# Properties

 Degree $8$ Conductor $1475789056$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 2·4-s + 6·5-s − 4·8-s + 3·9-s − 12·10-s + 8·16-s + 6·17-s − 6·18-s + 12·20-s + 11·25-s + 16·29-s − 8·32-s − 12·34-s + 6·36-s − 6·37-s − 24·40-s + 18·45-s − 22·50-s + 2·53-s − 32·58-s + 18·61-s + 8·64-s + 12·68-s − 12·72-s − 30·73-s + 12·74-s + ⋯
 L(s)  = 1 − 1.41·2-s + 4-s + 2.68·5-s − 1.41·8-s + 9-s − 3.79·10-s + 2·16-s + 1.45·17-s − 1.41·18-s + 2.68·20-s + 11/5·25-s + 2.97·29-s − 1.41·32-s − 2.05·34-s + 36-s − 0.986·37-s − 3.79·40-s + 2.68·45-s − 3.11·50-s + 0.274·53-s − 4.20·58-s + 2.30·61-s + 64-s + 1.45·68-s − 1.41·72-s − 3.51·73-s + 1.39·74-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\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 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{196} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.33175$$ $$L(\frac12)$$ $$\approx$$ $$1.33175$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
7 $$1$$
good3$C_2$$\times$$C_2^2$ $$( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )$$
5$C_2^2$ $$( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^3$ $$1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_2^2$$\times$$C_2^2$ $$( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} )$$
23$C_2^3$ $$1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
31$C_2^2$$\times$$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} )$$
37$C_2^2$ $$( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 82 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2^2$ $$( 1 + 54 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 15 T + 148 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2^3$ $$1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 106 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$