# Properties

 Degree 2 Conductor 13 Sign $0.399 - 0.916i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.58 + 2.58i)2-s + 2.16·3-s − 9.32i·4-s + (0.418 − 0.418i)5-s + (−5.58 + 5.58i)6-s + (−1.41 − 1.41i)7-s + (13.7 + 13.7i)8-s − 4.32·9-s + 2.16i·10-s + (−7.32 − 7.32i)11-s − 20.1i·12-s + (9.90 + 8.41i)13-s + 7.32·14-s + (0.905 − 0.905i)15-s − 33.6·16-s + 15.9i·17-s + ⋯
 L(s)  = 1 + (−1.29 + 1.29i)2-s + 0.720·3-s − 2.33i·4-s + (0.0837 − 0.0837i)5-s + (−0.930 + 0.930i)6-s + (−0.202 − 0.202i)7-s + (1.71 + 1.71i)8-s − 0.480·9-s + 0.216i·10-s + (−0.665 − 0.665i)11-s − 1.68i·12-s + (0.761 + 0.647i)13-s + 0.523·14-s + (0.0603 − 0.0603i)15-s − 2.10·16-s + 0.939i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.399 - 0.916i$ motivic weight = $$2$$ character : $\chi_{13} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :1),\ 0.399 - 0.916i)$ $L(\frac{3}{2})$ $\approx$ $0.424951 + 0.278524i$ $L(\frac12)$ $\approx$ $0.424951 + 0.278524i$ $L(2)$ 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 + (-9.90 - 8.41i)T$$
good2 $$1 + (2.58 - 2.58i)T - 4iT^{2}$$
3 $$1 - 2.16T + 9T^{2}$$
5 $$1 + (-0.418 + 0.418i)T - 25iT^{2}$$
7 $$1 + (1.41 + 1.41i)T + 49iT^{2}$$
11 $$1 + (7.32 + 7.32i)T + 121iT^{2}$$
17 $$1 - 15.9iT - 289T^{2}$$
19 $$1 + (3.16 - 3.16i)T - 361iT^{2}$$
23 $$1 + 27.4iT - 529T^{2}$$
29 $$1 - 25.8T + 841T^{2}$$
31 $$1 + (-19.4 + 19.4i)T - 961iT^{2}$$
37 $$1 + (4.23 + 4.23i)T + 1.36e3iT^{2}$$
41 $$1 + (-11.1 + 11.1i)T - 1.68e3iT^{2}$$
43 $$1 - 11.5iT - 1.84e3T^{2}$$
47 $$1 + (-35.3 - 35.3i)T + 2.20e3iT^{2}$$
53 $$1 + 4.18T + 2.80e3T^{2}$$
59 $$1 + (30.2 + 30.2i)T + 3.48e3iT^{2}$$
61 $$1 + 67.6T + 3.72e3T^{2}$$
67 $$1 + (81.0 - 81.0i)T - 4.48e3iT^{2}$$
71 $$1 + (-50.4 + 50.4i)T - 5.04e3iT^{2}$$
73 $$1 + (31.6 + 31.6i)T + 5.32e3iT^{2}$$
79 $$1 - 50.7T + 6.24e3T^{2}$$
83 $$1 + (-18.6 + 18.6i)T - 6.88e3iT^{2}$$
89 $$1 + (-91.1 - 91.1i)T + 7.92e3iT^{2}$$
97 $$1 + (-87.3 + 87.3i)T - 9.40e3iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}