# Properties

 Degree 16 Conductor $2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·2-s + 36·4-s − 120·8-s + 330·16-s − 8·23-s − 792·32-s + 64·46-s + 12·49-s − 8·53-s + 1.71e3·64-s + 48·79-s − 288·92-s − 96·98-s + 64·106-s + 16·107-s + 48·113-s + 80·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 384·158-s + 163-s + ⋯
 L(s)  = 1 − 5.65·2-s + 18·4-s − 42.4·8-s + 82.5·16-s − 1.66·23-s − 140.·32-s + 9.43·46-s + 12/7·49-s − 1.09·53-s + 214.5·64-s + 5.40·79-s − 30.0·92-s − 9.69·98-s + 6.21·106-s + 1.54·107-s + 4.51·113-s + 7.27·121-s + 0.0887·127-s − 303.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 30.5·158-s + 0.0783·163-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{3150} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 2^{8} \cdot 3^{16} \cdot 5^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$ $L(1)$ $\approx$ $0.9381679553$ $L(\frac12)$ $\approx$ $0.9381679553$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 16. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 $$( 1 + T )^{8}$$
3 $$1$$
5 $$1$$
7 $$( 1 - 6 T^{2} + p^{2} T^{4} )^{2}$$
good11 $$( 1 - 20 T^{2} + p^{2} T^{4} )^{4}$$
13 $$( 1 + 22 T^{2} + 259 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
17 $$( 1 - 29 T^{2} + p^{2} T^{4} )^{4}$$
19 $$( 1 - 16 T^{2} - 14 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
23 $$( 1 + T + p T^{2} )^{8}$$
29 $$( 1 - 30 T^{2} + 107 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
31 $$( 1 - 34 T^{2} + 1411 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
37 $$( 1 - 66 T^{2} + p^{2} T^{4} )^{4}$$
41 $$( 1 + 74 T^{2} + 3931 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
43 $$( 1 - 118 T^{2} + 6979 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
47 $$( 1 - 128 T^{2} + 7714 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
53 $$( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4}$$
59 $$( 1 + 46 T^{2} + 5691 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
61 $$( 1 - 214 T^{2} + 18691 T^{4} - 214 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
67 $$( 1 - p T^{2} )^{8}$$
71 $$( 1 - 20 T^{2} - 2618 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
73 $$( 1 + 72 T^{2} + 4754 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
79 $$( 1 - 12 T + 144 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4}$$
83 $$( 1 - 62 T^{2} + 9739 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
89 $$( 1 + 116 T^{2} + 6406 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
97 $$( 1 + 48 T^{2} - 9406 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}