# Properties

 Degree 16 Conductor $3^{8} \cdot 211^{8}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s − 3·4-s − 2·7-s + 9-s − 6·12-s + 2·13-s + 6·16-s − 2·19-s − 4·21-s − 25-s + 6·28-s − 4·31-s − 3·36-s + 4·39-s − 8·43-s + 12·48-s + 3·49-s − 6·52-s − 4·57-s − 6·61-s − 2·63-s − 10·64-s − 8·73-s − 2·75-s + 6·76-s + 12·84-s − 4·91-s + ⋯
 L(s)  = 1 + 2·3-s − 3·4-s − 2·7-s + 9-s − 6·12-s + 2·13-s + 6·16-s − 2·19-s − 4·21-s − 25-s + 6·28-s − 4·31-s − 3·36-s + 4·39-s − 8·43-s + 12·48-s + 3·49-s − 6·52-s − 4·57-s − 6·61-s − 2·63-s − 10·64-s − 8·73-s − 2·75-s + 6·76-s + 12·84-s − 4·91-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 211^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{8} \cdot 211^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$16$$ $$N$$ = $$3^{8} \cdot 211^{8}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{633} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(16,\ 3^{8} \cdot 211^{8} ,\ ( \ : [0]^{8} ),\ 1 )$ $L(\frac{1}{2})$ $\approx$ $0.02296546352$ $L(\frac12)$ $\approx$ $0.02296546352$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{3,\;211\}$, $$F_p$$ is a polynomial of degree 16. If $p \in \{3,\;211\}$, then $F_p$ is a polynomial of degree at most 15.
$p$$F_p$
bad3 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
211 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
good2 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
5 $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
7 $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
11 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
13 $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
17 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
19 $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
23 $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
29 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
31 $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
37 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
41 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
43 $$( 1 + T + T^{2} )^{8}$$
47 $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
53 $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 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 )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
67 $$( 1 + T^{2} )^{8}$$
71 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
73 $$( 1 + T + T^{2} )^{8}$$
79 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
83 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
89 $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
97 $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}