# Properties

 Degree 16 Conductor $2^{40} \cdot 5^{24}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·13-s − 2·17-s + 2·37-s + 2·53-s + 2·73-s + 81-s − 10·89-s + 2·97-s + 4·101-s + 8·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
 L(s)  = 1 − 2·13-s − 2·17-s + 2·37-s + 2·53-s + 2·73-s + 81-s − 10·89-s + 2·97-s + 4·101-s + 8·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{40} \cdot 5^{24}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{4000} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 2^{40} \cdot 5^{24} ,\ ( \ : [0]^{8} ),\ 1 )$ $L(\frac{1}{2})$ $\approx$ $0.8664634610$ $L(\frac12)$ $\approx$ $0.8664634610$ $L(1)$ 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$$
good3 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
7 $$( 1 + T^{4} )^{4}$$
11 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
13 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
17 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
19 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
23 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
29 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
31 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
37 $$( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
41 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
43 $$( 1 + T^{4} )^{4}$$
47 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
53 $$( 1 + T^{2} )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
59 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
61 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
67 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
71 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
73 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
79 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
83 $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
89 $$( 1 + T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
97 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}