# Properties

 Label 8-2352e4-1.1-c2e4-0-10 Degree $8$ Conductor $3.060\times 10^{13}$ Sign $1$ Analytic cond. $1.68690\times 10^{7}$ Root an. cond. $8.00545$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 12·5-s − 6·9-s − 36·17-s + 64·25-s + 48·29-s + 64·37-s + 132·41-s − 72·45-s + 192·53-s + 264·61-s − 24·73-s + 27·81-s − 432·85-s + 276·89-s + 600·97-s − 252·101-s − 280·109-s − 168·113-s + 256·121-s + 372·125-s + 127-s + 131-s + 137-s + 139-s + 576·145-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 12/5·5-s − 2/3·9-s − 2.11·17-s + 2.55·25-s + 1.65·29-s + 1.72·37-s + 3.21·41-s − 8/5·45-s + 3.62·53-s + 4.32·61-s − 0.328·73-s + 1/3·81-s − 5.08·85-s + 3.10·89-s + 6.18·97-s − 2.49·101-s − 2.56·109-s − 1.48·113-s + 2.11·121-s + 2.97·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.97·145-s + 0.00671·149-s + 0.00662·151-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{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: $$1.68690\times 10^{7}$$ Root analytic conductor: $$8.00545$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{2352} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$18.52378418$$ $$L(\frac12)$$ $$\approx$$ $$18.52378418$$ $$L(2)$$ 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$C_2$ $$( 1 + p T^{2} )^{2}$$
7 $$1$$
good5$D_{4}$ $$( 1 - 6 T + 22 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 256 T^{2} + 44334 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8}$$
13$C_2$ $$( 1 + p^{2} T^{2} )^{4}$$
17$D_{4}$ $$( 1 + 18 T + 622 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 28 p T^{2} + 310086 T^{4} - 28 p^{5} T^{6} + p^{8} T^{8}$$
23$D_4\times C_2$ $$1 - 1168 T^{2} + 739566 T^{4} - 1168 p^{4} T^{6} + p^{8} T^{8}$$
29$D_{4}$ $$( 1 - 24 T + 494 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 - 76 p T^{2} + 2701926 T^{4} - 76 p^{5} T^{6} + p^{8} T^{8}$$
37$D_{4}$ $$( 1 - 32 T + 1662 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
41$D_{4}$ $$( 1 - 66 T + 4414 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 2930 T^{2} + p^{4} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 4420 T^{2} + 11574534 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 - 96 T + 6590 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 5186 T^{2} + p^{4} T^{4} )^{2}$$
61$D_{4}$ $$( 1 - 132 T + 10466 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 9748 T^{2} + 62331846 T^{4} - 9748 p^{4} T^{6} + p^{8} T^{8}$$
71$D_4\times C_2$ $$1 - 14320 T^{2} + 100457262 T^{4} - 14320 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 12 T + 9362 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 1972 T^{2} - 41007642 T^{4} - 1972 p^{4} T^{6} + p^{8} T^{8}$$
83$D_4\times C_2$ $$1 - 20548 T^{2} + 188196006 T^{4} - 20548 p^{4} T^{6} + p^{8} T^{8}$$
89$D_{4}$ $$( 1 - 138 T + 14350 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$D_{4}$ $$( 1 - 300 T + 39986 T^{2} - 300 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$