# Properties

 Label 8-260e4-1.1-c1e4-0-1 Degree $8$ Conductor $4569760000$ Sign $1$ Analytic cond. $18.5781$ Root an. cond. $1.44087$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 2·4-s − 4·5-s + 4·8-s − 8·10-s − 6·13-s + 8·16-s − 4·17-s − 8·20-s + 5·25-s − 12·26-s − 30·29-s + 8·32-s − 8·34-s − 22·37-s − 16·40-s + 24·41-s + 10·50-s − 12·52-s − 10·53-s − 60·58-s − 12·61-s + 8·64-s + 24·65-s − 8·68-s + 22·73-s − 44·74-s + ⋯
 L(s)  = 1 + 1.41·2-s + 4-s − 1.78·5-s + 1.41·8-s − 2.52·10-s − 1.66·13-s + 2·16-s − 0.970·17-s − 1.78·20-s + 25-s − 2.35·26-s − 5.57·29-s + 1.41·32-s − 1.37·34-s − 3.61·37-s − 2.52·40-s + 3.74·41-s + 1.41·50-s − 1.66·52-s − 1.37·53-s − 7.87·58-s − 1.53·61-s + 64-s + 2.97·65-s − 0.970·68-s + 2.57·73-s − 5.11·74-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{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(2^{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: $$2^{8} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$18.5781$$ Root analytic conductor: $$1.44087$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{260} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.300859861$$ $$L(\frac12)$$ $$\approx$$ $$1.300859861$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4}$$
5$C_2^2$ $$1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
good3$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
7$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
17$C_2$$\times$$C_2^2$ $$( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )$$
19$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
29$C_2$$\times$$C_2^2$ $$( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} )$$
31$C_2$ $$( 1 + p T^{2} )^{4}$$
37$C_2$$\times$$C_2^2$ $$( 1 + 12 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )$$
41$C_2$$\times$$C_2^2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )$$
43$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
47$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
53$C_2^2$$\times$$C_2^2$ $$( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )$$
59$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
61$C_2$$\times$$C_2^2$ $$( 1 + 12 T + p T^{2} )^{2}( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} )$$
67$C_2^3$ $$1 - p^{2} T^{4} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$$\times$$C_2^2$ $$( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )$$
79$C_2$ $$( 1 + p T^{2} )^{4}$$
83$C_2^2$ $$( 1 + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$$\times$$C_2^2$ $$( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$