# Properties

 Label 8-390e4-1.1-c1e4-0-13 Degree $8$ Conductor $23134410000$ Sign $1$ Analytic cond. $94.0517$ Root an. cond. $1.76469$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s − 2·4-s + 10·9-s − 8·12-s + 6·13-s + 3·16-s + 8·17-s − 4·23-s − 2·25-s + 20·27-s + 8·29-s − 20·36-s + 24·39-s − 8·43-s + 12·48-s − 8·49-s + 32·51-s − 12·52-s − 4·53-s + 40·61-s − 4·64-s − 16·68-s − 16·69-s − 8·75-s + 32·79-s + 35·81-s + 32·87-s + ⋯
 L(s)  = 1 + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 1.66·13-s + 3/4·16-s + 1.94·17-s − 0.834·23-s − 2/5·25-s + 3.84·27-s + 1.48·29-s − 3.33·36-s + 3.84·39-s − 1.21·43-s + 1.73·48-s − 8/7·49-s + 4.48·51-s − 1.66·52-s − 0.549·53-s + 5.12·61-s − 1/2·64-s − 1.94·68-s − 1.92·69-s − 0.923·75-s + 3.60·79-s + 35/9·81-s + 3.43·87-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \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(2^{4} \cdot 3^{4} \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: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$94.0517$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$5.692411814$$ $$L(\frac12)$$ $$\approx$$ $$5.692411814$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_1$ $$( 1 - T )^{4}$$
5$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_2^2$ $$1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
good7$D_4\times C_2$ $$1 + 8 T^{2} + 46 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
19$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
23$D_{4}$ $$( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
31$D_4\times C_2$ $$1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 80 T^{2} + 3262 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 44 T^{2} + 3814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}$$
53$D_{4}$ $$( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 184 T^{2} + 14814 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2$ $$( 1 - 10 T + p T^{2} )^{4}$$
67$D_4\times C_2$ $$1 - 72 T^{2} + 4766 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 140 T^{2} + 13894 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 96 T^{2} + 7454 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2$ $$( 1 - 8 T + p T^{2} )^{4}$$
83$D_4\times C_2$ $$1 - 188 T^{2} + 21526 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 192 T^{2} + 22526 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$