# Properties

 Degree 8 Conductor $2^{12} \cdot 3^{8} \cdot 5^{8}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯
 L(s)  = 1 + 4-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{12} \cdot 3^{8} \cdot 5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{1800} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 2^{12} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$ $L(\frac{1}{2})$ $\approx$ $0.9506311319$ $L(\frac12)$ $\approx$ $0.9506311319$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5\}$, $$F_p$$ is a polynomial of degree 8. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad2$C_2^2$ $$1 - T^{2} + T^{4}$$
3$C_2^2$ $$1 - T^{2} + T^{4}$$
5 $$1$$
good7$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
13$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
17$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
19$C_1$ $$( 1 + T )^{8}$$
23$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
29$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
37$C_2$ $$( 1 + T^{2} )^{4}$$
41$C_2$ $$( 1 + T + T^{2} )^{4}$$
43$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
47$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
53$C_2$ $$( 1 + T^{2} )^{4}$$
59$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
61$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
67$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
73$C_2$ $$( 1 + T^{2} )^{4}$$
79$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
83$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
89$C_2$ $$( 1 - T + T^{2} )^{4}$$
97$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}