# Properties

 Degree 8 Conductor $2^{20} \cdot 7^{4}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·5-s − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 − 2·5-s − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{20} \cdot 7^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{224} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{20} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.1812040511$$ $$L(\frac12)$$ $$\approx$$ $$0.1812040511$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;7\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
7$C_2$ $$( 1 + T^{2} )^{2}$$
good3$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
5$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
11$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
13$C_2$ $$( 1 + T^{2} )^{4}$$
17$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
19$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
23$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
29$C_2$ $$( 1 + T^{2} )^{4}$$
31$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
37$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
41$C_2$ $$( 1 + T^{2} )^{4}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
47$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
53$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
59$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
61$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
67$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
73$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
79$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
83$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
89$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
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}