 871024121641 p^{11} T^{4} - 540620 p^{22} T^{5} + p^{33} T^{6} $
	13	$S_4\times C_2$	$ 1 - 64690 p T + 1597595357099 T^{2} - 4490565644417681884 T^{3} + 1597595357099 p^{11} T^{4} - 64690 p^{23} T^{5} + p^{33} T^{6} $
	17	$S_4\times C_2$	$ 1 - 15165038 T + 88864608966847 T^{2} - $$38\!\cdots\!44$$ T^{3} + 88864608966847 p^{11} T^{4} - 15165038 p^{22} T^{5} + p^{33} T^{6} $
	19	$S_4\times C_2$	$ 1 - 17743756 T + 229657463589617 T^{2} - $$22\!\cdots\!28$$ T^{3} + 229657463589617 p^{11} T^{4} - 17743756 p^{22} T^{5} + p^{33} T^{6} $
	23	$S_4\times C_2$	$ 1 + 28140816 T + 2322805026860565 T^{2} + $$38\!\cdots\!24$$ T^{3} + 2322805026860565 p^{11} T^{4} + 28140816 p^{22} T^{5} + p^{33} T^{6} $
	29	$S_4\times C_2$	$ 1 + 67382798 T - 2080427172177893 T^{2} - $$21\!\cdots\!16$$ T^{3} - 2080427172177893 p^{11} T^{4} + 67382798 p^{22} T^{5} + p^{33} T^{6} $
	31	$S_4\times C_2$	$ 1 + 206919496 T + 29249324330157597 T^{2} + $$26\!\cdots\!52$$ T^{3} + 29249324330157597 p^{11} T^{4} + 206919496 p^{22} T^{5} + p^{33} T^{6} $
	37	$S_4\times C_2$	$ 1 + 318337278 T + 488167879326286755 T^{2} + $$97\!\cdots\!08$$ T^{3} + 488167879326286755 p^{11} T^{4} + 318337278 p^{22} T^{5} + p^{33} T^{6} $
	41	$S_4\times C_2$	$ 1 - 2110085854 T + 3087038640326735767 T^{2} - $$26\!\cdots\!08$$ T^{3} + 3087038640326735767 p^{11} T^{4} - 2110085854 p^{22} T^{5} + p^{33} T^{6} $
	43	$S_4\times C_2$	$ 1 - 418259692 T + 1797645863194484297 T^{2} - $$99\!\cdots\!80$$ T^{3} + 1797645863194484297 p^{11} T^{4} - 418259692 p^{22} T^{5} + p^{33} T^{6} $
	47	$S_4\times C_2$	$ 1 + 1599668584 T + 5091663441833436973 T^{2} + $$52\!\cdots\!80$$ T^{3} + 5091663441833436973 p^{11} T^{4} + 1599668584 p^{22} T^{5} + p^{33} T^{6} $
	53	$S_4\times C_2$	$ 1 - 4489142234 T + 28713145956507182035 T^{2} - $$73\!\cdots\!56$$ T^{3} + 28713145956507182035 p^{11} T^{4} - 4489142234 p^{22} T^{5} + p^{33} T^{6} $
	59	$S_4\times C_2$	$ 1 - 11102167484 T + 61748067044497465177 T^{2} - $$31\!\cdots\!12$$ T^{3} + 61748067044497465177 p^{11} T^{4} - 11102167484 p^{22} T^{5} + p^{33} T^{6} $
	61	$S_4\times C_2$	$ 1 + 3568120958 T + 27391638978954256379 T^{2} + $$86\!\cdots\!04$$ p T^{3} + 27391638978954256379 p^{11} T^{4} + 3568120958 p^{22} T^{5} + p^{33} T^{6} $
	67	$S_4\times C_2$	$ 1 - 2229942788 T + $$29\!\cdots\!97$$ T^{2} - $$25\!\cdots\!44$$ T^{3} + $$29\!\cdots\!97$$ p^{11} T^{4} - 2229942788 p^{22} T^{5} + p^{33} T^{6} $
	71	$S_4\times C_2$	$ 1 - 49842766696 T + $$14\!\cdots\!85$$ T^{2} - $$26\!\cdots\!00$$ T^{3} + $$14\!\cdots\!85$$ p^{11} T^{4} - 49842766696 p^{22} T^{5} + p^{33} T^{6} $
