# Properties

 Degree 16 Conductor $3^{16} \cdot 5^{16} \cdot 7^{8}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·16-s + 8·31-s + 4·61-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
 L(s)  = 1 + 2·16-s + 8·31-s + 4·61-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$3^{16} \cdot 5^{16} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{1575} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(16,\ 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [0]^{8} ),\ 1 )$$ $$L(\frac{1}{2})$$ $$\approx$$ $$1.913063835$$ $$L(\frac12)$$ $$\approx$$ $$1.913063835$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1 - T^{4} + T^{8}$$
good2 $$( 1 - T^{4} + T^{8} )^{2}$$
11 $$( 1 - T^{2} + T^{4} )^{4}$$
13 $$( 1 - T^{4} + T^{8} )^{2}$$
17 $$( 1 - T^{4} + T^{8} )^{2}$$
19 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
23 $$( 1 - T^{4} + T^{8} )^{2}$$
29 $$( 1 - T )^{8}( 1 + T )^{8}$$
31 $$( 1 - T + T^{2} )^{8}$$
37 $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
41 $$( 1 + T^{2} )^{8}$$
43 $$( 1 + T^{4} )^{4}$$
47 $$( 1 - T^{4} + T^{8} )^{2}$$
53 $$( 1 - T^{4} + T^{8} )^{2}$$
59 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
61 $$( 1 - T )^{8}( 1 + T + T^{2} )^{4}$$
67 $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
71 $$( 1 + T^{2} )^{8}$$
73 $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
79 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
83 $$( 1 + T^{4} )^{4}$$
89 $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
97 $$( 1 - T^{4} + T^{8} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}