# Properties

 Degree $8$ Conductor $1.100\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·17-s − 32·47-s − 4·49-s − 32·79-s + 14·81-s − 8·97-s − 8·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
 L(s)  = 1 + 1.94·17-s − 4.66·47-s − 4/7·49-s − 3.60·79-s + 14/9·81-s − 0.812·97-s − 0.752·113-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{40}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1024} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{40} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.8901810629$$ $$L(\frac12)$$ $$\approx$$ $$0.8901810629$$ $$L(\frac{3}{2})$$ 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 $$1$$
good3$C_2^2$$\times$$C_2^2$ $$( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )$$
5$C_2^2$$\times$$C_2^2$ $$( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} )$$
7$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^3$ $$1 + 82 T^{4} + p^{4} T^{8}$$
13$C_2^3$ $$1 + 146 T^{4} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
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$C_2^2$ $$( 1 - 30 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^3$ $$1 - 1198 T^{4} + p^{4} T^{8}$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2^3$ $$1 - 2062 T^{4} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^3$ $$1 - 1198 T^{4} + p^{4} T^{8}$$
47$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
53$C_2^3$ $$1 - 718 T^{4} + p^{4} T^{8}$$
59$C_2^3$ $$1 - 878 T^{4} + p^{4} T^{8}$$
61$C_2^3$ $$1 + 6482 T^{4} + p^{4} T^{8}$$
67$C_2^3$ $$1 - 7822 T^{4} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 + 8 T + p T^{2} )^{4}$$
83$C_2^3$ $$1 + 3122 T^{4} + p^{4} T^{8}$$
89$C_2^2$ $$( 1 - 174 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$