# Properties

 Degree 2 Conductor $5 \cdot 13$ Sign $0.868 - 0.496i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + 2i·3-s − 4-s + (1 − 2i)5-s + 2i·6-s − 3·8-s − 9-s + (1 − 2i)10-s − 2i·11-s − 2i·12-s + (−3 − 2i)13-s + (4 + 2i)15-s − 16-s − 18-s + 6i·19-s + (−1 + 2i)20-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.15i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.816i·6-s − 1.06·8-s − 0.333·9-s + (0.316 − 0.632i)10-s − 0.603i·11-s − 0.577i·12-s + (−0.832 − 0.554i)13-s + (1.03 + 0.516i)15-s − 0.250·16-s − 0.235·18-s + 1.37i·19-s + (−0.223 + 0.447i)20-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$65$$    =    $$5 \cdot 13$$ $$\varepsilon$$ = $0.868 - 0.496i$ motivic weight = $$1$$ character : $\chi_{65} (64, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 65,\ (\ :1/2),\ 0.868 - 0.496i)$ $L(1)$ $\approx$ $1.06832 + 0.283707i$ $L(\frac12)$ $\approx$ $1.06832 + 0.283707i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{5,\;13\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 $$1 + (-1 + 2i)T$$
13 $$1 + (3 + 2i)T$$
good2 $$1 - T + 2T^{2}$$
3 $$1 - 2iT - 3T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 + 2iT - 11T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 6iT - 19T^{2}$$
23 $$1 - 6iT - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + 6iT - 31T^{2}$$
37 $$1 + 6T + 37T^{2}$$
41 $$1 - 8iT - 41T^{2}$$
43 $$1 + 6iT - 43T^{2}$$
47 $$1 - 8T + 47T^{2}$$
53 $$1 + 12iT - 53T^{2}$$
59 $$1 + 2iT - 59T^{2}$$
61 $$1 - 6T + 61T^{2}$$
67 $$1 + 12T + 67T^{2}$$
71 $$1 - 2iT - 71T^{2}$$
73 $$1 - 6T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 4T + 83T^{2}$$
89 $$1 - 8iT - 89T^{2}$$
97 $$1 - 6T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}