	73	$S_4\times C_2$	$ 1 - 40752219934 T + $$12\!\cdots\!95$$ T^{2} - $$26\!\cdots\!36$$ T^{3} + $$12\!\cdots\!95$$ p^{11} T^{4} - 40752219934 p^{22} T^{5} + p^{33} T^{6} $
	79	$S_4\times C_2$	$ 1 - 113159960 T + $$18\!\cdots\!37$$ T^{2} + $$13\!\cdots\!20$$ T^{3} + $$18\!\cdots\!37$$ p^{11} T^{4} - 113159960 p^{22} T^{5} + p^{33} T^{6} $
	83	$S_4\times C_2$	$ 1 - 6259660308 T + $$37\!\cdots\!21$$ T^{2} - $$16\!\cdots\!88$$ T^{3} + $$37\!\cdots\!21$$ p^{11} T^{4} - 6259660308 p^{22} T^{5} + p^{33} T^{6} $
	89	$S_4\times C_2$	$ 1 + 59972401554 T + $$29\!\cdots\!27$$ T^{2} + $$84\!\cdots\!12$$ T^{3} + $$29\!\cdots\!27$$ p^{11} T^{4} + 59972401554 p^{22} T^{5} + p^{33} T^{6} $
	97	$S_4\times C_2$	$ 1 + 207831285882 T + $$28\!\cdots\!67$$ T^{2} + $$26\!\cdots\!76$$ T^{3} + $$28\!\cdots\!67$$ p^{11} T^{4} + 207831285882 p^{22} T^{5} + p^{33} T^{6} $
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.71929773106409637050618574288, −14.01721976863694618361139233977, −14.01271793697483430622556587521, −13.91922694574203468931879600809, −12.85982373338608671803792737886, −12.81831625076203930733094673022, −12.20243374244652704098083399569, −11.49595669013411873873776368447, −10.74654215327138089288277065645, −10.00772480036113087759238202734, −9.592079496797552020896706665393, −9.399568815553948585832012583558, −9.122848298850613298242115854820, −8.025314805281953920685288232571, −7.995520899264134392699618666307, −7.10949755406581413442626098762, −6.49390758188442883354091661666, −5.51790093004423822430495691707, −5.39161432704293351256383159161, −3.81134194280549668901374874555, −3.72981044516741783007100280936, −2.92046330832211970092033260221, −2.16047363836934490419594198924, −1.33084012987036030831987407700, −0.981473718487176865459024094512, 0.981473718487176865459024094512, 1.33084012987036030831987407700, 2.16047363836934490419594198924, 2.92046330832211970092033260221, 3.72981044516741783007100280936, 3.81134194280549668901374874555, 5.39161432704293351256383159161, 5.51790093004423822430495691707, 6.49390758188442883354091661666, 7.10949755406581413442626098762, 7.995520899264134392699618666307, 8.025314805281953920685288232571, 9.122848298850613298242115854820, 9.399568815553948585832012583558, 9.592079496797552020896706665393, 10.00772480036113087759238202734, 10.74654215327138089288277065645, 11.49595669013411873873776368447, 12.20243374244652704098083399569, 12.81831625076203930733094673022, 12.85982373338608671803792737886, 13.91922694574203468931879600809, 14.01271793697483430622556587521, 14.01721976863694618361139233977, 14.71929773106409637050618574288

