# Properties

 Degree 2 Conductor 13 Sign $0.751 - 0.659i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.397 + 1.48i)2-s + (−0.503 − 0.872i)3-s + (53.3 + 30.8i)4-s + (84.5 + 84.5i)5-s + (1.49 − 0.400i)6-s + (41.0 + 153. i)7-s + (−136. + 136. i)8-s + (363. − 630. i)9-s + (−159. + 91.8i)10-s + (−1.01e3 − 270. i)11-s − 62.0i·12-s + (−1.38e3 − 1.70e3i)13-s − 243.·14-s + (31.1 − 116. i)15-s + (1.82e3 + 3.15e3i)16-s + (−1.49e3 − 863. i)17-s + ⋯
 L(s)  = 1 + (−0.0497 + 0.185i)2-s + (−0.0186 − 0.0323i)3-s + (0.834 + 0.481i)4-s + (0.676 + 0.676i)5-s + (0.00692 − 0.00185i)6-s + (0.119 + 0.446i)7-s + (−0.266 + 0.266i)8-s + (0.499 − 0.864i)9-s + (−0.159 + 0.0918i)10-s + (−0.759 − 0.203i)11-s − 0.0359i·12-s + (−0.629 − 0.776i)13-s − 0.0887·14-s + (0.00923 − 0.0344i)15-s + (0.445 + 0.771i)16-s + (−0.304 − 0.175i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13$$ $$\varepsilon$$ = $0.751 - 0.659i$ motivic weight = $$6$$ character : $\chi_{13} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 13,\ (\ :3),\ 0.751 - 0.659i)$ $L(\frac{7}{2})$ $\approx$ $1.52907 + 0.575408i$ $L(\frac12)$ $\approx$ $1.52907 + 0.575408i$ $L(4)$ 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 + (1.38e3 + 1.70e3i)T$$
good2 $$1 + (0.397 - 1.48i)T + (-55.4 - 32i)T^{2}$$
3 $$1 + (0.503 + 0.872i)T + (-364.5 + 631. i)T^{2}$$
5 $$1 + (-84.5 - 84.5i)T + 1.56e4iT^{2}$$
7 $$1 + (-41.0 - 153. i)T + (-1.01e5 + 5.88e4i)T^{2}$$
11 $$1 + (1.01e3 + 270. i)T + (1.53e6 + 8.85e5i)T^{2}$$
17 $$1 + (1.49e3 + 863. i)T + (1.20e7 + 2.09e7i)T^{2}$$
19 $$1 + (-6.57e3 + 1.76e3i)T + (4.07e7 - 2.35e7i)T^{2}$$
23 $$1 + (3.37e3 - 1.94e3i)T + (7.40e7 - 1.28e8i)T^{2}$$
29 $$1 + (1.75e4 + 3.04e4i)T + (-2.97e8 + 5.15e8i)T^{2}$$
31 $$1 + (1.21e4 + 1.21e4i)T + 8.87e8iT^{2}$$
37 $$1 + (-5.55e4 - 1.48e4i)T + (2.22e9 + 1.28e9i)T^{2}$$
41 $$1 + (1.95e4 - 7.29e4i)T + (-4.11e9 - 2.37e9i)T^{2}$$
43 $$1 + (-5.62e4 - 3.24e4i)T + (3.16e9 + 5.47e9i)T^{2}$$
47 $$1 + (6.42e4 - 6.42e4i)T - 1.07e10iT^{2}$$
53 $$1 + 2.72e5T + 2.21e10T^{2}$$
59 $$1 + (-7.41e4 - 2.76e5i)T + (-3.65e10 + 2.10e10i)T^{2}$$
61 $$1 + (-9.89e4 + 1.71e5i)T + (-2.57e10 - 4.46e10i)T^{2}$$
67 $$1 + (1.49e5 - 5.57e5i)T + (-7.83e10 - 4.52e10i)T^{2}$$
71 $$1 + (-3.66e5 + 9.83e4i)T + (1.10e11 - 6.40e10i)T^{2}$$
73 $$1 + (-1.88e5 + 1.88e5i)T - 1.51e11iT^{2}$$
79 $$1 + 7.16e5T + 2.43e11T^{2}$$
83 $$1 + (4.06e5 + 4.06e5i)T + 3.26e11iT^{2}$$
89 $$1 + (-2.70e5 - 7.25e4i)T + (4.30e11 + 2.48e11i)T^{2}$$
97 $$1 + (2.54e5 - 6.82e4i)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}