# Properties

 Degree 2 Conductor $2^{3} \cdot 13$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s − 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 20-s + 21-s − 24-s + 26-s + 27-s − 28-s + 30-s + 2·31-s + 32-s − 34-s + 35-s − 37-s − 39-s + ⋯
 L(s)  = 1 + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s − 10-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 20-s + 21-s − 24-s + 26-s + 27-s − 28-s + 30-s + 2·31-s + 32-s − 34-s + 35-s − 37-s − 39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$104$$    =    $$2^{3} \cdot 13$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{104} (51, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 104,\ (\ :0),\ 1)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.6434240793$$ $$L(\frac12)$$ $$\approx$$ $$0.6434240793$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;13\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - T$$
13 $$1 - T$$
good3 $$1 + T + T^{2}$$
5 $$1 + T + T^{2}$$
7 $$1 + T + T^{2}$$
11 $$( 1 - T )( 1 + T )$$
17 $$1 + T + T^{2}$$
19 $$( 1 - T )( 1 + T )$$
23 $$( 1 - T )( 1 + T )$$
29 $$( 1 - T )( 1 + T )$$
31 $$( 1 - T )^{2}$$
37 $$1 + T + T^{2}$$
41 $$( 1 - T )( 1 + T )$$
43 $$1 + T + T^{2}$$
47 $$1 + T + T^{2}$$
53 $$( 1 - T )( 1 + T )$$
59 $$( 1 - T )( 1 + T )$$
61 $$( 1 - T )( 1 + T )$$
67 $$( 1 - T )( 1 + T )$$
71 $$1 + T + T^{2}$$
73 $$( 1 - T )( 1 + T )$$
79 $$( 1 - T )( 1 + T )$$
83 $$( 1 - T )( 1 + T )$$
89 $$( 1 - T )( 1 + T )$$
97 $$( 1 - T )( 1 + T )$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}