# 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 = $(2,\ 104,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $0.3506445621$ $L(\frac12)$ $\approx$ $0.3506445621$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;13\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$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}