# Properties

 Degree 2 Conductor 13 Sign $0.984 - 0.176i$ Motivic weight 2 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.581 + 0.581i)2-s − 4.16·3-s − 3.32i·4-s + (3.58 + 3.58i)5-s + (−2.41 − 2.41i)6-s + (−4.58 + 4.58i)7-s + (4.25 − 4.25i)8-s + 8.32·9-s + 4.16i·10-s + (5.32 − 5.32i)11-s + 13.8i·12-s + (−5.90 − 11.5i)13-s − 5.32·14-s + (−14.9 − 14.9i)15-s − 8.35·16-s + 21.9i·17-s + ⋯
 L(s)  = 1 + (0.290 + 0.290i)2-s − 1.38·3-s − 0.831i·4-s + (0.716 + 0.716i)5-s + (−0.403 − 0.403i)6-s + (−0.654 + 0.654i)7-s + (0.532 − 0.532i)8-s + 0.924·9-s + 0.416i·10-s + (0.484 − 0.484i)11-s + 1.15i·12-s + (−0.454 − 0.890i)13-s − 0.380·14-s + (−0.993 − 0.993i)15-s − 0.521·16-s + 1.29i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.984 - 0.176i$ motivic weight = $$2$$ character : $\chi_{13} (8, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :1),\ 0.984 - 0.176i)$ $L(\frac{3}{2})$ $\approx$ $0.648809 + 0.0577555i$ $L(\frac12)$ $\approx$ $0.648809 + 0.0577555i$ $L(2)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 13$, $$F_p(T)$$ is a polynomial of degree 2. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 $$1 + (5.90 + 11.5i)T$$
good2 $$1 + (-0.581 - 0.581i)T + 4iT^{2}$$
3 $$1 + 4.16T + 9T^{2}$$
5 $$1 + (-3.58 - 3.58i)T + 25iT^{2}$$
7 $$1 + (4.58 - 4.58i)T - 49iT^{2}$$
11 $$1 + (-5.32 + 5.32i)T - 121iT^{2}$$
17 $$1 - 21.9iT - 289T^{2}$$
19 $$1 + (-3.16 - 3.16i)T + 361iT^{2}$$
23 $$1 - 8.51iT - 529T^{2}$$
29 $$1 + 5.81T + 841T^{2}$$
31 $$1 + (-0.513 - 0.513i)T + 961iT^{2}$$
37 $$1 + (-24.2 + 24.2i)T - 1.36e3iT^{2}$$
41 $$1 + (-4.83 - 4.83i)T + 1.68e3iT^{2}$$
43 $$1 + 30.4iT - 1.84e3T^{2}$$
47 $$1 + (37.3 - 37.3i)T - 2.20e3iT^{2}$$
53 $$1 + 35.8T + 2.80e3T^{2}$$
59 $$1 + (-58.2 + 58.2i)T - 3.48e3iT^{2}$$
61 $$1 + 80.3T + 3.72e3T^{2}$$
67 $$1 + (-39.0 - 39.0i)T + 4.48e3iT^{2}$$
71 $$1 + (-91.5 - 91.5i)T + 5.04e3iT^{2}$$
73 $$1 + (-31.6 + 31.6i)T - 5.32e3iT^{2}$$
79 $$1 + 18.7T + 6.24e3T^{2}$$
83 $$1 + (44.6 + 44.6i)T + 6.88e3iT^{2}$$
89 $$1 + (-8.89 + 8.89i)T - 7.92e3iT^{2}$$
97 $$1 + (121. + 121. i)T + 9.40e3iT^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}