# Properties

 Label 8-585e4-1.1-c1e4-0-1 Degree $8$ Conductor $117117950625$ Sign $1$ Analytic cond. $476.136$ Root an. cond. $2.16130$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·4-s + 40·16-s − 10·25-s − 2·49-s − 28·61-s − 160·64-s + 44·79-s + 80·100-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 16·196-s + 197-s + 199-s + ⋯
 L(s)  = 1 − 4·4-s + 10·16-s − 2·25-s − 2/7·49-s − 3.58·61-s − 20·64-s + 4.95·79-s + 8·100-s + 3.09·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 8/7·196-s + 0.0712·197-s + 0.0708·199-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$476.136$$ Root analytic conductor: $$2.16130$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4409726671$$ $$L(\frac12)$$ $$\approx$$ $$0.4409726671$$ $$L(\frac{3}{2})$$ 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)$
bad3 $$1$$
5$C_2$ $$( 1 + p T^{2} )^{2}$$
13$C_2$ $$( 1 - p T^{2} )^{2}$$
good2$C_2$ $$( 1 + p T^{2} )^{4}$$
7$C_2^2$ $$( 1 + T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 17 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 + 31 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 - p T^{2} )^{4}$$
23$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
31$C_2$ $$( 1 - p T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 61 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 43 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - p T^{2} )^{4}$$
47$C_2$ $$( 1 + p T^{2} )^{4}$$
53$C_2^2$ $$( 1 - 41 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 38 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 + 7 T + p T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 74 T^{2} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 103 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 94 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 - 11 T + p T^{2} )^{4}$$
83$C_2$ $$( 1 + p T^{2} )^{4}$$
89$C_2^2$ $$( 1 - 173 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 181 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$