# Properties

 Degree 2 Conductor 13 Sign $0.964 + 0.265i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.5 − 0.866i)2-s + (−1 + 1.73i)3-s + (0.5 + 0.866i)4-s − 1.73i·5-s + (3 − 1.73i)6-s + 1.73i·8-s + (−0.499 − 0.866i)9-s + (−1.49 + 2.59i)10-s − 2·12-s + (−2.5 − 2.59i)13-s + (2.99 + 1.73i)15-s + (2.49 − 4.33i)16-s + (1.5 + 2.59i)17-s + 1.73i·18-s + (−3 + 1.73i)19-s + (1.50 − 0.866i)20-s + ⋯
 L(s)  = 1 + (−1.06 − 0.612i)2-s + (−0.577 + 0.999i)3-s + (0.250 + 0.433i)4-s − 0.774i·5-s + (1.22 − 0.707i)6-s + 0.612i·8-s + (−0.166 − 0.288i)9-s + (−0.474 + 0.821i)10-s − 0.577·12-s + (−0.693 − 0.720i)13-s + (0.774 + 0.447i)15-s + (0.624 − 1.08i)16-s + (0.363 + 0.630i)17-s + 0.408i·18-s + (−0.688 + 0.397i)19-s + (0.335 − 0.193i)20-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.964 + 0.265i$ motivic weight = $$1$$ character : $\chi_{13} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :1/2),\ 0.964 + 0.265i)$ $L(1)$ $\approx$ $0.298115 - 0.0402203i$ $L(\frac12)$ $\approx$ $0.298115 - 0.0402203i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $$F_p$$ is a polynomial of degree 2. If $p = 13$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad13 $$1 + (2.5 + 2.59i)T$$
good2 $$1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2}$$
3 $$1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + 1.73iT - 5T^{2}$$
7 $$1 + (3.5 - 6.06i)T^{2}$$
11 $$1 + (5.5 + 9.52i)T^{2}$$
17 $$1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 3.46iT - 31T^{2}$$
37 $$1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 3.46iT - 47T^{2}$$
53 $$1 + 3T + 53T^{2}$$
59 $$1 + (-6 + 3.46i)T + (29.5 - 51.0i)T^{2}$$
61 $$1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-3 - 1.73i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + (-3 + 1.73i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 - 1.73iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 - 13.8iT - 83T^{2}$$
89 $$1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}