# Properties

 Degree $8$ Conductor $1.032\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·5-s − 18·17-s + 11·25-s − 14·37-s + 14·49-s + 6·53-s + 6·61-s − 18·73-s − 108·85-s − 6·89-s − 18·101-s + 14·109-s − 48·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 2.68·5-s − 4.36·17-s + 11/5·25-s − 2.30·37-s + 2·49-s + 0.824·53-s + 0.768·61-s − 2.10·73-s − 11.7·85-s − 0.635·89-s − 1.79·101-s + 1.34·109-s − 4.51·113-s − 0.0909·121-s − 0.536·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1008} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2746018601$$ $$L(\frac12)$$ $$\approx$$ $$0.2746018601$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
good5$C_2^2$ $$( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^3$ $$1 + T^{2} - 120 T^{4} + p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2^3$ $$1 - 31 T^{2} + 600 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^3$ $$1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2^3$ $$1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2^2$ $$( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 31 T^{2} - 1248 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 113 T^{2} + 8280 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^3$ $$1 + 137 T^{2} + 12528 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2^2$ $$( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 182 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$