Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

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$
bad3$C_1$ \( ( 1 - p^{5} T )^{3} \)
5$C_1$ \( ( 1 - p^{5} T )^{3} \)
good2$S_4\times C_2$ \( 1 + T + 347 p T^{2} + 709 p^{4} T^{3} + 347 p^{12} T^{4} + p^{22} T^{5} + p^{33} T^{6} \)
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$S_4\times C_2$ \( 1 + 207831285882 T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(28\!\cdots\!67\)\( p^{11} T^{4} + 207831285882 p^{22} T^{5} + p^{33} T^{6} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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

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