# Properties

 Label 8-585e4-1.1-c1e4-0-9 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 + 6·3-s + 21·9-s + 6·13-s + 4·16-s − 4·17-s − 2·23-s + 25-s + 54·27-s − 4·29-s + 36·39-s − 10·43-s + 24·48-s − 10·49-s − 24·51-s + 36·53-s + 10·61-s − 12·69-s + 6·75-s − 2·79-s + 108·81-s − 24·87-s − 14·101-s − 32·103-s + 68·107-s − 38·113-s + 126·117-s + 14·121-s + ⋯
 L(s)  = 1 + 3.46·3-s + 7·9-s + 1.66·13-s + 16-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 10.3·27-s − 0.742·29-s + 5.76·39-s − 1.52·43-s + 3.46·48-s − 1.42·49-s − 3.36·51-s + 4.94·53-s + 1.28·61-s − 1.44·69-s + 0.692·75-s − 0.225·79-s + 12·81-s − 2.57·87-s − 1.39·101-s − 3.15·103-s + 6.57·107-s − 3.57·113-s + 11.6·117-s + 1.27·121-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

 $$L(1)$$ $$\approx$$ $$13.70439053$$ $$L(\frac12)$$ $$\approx$$ $$13.70439053$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 - p T + p T^{2} )^{2}$$
5$C_2^2$ $$1 - T^{2} + T^{4}$$
13$C_2^2$ $$1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
good2$C_2^2$$\times$$C_2^2$ $$( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )$$
7$C_2^3$ $$1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^3$ $$1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 + T + p T^{2} )^{4}$$
19$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^3$ $$1 - 2 T^{2} - 957 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2$ $$( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2}$$
41$C_2^3$ $$1 + 78 T^{2} + 4403 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8}$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}$$
47$C_2^3$ $$1 - 6 T^{2} - 2173 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2$ $$( 1 - 9 T + p T^{2} )^{4}$$
59$C_2^3$ $$1 + 82 T^{2} + 3243 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 42 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 130 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
89$C_2$ $$( 1 - p T^{2} )^{4}$$
97$C_2^3$ $$1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$