# Properties

 Degree 2 Conductor $5 \cdot 13$ Sign $0.979 - 0.202i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s + (1 + i)3-s − 4-s + (−1 − 2i)5-s + (1 + i)6-s + 2i·7-s − 3·8-s − i·9-s + (−1 − 2i)10-s + (−1 + i)11-s + (−1 − i)12-s + (3 − 2i)13-s + 2i·14-s + (1 − 3i)15-s − 16-s + (1 + i)17-s + ⋯
 L(s)  = 1 + 0.707·2-s + (0.577 + 0.577i)3-s − 0.5·4-s + (−0.447 − 0.894i)5-s + (0.408 + 0.408i)6-s + 0.755i·7-s − 1.06·8-s − 0.333i·9-s + (−0.316 − 0.632i)10-s + (−0.301 + 0.301i)11-s + (−0.288 − 0.288i)12-s + (0.832 − 0.554i)13-s + 0.534i·14-s + (0.258 − 0.774i)15-s − 0.250·16-s + (0.242 + 0.242i)17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$65$$    =    $$5 \cdot 13$$ $$\varepsilon$$ = $0.979 - 0.202i$ motivic weight = $$1$$ character : $\chi_{65} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 65,\ (\ :1/2),\ 0.979 - 0.202i)$ $L(1)$ $\approx$ $1.15677 + 0.118445i$ $L(\frac12)$ $\approx$ $1.15677 + 0.118445i$ $L(\frac{3}{2})$ 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 + (-1 - i)T + 3iT^{2}$$
7 $$1 - 2iT - 7T^{2}$$
11 $$1 + (1 - i)T - 11iT^{2}$$
17 $$1 + (-1 - i)T + 17iT^{2}$$
19 $$1 + (5 - 5i)T - 19iT^{2}$$
23 $$1 + (-3 + 3i)T - 23iT^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + (-5 - 5i)T + 31iT^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 + (7 + 7i)T + 41iT^{2}$$
43 $$1 + (1 - i)T - 43iT^{2}$$
47 $$1 + 6iT - 47T^{2}$$
53 $$1 + (-5 - 5i)T + 53iT^{2}$$
59 $$1 + (7 + 7i)T + 59iT^{2}$$
61 $$1 + 14T + 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 + (-1 - i)T + 71iT^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 - 2iT - 79T^{2}$$
83 $$1 - 6iT - 83T^{2}$$
89 $$1 + (-5 - 5i)T + 89iT^{2}$$
97 $$1 - 2T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}