# Properties

 Degree 16 Conductor $2^{48} \cdot 5^{8}$ Sign $1$ Motivic weight 5 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 480·9-s + 1.19e4·25-s + 9.11e4·41-s + 7.54e4·49-s − 9.21e4·81-s − 7.94e5·89-s + 9.33e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 5.71e6·225-s + ⋯
 L(s)  = 1 − 1.97·9-s + 3.80·25-s + 8.46·41-s + 4.48·49-s − 1.56·81-s − 10.6·89-s + 5.79·121-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 − 5.45·169-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 − 7.52·225-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{48} \cdot 5^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : induced by $\chi_{320} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )$$ $$L(3)$$ $$\approx$$ $$4.764931597$$ $$L(\frac12)$$ $$\approx$$ $$4.764931597$$ $$L(\frac{7}{2})$$ 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 16. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$1$$
5 $$( 1 - 238 p^{2} T^{2} + p^{10} T^{4} )^{2}$$
good3 $$( 1 + 40 p T^{2} + p^{10} T^{4} )^{4}$$
7 $$( 1 - 18852 T^{2} + p^{10} T^{4} )^{4}$$
11 $$( 1 - 233298 T^{2} + p^{10} T^{4} )^{4}$$
13 $$( 1 + 506394 T^{2} + p^{10} T^{4} )^{4}$$
17 $$( 1 - 2662570 T^{2} + p^{10} T^{4} )^{4}$$
19 $$( 1 + 1673278 T^{2} + p^{10} T^{4} )^{4}$$
23 $$( 1 - 12357236 T^{2} + p^{10} T^{4} )^{4}$$
29 $$( 1 - 27077290 T^{2} + p^{10} T^{4} )^{4}$$
31 $$( 1 + 19214 p^{2} T^{2} + p^{10} T^{4} )^{4}$$
37 $$( 1 + 77394 p^{2} T^{2} + p^{10} T^{4} )^{4}$$
41 $$( 1 - 11396 T + p^{5} T^{2} )^{8}$$
43 $$( 1 - 72715480 T^{2} + p^{10} T^{4} )^{4}$$
47 $$( 1 - 453462436 T^{2} + p^{10} T^{4} )^{4}$$
53 $$( 1 + 539491786 T^{2} + p^{10} T^{4} )^{4}$$
59 $$( 1 - 1161609714 T^{2} + p^{10} T^{4} )^{4}$$
61 $$( 1 + 686048530 T^{2} + p^{10} T^{4} )^{4}$$
67 $$( 1 + 2629714328 T^{2} + p^{10} T^{4} )^{4}$$
71 $$( 1 + 2762141854 T^{2} + p^{10} T^{4} )^{4}$$
73 $$( 1 - 4145966042 T^{2} + p^{10} T^{4} )^{4}$$
79 $$( 1 + 5182544750 T^{2} + p^{10} T^{4} )^{4}$$
83 $$( 1 - 3415955864 T^{2} + p^{10} T^{4} )^{4}$$
89 $$( 1 + 99362 T + p^{5} T^{2} )^{8}$$
97 $$( 1 + 15344227702 T^{2} + p^{10} T^{4} )^{4}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}