# Properties

 Degree 2 Conductor $2^{2} \cdot 13$ Sign $0.711 + 0.702i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 5-s + 0.999·8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)26-s + (0.5 + 0.866i)29-s + (−0.499 + 0.866i)32-s − 0.999·34-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$52$$    =    $$2^{2} \cdot 13$$ $$\varepsilon$$ = $0.711 + 0.702i$ motivic weight = $$0$$ character : $\chi_{52} (3, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 52,\ (\ :0),\ 0.711 + 0.702i)$ $L(\frac{1}{2})$ $\approx$ $0.3248332447$ $L(\frac12)$ $\approx$ $0.3248332447$ $L(1)$ 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 + (0.5 + 0.866i)T$$
13 $$1 + (0.5 + 0.866i)T$$
good3 $$1 + (0.5 - 0.866i)T^{2}$$
5 $$1 + T + T^{2}$$
7 $$1 + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.5 - 0.866i)T^{2}$$
17 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T^{2}$$
23 $$1 + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
41 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T + T^{2}$$
59 $$1 + (0.5 + 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.5 - 0.866i)T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 + T + T^{2}$$
79 $$1 - T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}$$
97 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}