# Properties

 Degree 8 Conductor $2^{24} \cdot 5^{4}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 972·9-s + 6.25e3·25-s + 6.08e4·41-s − 5.37e4·49-s + 5.90e5·81-s + 5.15e5·89-s − 6.42e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.36e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 6.07e6·225-s + ⋯
 L(s)  = 1 − 4·9-s + 2·25-s + 5.64·41-s − 3.19·49-s + 10·81-s + 6.89·89-s − 3.98·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 3.67·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 8·225-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$8$$ $$N$$ = $$2^{24} \cdot 5^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{320} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )$$ $$L(3)$$ $$\approx$$ $$1.508028091$$ $$L(\frac12)$$ $$\approx$$ $$1.508028091$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 8. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
5$C_2$ $$( 1 - p^{5} T^{2} )^{2}$$
good3$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
7$C_2^2$ $$( 1 + 26886 T^{2} + p^{10} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 321102 T^{2} + p^{10} T^{4} )^{2}$$
13$C_2^2$ $$( 1 + 682086 T^{2} + p^{10} T^{4} )^{2}$$
17$C_2$ $$( 1 - p^{5} T^{2} )^{4}$$
19$C_2^2$ $$( 1 - 1288802 T^{2} + p^{10} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 1772186 T^{2} + p^{10} T^{4} )^{2}$$
29$C_2$ $$( 1 - p^{5} T^{2} )^{4}$$
31$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
37$C_2^2$ $$( 1 - 112652586 T^{2} + p^{10} T^{4} )^{2}$$
41$C_2$ $$( 1 - 15202 T + p^{5} T^{2} )^{4}$$
43$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
47$C_2^2$ $$( 1 - 222705514 T^{2} + p^{10} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 552886486 T^{2} + p^{10} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 646632402 T^{2} + p^{10} T^{4} )^{2}$$
61$C_2$ $$( 1 - p^{5} T^{2} )^{4}$$
67$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
71$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
73$C_2$ $$( 1 - p^{5} T^{2} )^{4}$$
79$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
83$C_2$ $$( 1 + p^{5} T^{2} )^{4}$$
89$C_2$ $$( 1 - 128786 T + p^{5} T^{2} )^{4}$$
97$C_2$ $$( 1 - p^{5} T^{2} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}