# Properties

 Label 8-2352e4-1.1-c1e4-0-3 Degree $8$ Conductor $3.060\times 10^{13}$ Sign $1$ Analytic cond. $124410.$ Root an. cond. $4.33368$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 4·5-s + 9-s − 4·11-s + 8·15-s + 12·17-s − 8·19-s − 4·23-s + 12·25-s − 2·27-s − 8·33-s − 8·37-s − 24·41-s + 4·45-s − 8·47-s + 24·51-s + 4·53-s − 16·55-s − 16·57-s + 8·61-s − 8·69-s + 8·71-s + 24·73-s + 24·75-s + 16·79-s − 4·81-s − 16·83-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.78·5-s + 1/3·9-s − 1.20·11-s + 2.06·15-s + 2.91·17-s − 1.83·19-s − 0.834·23-s + 12/5·25-s − 0.384·27-s − 1.39·33-s − 1.31·37-s − 3.74·41-s + 0.596·45-s − 1.16·47-s + 3.36·51-s + 0.549·53-s − 2.15·55-s − 2.11·57-s + 1.02·61-s − 0.963·69-s + 0.949·71-s + 2.80·73-s + 2.77·75-s + 1.80·79-s − 4/9·81-s − 1.75·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \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^{16} \cdot 3^{4} \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: $$2^{16} \cdot 3^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$124410.$$ Root analytic conductor: $$4.33368$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.623148297$$ $$L(\frac12)$$ $$\approx$$ $$1.623148297$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_2$ $$( 1 - T + T^{2} )^{2}$$
7 $$1$$
good5$D_4\times C_2$ $$1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 + 8 T^{2} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 12 T + 76 T^{2} - 24 p T^{3} + 111 p T^{4} - 24 p^{2} T^{5} + 76 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
19$D_4\times C_2$ $$1 + 8 T + 18 T^{2} + 64 T^{3} + 539 T^{4} + 64 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
23$C_4\times C_2$ $$1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^3$ $$1 - 54 T^{2} + 1955 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 42 T^{2} + p^{2} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 8 T + 26 T^{2} - 448 T^{3} - 3773 T^{4} - 448 p T^{5} + 26 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 - 46 T^{2} - 1365 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 - 8 T - 24 T^{2} + 272 T^{3} + 119 T^{4} + 272 p T^{5} - 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2^3$ $$1 - 6 T^{2} - 4453 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 24 T + 304 T^{2} - 3024 T^{3} + 26607 T^{4} - 3024 p T^{5} + 304 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 16 T + 66 T^{2} - 512 T^{3} + 9635 T^{4} - 512 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
89$D_4\times C_2$ $$1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 1640 p T^{5} + 140 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
97$D_{4}$ $$( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$