# Properties

 Degree $8$ Conductor $1944810000$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s − 2·4-s + 4·5-s + 6·7-s + 2·9-s + 4·12-s − 8·15-s + 3·16-s + 12·17-s − 8·20-s − 12·21-s + 10·25-s − 6·27-s − 12·28-s + 24·35-s − 4·36-s + 12·37-s + 8·41-s + 16·43-s + 8·45-s − 8·47-s − 6·48-s + 18·49-s − 24·51-s + 16·60-s + 12·63-s − 4·64-s + ⋯
 L(s)  = 1 − 1.15·3-s − 4-s + 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s − 2.06·15-s + 3/4·16-s + 2.91·17-s − 1.78·20-s − 2.61·21-s + 2·25-s − 1.15·27-s − 2.26·28-s + 4.05·35-s − 2/3·36-s + 1.97·37-s + 1.24·41-s + 2.43·43-s + 1.19·45-s − 1.16·47-s − 0.866·48-s + 18/7·49-s − 3.36·51-s + 2.06·60-s + 1.51·63-s − 1/2·64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{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 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{210} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.85122$$ $$L(\frac12)$$ $$\approx$$ $$1.85122$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
5$C_1$ $$( 1 - T )^{4}$$
7$C_2^2$ $$1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
good11$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )^{2}$$
13$D_4\times C_2$ $$1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}$$
17$D_{4}$ $$( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
19$C_4\times C_2$ $$1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 - 30 T^{2} + p^{2} T^{4} )^{2}$$
29$D_4\times C_2$ $$1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8}$$
37$D_{4}$ $$( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
41$D_{4}$ $$( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_{4}$ $$( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
47$D_{4}$ $$( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
53$D_4\times C_2$ $$1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 + 98 T^{2} + p^{2} T^{4} )^{2}$$
61$D_4\times C_2$ $$1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2$ $$( 1 + 12 T + p T^{2} )^{4}$$
71$D_4\times C_2$ $$1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 + 78 T^{2} + p^{2} T^{4} )^{2}$$
83$D_{4}$ $$( 1 + 2 T - 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
89$D_{4}$ $$( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2}$$
97$D_4\times C_2$ $$1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$