# Properties

 Degree 8 Conductor $2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 4

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·5-s − 10·7-s − 3·9-s − 6·11-s − 12·13-s + 3·16-s + 6·19-s + 4·20-s − 12·23-s + 25-s + 20·28-s − 12·29-s + 20·35-s + 6·36-s − 2·37-s + 18·41-s + 4·43-s + 12·44-s + 6·45-s − 36·47-s + 61·49-s + 24·52-s − 6·53-s + 12·55-s − 12·59-s + 30·63-s + ⋯
 L(s)  = 1 − 4-s − 0.894·5-s − 3.77·7-s − 9-s − 1.80·11-s − 3.32·13-s + 3/4·16-s + 1.37·19-s + 0.894·20-s − 2.50·23-s + 1/5·25-s + 3.77·28-s − 2.22·29-s + 3.38·35-s + 36-s − 0.328·37-s + 2.81·41-s + 0.609·43-s + 1.80·44-s + 0.894·45-s − 5.25·47-s + 61/7·49-s + 3.32·52-s − 0.824·53-s + 1.61·55-s − 1.56·59-s + 3.77·63-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{630} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$4$$ Selberg data = $$(8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 + T + T^{2} )^{2}$$
7$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
good11$D_4\times C_2$ $$1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 12 T + 70 T^{2} + 264 T^{3} + 1059 T^{4} + 264 p T^{5} + 70 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 12 T + 109 T^{2} + 732 T^{3} + 4272 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 52 T^{2} + 1626 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 2 T - 44 T^{2} - 52 T^{3} + 787 T^{4} - 52 p T^{5} - 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 4 T - 47 T^{2} + 92 T^{3} + 1432 T^{4} + 92 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2$ $$( 1 + 9 T + p T^{2} )^{4}$$
53$D_4\times C_2$ $$1 + 6 T + 40 T^{2} + 168 T^{3} - 1389 T^{4} + 168 p T^{5} + 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
71$D_4\times C_2$ $$1 - 260 T^{2} + 26874 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
79$D_{4}$ $$( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 + 12 T - 31 T^{2} + 108 T^{3} + 11784 T^{4} + 108 p T^{5} - 31 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 + 12 T + 38 T^{2} - 864 T^{3} - 9501 T^{4} - 864 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 12 T + 110 T^{2} - 744 T^{3} - 909 T^{4} - 744 p T^{5} + 110 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}