# Properties

 Degree 2 Conductor 3 Sign $0.851 + 0.524i$ Motivic weight 38 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 6.48e5i·2-s + (−9.89e8 − 6.09e8i)3-s − 1.45e11·4-s + 1.81e13i·5-s + (3.95e14 − 6.41e14i)6-s − 1.77e16·7-s + 8.40e16i·8-s + (6.07e17 + 1.20e18i)9-s − 1.17e19·10-s + 1.89e19i·11-s + (1.43e20 + 8.84e19i)12-s − 1.72e21·13-s − 1.15e22i·14-s + (1.10e22 − 1.79e22i)15-s − 9.43e22·16-s − 2.70e23i·17-s + ⋯
 L(s)  = 1 + 1.23i·2-s + (−0.851 − 0.524i)3-s − 0.528·4-s + 0.952i·5-s + (0.648 − 1.05i)6-s − 1.55·7-s + 0.583i·8-s + (0.449 + 0.893i)9-s − 1.17·10-s + 0.309i·11-s + (0.449 + 0.277i)12-s − 1.18·13-s − 1.92i·14-s + (0.499 − 0.810i)15-s − 1.24·16-s − 1.13i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(39-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3$$ $$\varepsilon$$ = $0.851 + 0.524i$ motivic weight = $$38$$ character : $\chi_{3} (2, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 3,\ (\ :19),\ 0.851 + 0.524i)$$ $$L(\frac{39}{2})$$ $$\approx$$ $$0.1896044109$$ $$L(\frac12)$$ $$\approx$$ $$0.1896044109$$ $$L(20)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 3$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (9.89e8 + 6.09e8i)T$$
good2 $$1 - 6.48e5iT - 2.74e11T^{2}$$
5 $$1 - 1.81e13iT - 3.63e26T^{2}$$
7 $$1 + 1.77e16T + 1.29e32T^{2}$$
11 $$1 - 1.89e19iT - 3.74e39T^{2}$$
13 $$1 + 1.72e21T + 2.13e42T^{2}$$
17 $$1 + 2.70e23iT - 5.71e46T^{2}$$
19 $$1 - 1.47e24T + 3.91e48T^{2}$$
23 $$1 - 7.60e25iT - 5.56e51T^{2}$$
29 $$1 - 5.84e26iT - 3.72e55T^{2}$$
31 $$1 - 3.31e28T + 4.69e56T^{2}$$
37 $$1 + 2.91e29T + 3.90e59T^{2}$$
41 $$1 + 3.49e30iT - 1.93e61T^{2}$$
43 $$1 + 3.83e30T + 1.17e62T^{2}$$
47 $$1 + 8.27e31iT - 3.46e63T^{2}$$
53 $$1 + 1.02e33iT - 3.33e65T^{2}$$
59 $$1 - 7.44e33iT - 1.96e67T^{2}$$
61 $$1 + 1.73e33T + 6.95e67T^{2}$$
67 $$1 + 4.97e34T + 2.45e69T^{2}$$
71 $$1 - 2.73e34iT - 2.22e70T^{2}$$
73 $$1 - 2.76e35T + 6.40e70T^{2}$$
79 $$1 + 1.24e36T + 1.28e72T^{2}$$
83 $$1 - 4.54e36iT - 8.41e72T^{2}$$
89 $$1 - 1.42e36iT - 1.19e74T^{2}$$
97 $$1 + 3.92e37T + 3.14e75T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}