Introduction	Features
Universe	News

Degree	6
Conductor	$ 3^{3} \cdot 5^{3} $
Sign	$1$
Motivic weight	11
Primitive	no
Self-dual	yes
Analytic rank	0

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Dirichlet series

Functional equation

Invariants

Euler product

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

\( d \)	=	\(6\)
\( N \)	=	\(3375\) = \(3^{3} \cdot 5^{3}\)
\( \varepsilon \)	=	$1$
motivic weight	=	\(11\)
character	:	induced by $\chi_{15} (1, \cdot )$
primitive	:	no
self-dual	:	yes
analytic rank	=	0
Selberg data	=	$(6,\ 3375,\ (\ :11/2, 11/2, 11/2),\ 1)$
$L(6)$	$\approx$	$10.6184$
$L(\frac12)$	$\approx$	$10.6184$
$L(\frac{13}{2})$		not available
$L(1)$		not available

	$p$	$\Gal(F_p)$	$F_p$
bad	3	$C_1$	\( ( 1 - p^{5} T )^{3} \)
	5	$C_1$	\( ( 1 - p^{5} T )^{3} \)
good	2	$S_4\times C_2$	\( 1 + T + 347 p T^{2} + 709 p^{4} T^{3} + 347 p^{12} T^{4} + p^{22} T^{5} + p^{33} T^{6} \)
	7	$S_4\times C_2$	\( 1 + 14608 T + 461534355 p T^{2} + 1634953686752 p^{2} T^{3} + 461534355 p^{12} T^{4} + 14608 p^{22} T^{5} + p^{33} T^{6} \)
	11	$S_4\times C_2$	\( 1 - 540620 T + 871024121641 T^{2} - 293660175406171784 T^{3} + 871024121641 p^{11} T^{4} - 540620 p^{22} T^{5} + p^{33} T^{6} \)
	13	$S_4\times C_2$	\( 1 - 64690 p T + 1597595357099 T^{2} - 4490565644417681884 T^{3} + 1597595357099 p^{11} T^{4} - 64690 p^{23} T^{5} + p^{33} T^{6} \)
	17	$S_4\times C_2$	\( 1 - 15165038 T + 88864608966847 T^{2} - \)\(38\!\cdots\!44\)\( T^{3} + 88864608966847 p^{11} T^{4} - 15165038 p^{22} T^{5} + p^{33} T^{6} \)
	19	$S_4\times C_2$	\( 1 - 17743756 T + 229657463589617 T^{2} - \)\(22\!\cdots\!28\)\( T^{3} + 229657463589617 p^{11} T^{4} - 17743756 p^{22} T^{5} + p^{33} T^{6} \)
	23	$S_4\times C_2$	\( 1 + 28140816 T + 2322805026860565 T^{2} + \)\(38\!\cdots\!24\)\( T^{3} + 2322805026860565 p^{11} T^{4} + 28140816 p^{22} T^{5} + p^{33} T^{6} \)
	29	$S_4\times C_2$	\( 1 + 67382798 T - 2080427172177893 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} - 2080427172177893 p^{11} T^{4} + 67382798 p^{22} T^{5} + p^{33} T^{6} \)
	31	$S_4\times C_2$	\( 1 + 206919496 T + 29249324330157597 T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + 29249324330157597 p^{11} T^{4} + 206919496 p^{22} T^{5} + p^{33} T^{6} \)
	37	$S_4\times C_2$	\( 1 + 318337278 T + 488167879326286755 T^{2} + \)\(97\!\cdots\!08\)\( T^{3} + 488167879326286755 p^{11} T^{4} + 318337278 p^{22} T^{5} + p^{33} T^{6} \)
	41	$S_4\times C_2$	\( 1 - 2110085854 T + 3087038640326735767 T^{2} - \)\(26\!\cdots\!08\)\( T^{3} + 3087038640326735767 p^{11} T^{4} - 2110085854 p^{22} T^{5} + p^{33} T^{6} \)
	43	$S_4\times C_2$	\( 1 - 418259692 T + 1797645863194484297 T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + 1797645863194484297 p^{11} T^{4} - 418259692 p^{22} T^{5} + p^{33} T^{6} \)
	47	$S_4\times C_2$	\( 1 + 1599668584 T + 5091663441833436973 T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + 5091663441833436973 p^{11} T^{4} + 1599668584 p^{22} T^{5} + p^{33} T^{6} \)
	53	$S_4\times C_2$	\( 1 - 4489142234 T + 28713145956507182035 T^{2} - \)\(73\!\cdots\!56\)\( T^{3} + 28713145956507182035 p^{11} T^{4} - 4489142234 p^{22} T^{5} + p^{33} T^{6} \)
	59	$S_4\times C_2$	\( 1 - 11102167484 T + 61748067044497465177 T^{2} - \)\(31\!\cdots\!12\)\( T^{3} + 61748067044497465177 p^{11} T^{4} - 11102167484 p^{22} T^{5} + p^{33} T^{6} \)
	61	$S_4\times C_2$	\( 1 + 3568120958 T + 27391638978954256379 T^{2} + \)\(86\!\cdots\!04\)\( p T^{3} + 27391638978954256379 p^{11} T^{4} + 3568120958 p^{22} T^{5} + p^{33} T^{6} \)
	67	$S_4\times C_2$	\( 1 - 2229942788 T + \)\(29\!\cdots\!97\)\( T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!97\)\( p^{11} T^{4} - 2229942788 p^{22} T^{5} + p^{33} T^{6} \)
	71	$S_4\times C_2$	\( 1 - 49842766696 T + \)\(14\!\cdots\!85\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!85\)\( p^{11} T^{4} - 49842766696 p^{22} T^{5} + p^{33} T^{6} \)
	73	$S_4\times C_2$	\( 1 - 40752219934 T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!95\)\( p^{11} T^{4} - 40752219934 p^{22} T^{5} + p^{33} T^{6} \)
	79	$S_4\times C_2$	\( 1 - 113159960 T + \)\(18\!\cdots\!37\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!37\)\( p^{11} T^{4} - 113159960 p^{22} T^{5} + p^{33} T^{6} \)
	83	$S_4\times C_2$	\( 1 - 6259660308 T + \)\(37\!\cdots\!21\)\( T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(37\!\cdots\!21\)\( p^{11} T^{4} - 6259660308 p^{22} T^{5} + p^{33} T^{6} \)
	89	$S_4\times C_2$	\( 1 + 59972401554 T + \)\(29\!\cdots\!27\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{3} + \)\(29\!\cdots\!27\)\( p^{11} T^{4} + 59972401554 p^{22} T^{5} + p^{33} T^{6} \)
	97	$S_4\times C_2$	\( 1 + 207831285882 T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(28\!\cdots\!67\)\( p^{11} T^{4} + 207831285882 p^{22} T^{5} + p^{33} T^{6} \)
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