# Properties

 Degree 8 Conductor $5^{8}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 12·11-s + 7·16-s − 152·31-s + 228·41-s − 112·61-s + 168·71-s − 63·81-s − 192·101-s − 394·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 84·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
 L(s)  = 1 − 1.09·11-s + 7/16·16-s − 4.90·31-s + 5.56·41-s − 1.83·61-s + 2.36·71-s − 7/9·81-s − 1.90·101-s − 3.25·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s − 0.477·176-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$390625$$    =    $$5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{25} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(8,\ 390625,\ (\ :1, 1, 1, 1),\ 1)$ $L(\frac{3}{2})$ $\approx$ $0.711859$ $L(\frac12)$ $\approx$ $0.711859$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 5$, $$F_p$$ is a polynomial of degree 8. If $p = 5$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad5 $$1$$
good2$C_2^3$ $$1 - 7 T^{4} + p^{8} T^{8}$$
3$C_2^3$ $$1 + 7 p^{2} T^{4} + p^{8} T^{8}$$
7$C_2^3$ $$1 - 2302 T^{4} + p^{8} T^{8}$$
11$C_2$ $$( 1 + 3 T + p^{2} T^{2} )^{4}$$
13$C_2^3$ $$1 - 4222 T^{4} + p^{8} T^{8}$$
17$C_2^3$ $$1 - 120817 T^{4} + p^{8} T^{8}$$
19$C_2^2$ $$( 1 - 697 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^3$ $$1 - 338782 T^{4} + p^{8} T^{8}$$
29$C_2^2$ $$( 1 - 782 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2$ $$( 1 + 38 T + p^{2} T^{2} )^{4}$$
37$C_2^3$ $$1 + 132578 T^{4} + p^{8} T^{8}$$
41$C_2$ $$( 1 - 57 T + p^{2} T^{2} )^{4}$$
43$C_2^3$ $$1 + 6484898 T^{4} + p^{8} T^{8}$$
47$C_2^3$ $$1 + 8816738 T^{4} + p^{8} T^{8}$$
53$C_2^3$ $$1 - 2892862 T^{4} + p^{8} T^{8}$$
59$C_2^2$ $$( 1 + 1138 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2$ $$( 1 + 28 T + p^{2} T^{2} )^{4}$$
67$C_2^3$ $$1 - 20810017 T^{4} + p^{8} T^{8}$$
71$C_2$ $$( 1 - 42 T + p^{2} T^{2} )^{4}$$
73$C_2^3$ $$1 + 49190543 T^{4} + p^{8} T^{8}$$
79$C_2^2$ $$( 1 - 6082 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^3$ $$1 + 27517583 T^{4} + p^{8} T^{8}$$
89$C_2^2$ $$( 1 - 15617 T^{2} + p^{4} T^{4} )^{2}$$
97$C_2^3$ $$1 - 7798462 T^{4} + p^{8} T^{8}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}