# Properties

 Degree 8 Conductor $5^{4} \cdot 13^{4} \cdot 31^{4}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·4-s + 10·16-s − 2·25-s − 4·49-s − 20·64-s − 2·81-s + 8·100-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 16·196-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 − 4·4-s + 10·16-s − 2·25-s − 4·49-s − 20·64-s − 2·81-s + 8·100-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 16·196-s + 197-s + 199-s + 211-s + 223-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$5^{4} \cdot 13^{4} \cdot 31^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{2015} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 5^{4} \cdot 13^{4} \cdot 31^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$ $L(\frac{1}{2})$ $\approx$ $0.002897918371$ $L(\frac12)$ $\approx$ $0.002897918371$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;13,\;31\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_2^2$ $$1 + T^{4}$$
31$C_2$ $$( 1 + T^{2} )^{2}$$
good2$C_2$ $$( 1 + T^{2} )^{4}$$
3$C_2^2$ $$( 1 + T^{4} )^{2}$$
7$C_2$ $$( 1 + T^{2} )^{4}$$
11$C_2$ $$( 1 + T^{2} )^{4}$$
17$C_2^2$ $$( 1 + T^{4} )^{2}$$
19$C_2$ $$( 1 + T^{2} )^{4}$$
23$C_2^2$ $$( 1 + T^{4} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
37$C_2^2$ $$( 1 + T^{4} )^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
43$C_2^2$ $$( 1 + T^{4} )^{2}$$
47$C_2$ $$( 1 + T^{2} )^{4}$$
53$C_2^2$ $$( 1 + T^{4} )^{2}$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
61$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
67$C_2$ $$( 1 + T^{2} )^{4}$$
71$C_2$ $$( 1 + T^{2} )^{4}$$
73$C_2^2$ $$( 1 + T^{4} )^{2}$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
83$C_2^2$ $$( 1 + T^{4} )^{2}$$
89$C_2$ $$( 1 + T^{2} )^{4}$$
97$C_2$ $$( 1 + T^{2} )^{4}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}