# Properties

 Label 8-2352e4-1.1-c2e4-0-4 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 − 24·5-s − 6·9-s − 48·13-s − 24·17-s + 264·25-s + 96·29-s + 72·41-s + 144·45-s − 88·53-s − 96·61-s + 1.15e3·65-s − 48·73-s + 27·81-s + 576·85-s + 120·89-s − 432·97-s + 72·101-s − 48·109-s + 56·113-s + 288·117-s + 76·121-s − 1.46e3·125-s + 127-s + 131-s + 137-s + 139-s − 2.30e3·145-s + ⋯
 L(s)  = 1 − 4.79·5-s − 2/3·9-s − 3.69·13-s − 1.41·17-s + 10.5·25-s + 3.31·29-s + 1.75·41-s + 16/5·45-s − 1.66·53-s − 1.57·61-s + 17.7·65-s − 0.657·73-s + 1/3·81-s + 6.77·85-s + 1.34·89-s − 4.45·97-s + 0.712·101-s − 0.440·109-s + 0.495·113-s + 2.46·117-s + 0.628·121-s − 11.7·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 15.8·145-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

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.1413504263$$ $$L(\frac12)$$ $$\approx$$ $$0.1413504263$$ $$L(2)$$ 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$D_{4}$ $$( 1 + 12 T + 84 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 76 T^{2} - 10746 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8}$$
13$D_{4}$ $$( 1 + 24 T + 384 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
17$D_{4}$ $$( 1 + 12 T + 564 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - 506 T^{2} + p^{4} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 + 20 T^{2} + 186534 T^{4} + 20 p^{4} T^{6} + p^{8} T^{8}$$
29$D_{4}$ $$( 1 - 48 T + 2186 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 140 T^{2} + 1478694 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8}$$
37$C_2^2$ $$( 1 + 1586 T^{2} + p^{4} T^{4} )^{2}$$
41$D_{4}$ $$( 1 - 36 T + 3588 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 - 1732 T^{2} + 3440358 T^{4} - 1732 p^{4} T^{6} + p^{8} T^{8}$$
47$D_4\times C_2$ $$1 - 8020 T^{2} + 25673574 T^{4} - 8020 p^{4} T^{6} + p^{8} T^{8}$$
53$D_{4}$ $$( 1 + 44 T + 3510 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 5332 T^{2} + 13053126 T^{4} - 5332 p^{4} T^{6} + p^{8} T^{8}$$
61$D_{4}$ $$( 1 + 48 T + 4656 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 - 16900 T^{2} + 111538854 T^{4} - 16900 p^{4} T^{6} + p^{8} T^{8}$$
71$D_4\times C_2$ $$1 - 11308 T^{2} + 69354150 T^{4} - 11308 p^{4} T^{6} + p^{8} T^{8}$$
73$D_{4}$ $$( 1 + 24 T + 8352 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
79$D_4\times C_2$ $$1 - 16708 T^{2} + 131100678 T^{4} - 16708 p^{4} T^{6} + p^{8} T^{8}$$
83$D_4\times C_2$ $$1 - 23428 T^{2} + 227987238 T^{4} - 23428 p^{4} T^{6} + p^{8} T^{8}$$
89$D_{4}$ $$( 1 - 60 T + 180 p T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
97$D_{4}$ $$( 1 + 216 T + 30240 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$