# Properties

 Degree 20 Conductor $2^{20} \cdot 3^{10} \cdot 167^{10}$ Sign $1$ Motivic weight 0 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2-s − 3-s + 6-s + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯
 L(s)  = 1 − 2-s − 3-s + 6-s + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$20$$ $$N$$ = $$2^{20} \cdot 3^{10} \cdot 167^{10}$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : induced by $\chi_{2004} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(20,\ 2^{20} \cdot 3^{10} \cdot 167^{10} ,\ ( \ : [0]^{10} ),\ 1 )$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.2463203317$$ $$L(\frac12)$$ $$\approx$$ $$0.2463203317$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;167\}$,$$F_p(T)$$ is a polynomial of degree 20. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$$
3 $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$$
167 $$( 1 - T )^{10}$$
good5 $$( 1 + T^{2} )^{10}$$
7 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
11 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
13 $$( 1 - T )^{10}( 1 + T )^{10}$$
17 $$( 1 + T^{2} )^{10}$$
19 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
23 $$( 1 - T )^{10}( 1 + T )^{10}$$
29 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
31 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
37 $$( 1 - T )^{10}( 1 + T )^{10}$$
41 $$( 1 + T^{2} )^{10}$$
43 $$( 1 + T^{2} )^{10}$$
47 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
53 $$( 1 + T^{2} )^{10}$$
59 $$( 1 - T )^{10}( 1 + T )^{10}$$
61 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
67 $$( 1 + T^{2} )^{10}$$
71 $$( 1 - T )^{10}( 1 + T )^{10}$$
73 $$( 1 - T )^{10}( 1 + T )^{10}$$
79 $$( 1 + T^{2} )^{10}$$
83 $$( 1 - T )^{10}( 1 + T )^{10}$$
89 $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
97 $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}