# Properties

 Degree $8$ Conductor $9.683\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·2-s + 8·4-s − 8·8-s − 4·16-s + 14·25-s − 16·29-s + 32·32-s + 12·37-s − 56·50-s + 4·53-s + 64·58-s − 64·64-s − 48·74-s + 112·100-s − 16·106-s + 36·109-s + 64·113-s − 128·116-s + 42·121-s + 127-s + 64·128-s + 131-s + 137-s + 139-s + 96·148-s + 149-s + 151-s + ⋯
 L(s)  = 1 − 2.82·2-s + 4·4-s − 2.82·8-s − 16-s + 14/5·25-s − 2.97·29-s + 5.65·32-s + 1.97·37-s − 7.91·50-s + 0.549·53-s + 8.40·58-s − 8·64-s − 5.57·74-s + 56/5·100-s − 1.55·106-s + 3.44·109-s + 6.02·113-s − 11.8·116-s + 3.81·121-s + 0.0887·127-s + 5.65·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 7.89·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{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 3^{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: $$2^{8} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1764} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5918901884$$ $$L(\frac12)$$ $$\approx$$ $$0.5918901884$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + p T + p T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$C_2^2$ $$( 1 - 7 T^{2} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 21 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 31 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 11 T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 45 T^{2} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 + 59 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2$ $$( 1 - 3 T + p T^{2} )^{4}$$
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$C_2^2$ $$( 1 + 19 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2$ $$( 1 - T + p T^{2} )^{4}$$
59$C_2^2$ $$( 1 + 91 T^{2} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 95 T^{2} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 125 T^{2} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 54 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 71 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 77 T^{2} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 26 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 65 T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 + 106 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$