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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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]