# Properties

 Degree $8$ Conductor $2.818\times 10^{13}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·5-s + 12·13-s + 24·17-s + 32·25-s − 16·29-s + 12·37-s − 8·49-s − 16·53-s + 12·61-s − 96·65-s − 192·85-s − 16·97-s − 32·101-s − 28·109-s − 24·113-s − 104·125-s + 127-s + 131-s + 137-s + 139-s + 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯
 L(s)  = 1 − 3.57·5-s + 3.32·13-s + 5.82·17-s + 32/5·25-s − 2.97·29-s + 1.97·37-s − 8/7·49-s − 2.19·53-s + 1.53·61-s − 11.9·65-s − 20.8·85-s − 1.62·97-s − 3.18·101-s − 2.68·109-s − 2.25·113-s − 9.30·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{32} \cdot 3^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{2304} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.870932961$$ $$L(\frac12)$$ $$\approx$$ $$1.870932961$$ $$L(\frac{3}{2})$$ 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)$
bad2 $$1$$
3 $$1$$
good5$C_2^2$ $$( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + 4 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^3$ $$1 - 206 T^{4} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
19$C_2^2$$\times$$C_2^2$ $$( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )$$
23$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 44 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
43$C_2^3$ $$1 - 1198 T^{4} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + 86 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 4946 T^{4} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - 134 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2$ $$( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2}$$
79$C_2^2$ $$( 1 + 140 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^3$ $$1 - 5678 T^{4} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 + 18 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$