# Properties

 Label 8-15e8-1.1-c3e4-0-1 Degree $8$ Conductor $2562890625$ Sign $1$ Analytic cond. $31059.4$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 36·7-s − 252·13-s + 79·16-s + 784·31-s − 828·37-s + 576·43-s + 648·49-s − 1.28e3·61-s − 1.51e3·67-s − 1.51e3·73-s − 9.07e3·91-s + 1.00e3·97-s − 3.78e3·103-s + 2.84e3·112-s + 2.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.17e4·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 1.94·7-s − 5.37·13-s + 1.23·16-s + 4.54·31-s − 3.67·37-s + 2.04·43-s + 1.88·49-s − 2.70·61-s − 2.75·67-s − 2.42·73-s − 10.4·91-s + 1.05·97-s − 3.61·103-s + 2.39·112-s + 1.80·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 14.4·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$31059.4$$ Root analytic conductor: $$3.64354$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{225} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.2911067696$$ $$L(\frac12)$$ $$\approx$$ $$0.2911067696$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
5 $$1$$
good2$C_2^3$ $$1 - 79 T^{4} + p^{12} T^{8}$$
7$C_2^2$ $$( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 1204 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 126 T + 7938 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
17$C_2^3$ $$1 + 16720706 T^{4} + p^{12} T^{8}$$
19$C_2^2$ $$( 1 - 8818 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2^3$ $$1 - 102026878 T^{4} + p^{12} T^{8}$$
29$C_2^2$ $$( 1 - 3710 T^{2} + p^{6} T^{4} )^{2}$$
31$C_2$ $$( 1 - 196 T + p^{3} T^{2} )^{4}$$
37$C_2^2$ $$( 1 + 414 T + 85698 T^{2} + 414 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
41$C_2^2$ $$( 1 - 66400 T^{2} + p^{6} T^{4} )^{2}$$
43$C_2^2$ $$( 1 - 288 T + 41472 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$C_2^3$ $$1 - 19408253758 T^{4} + p^{12} T^{8}$$
53$C_2^3$ $$1 - 718 p^{4} T^{4} + p^{12} T^{8}$$
59$C_2^2$ $$( 1 + 339316 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2$ $$( 1 + 322 T + p^{3} T^{2} )^{4}$$
67$C_2^2$ $$( 1 + 756 T + 285768 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 10150 T^{2} + p^{6} T^{4} )^{2}$$
73$C_2^2$ $$( 1 + 756 T + 285768 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 747934 T^{2} + p^{6} T^{4} )^{2}$$
83$C_2^3$ $$1 - 651490514638 T^{4} + p^{12} T^{8}$$
89$C_2^2$ $$( 1 + 1338496 T^{2} + p^{6} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 504 T + 127008 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$