# Properties

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

# Origins of factors

## 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$S_4\times C_2$ $$1 + 207831285882 T +$$$$28\!\cdots\!67$$$$T^{2} +$$$$26\!\cdots\!76$$$$T^{3} +$$$$28\!\cdots\!67$$$$p^{11} T^{4} + 207831285882 p^{22} T^{5} + p^{33} T^{6}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}