# Properties

 Degree $8$ Conductor $6.154\times 10^{12}$ Sign $1$ Motivic weight $0$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯
 L(s)  = 1 + 2·4-s − 2·9-s + 2·11-s + 16-s + 4·29-s − 4·36-s + 4·44-s + 49-s − 2·64-s − 4·71-s − 2·79-s + 3·81-s − 4·99-s + 4·109-s + 8·116-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$0$$ Character: induced by $\chi_{1575} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.063654534$$ $$L(\frac12)$$ $$\approx$$ $$2.063654534$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ $$( 1 + T^{2} )^{2}$$
5 $$1$$
7$C_2^2$ $$1 - T^{2} + T^{4}$$
good2$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{4}( 1 + T + T^{2} )^{2}$$
13$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
17$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
23$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
29$C_2$ $$( 1 - T + T^{2} )^{4}$$
31$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
37$C_2$ $$( 1 + T^{2} )^{4}$$
41$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
43$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
47$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
53$C_2$ $$( 1 + T^{2} )^{4}$$
59$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
61$C_2$ $$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$
67$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
71$C_2$ $$( 1 + T + T^{2} )^{4}$$
73$C_2^2$ $$( 1 - T^{2} + T^{4} )^{2}$$
79$C_1$$\times$$C_2$ $$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$
83$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{4}( 1 + T )^{4}$$
97$C_2$$\times$$C_2^2$ $$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$