# Properties

 Degree 2 Conductor 13 Sign $0.804 - 0.593i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−9.37 + 2.51i)2-s + (10.4 − 18.0i)3-s + (26.0 − 15.0i)4-s + (160. + 160. i)5-s + (−52.2 + 195. i)6-s + (162. + 43.4i)7-s + (232. − 232. i)8-s + (147. + 255. i)9-s + (−1.90e3 − 1.09e3i)10-s + (−264. − 986. i)11-s − 627. i·12-s + (189. + 2.18e3i)13-s − 1.62e3·14-s + (4.55e3 − 1.21e3i)15-s + (−2.55e3 + 4.43e3i)16-s + (1.83e3 − 1.06e3i)17-s + ⋯
 L(s)  = 1 + (−1.17 + 0.313i)2-s + (0.385 − 0.667i)3-s + (0.407 − 0.235i)4-s + (1.28 + 1.28i)5-s + (−0.242 + 0.903i)6-s + (0.472 + 0.126i)7-s + (0.453 − 0.453i)8-s + (0.202 + 0.351i)9-s + (−1.90 − 1.09i)10-s + (−0.198 − 0.741i)11-s − 0.363i·12-s + (0.0861 + 0.996i)13-s − 0.593·14-s + (1.34 − 0.361i)15-s + (−0.624 + 1.08i)16-s + (0.373 − 0.215i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.804 - 0.593i$ motivic weight = $$6$$ character : $\chi_{13} (7, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :3),\ 0.804 - 0.593i)$ $L(\frac{7}{2})$ $\approx$ $0.986169 + 0.324440i$ $L(\frac12)$ $\approx$ $0.986169 + 0.324440i$ $L(4)$ 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 + (-189. - 2.18e3i)T$$
good2 $$1 + (9.37 - 2.51i)T + (55.4 - 32i)T^{2}$$
3 $$1 + (-10.4 + 18.0i)T + (-364.5 - 631. i)T^{2}$$
5 $$1 + (-160. - 160. i)T + 1.56e4iT^{2}$$
7 $$1 + (-162. - 43.4i)T + (1.01e5 + 5.88e4i)T^{2}$$
11 $$1 + (264. + 986. i)T + (-1.53e6 + 8.85e5i)T^{2}$$
17 $$1 + (-1.83e3 + 1.06e3i)T + (1.20e7 - 2.09e7i)T^{2}$$
19 $$1 + (-334. + 1.24e3i)T + (-4.07e7 - 2.35e7i)T^{2}$$
23 $$1 + (1.19e4 + 6.88e3i)T + (7.40e7 + 1.28e8i)T^{2}$$
29 $$1 + (-1.20e4 + 2.09e4i)T + (-2.97e8 - 5.15e8i)T^{2}$$
31 $$1 + (2.63e4 + 2.63e4i)T + 8.87e8iT^{2}$$
37 $$1 + (7.41e3 + 2.76e4i)T + (-2.22e9 + 1.28e9i)T^{2}$$
41 $$1 + (-8.34e4 + 2.23e4i)T + (4.11e9 - 2.37e9i)T^{2}$$
43 $$1 + (1.02e5 - 5.93e4i)T + (3.16e9 - 5.47e9i)T^{2}$$
47 $$1 + (-2.43e3 + 2.43e3i)T - 1.07e10iT^{2}$$
53 $$1 + 2.62e5T + 2.21e10T^{2}$$
59 $$1 + (-1.48e5 - 3.97e4i)T + (3.65e10 + 2.10e10i)T^{2}$$
61 $$1 + (-1.47e4 - 2.55e4i)T + (-2.57e10 + 4.46e10i)T^{2}$$
67 $$1 + (-8.60e4 + 2.30e4i)T + (7.83e10 - 4.52e10i)T^{2}$$
71 $$1 + (-1.72e5 + 6.44e5i)T + (-1.10e11 - 6.40e10i)T^{2}$$
73 $$1 + (-4.65e4 + 4.65e4i)T - 1.51e11iT^{2}$$
79 $$1 - 1.69e5T + 2.43e11T^{2}$$
83 $$1 + (-3.78e5 - 3.78e5i)T + 3.26e11iT^{2}$$
89 $$1 + (-3.73e4 - 1.39e5i)T + (-4.30e11 + 2.48e11i)T^{2}$$
97 $$1 + (5.46e4 - 2.04e5i)T + (-7.21e11 - 4.16e11i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}