# Properties

 Degree $2$ Conductor $3$ Sign $0.735 + 0.677i$ Motivic weight $38$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.26e5i·2-s + (−8.54e8 − 7.87e8i)3-s + 2.23e11·4-s − 6.89e12i·5-s + (−1.78e14 + 1.93e14i)6-s + 1.19e16·7-s − 1.12e17i·8-s + (1.09e17 + 1.34e18i)9-s − 1.55e18·10-s + 1.21e20i·11-s + (−1.91e20 − 1.76e20i)12-s + 1.57e21·13-s − 2.70e21i·14-s + (−5.42e21 + 5.88e21i)15-s + 3.60e22·16-s + 2.37e23i·17-s + ⋯
 L(s)  = 1 − 0.431i·2-s + (−0.735 − 0.677i)3-s + 0.814·4-s − 0.361i·5-s + (−0.292 + 0.317i)6-s + 1.05·7-s − 0.782i·8-s + (0.0813 + 0.996i)9-s − 0.155·10-s + 1.98i·11-s + (−0.598 − 0.551i)12-s + 1.07·13-s − 0.453i·14-s + (−0.244 + 0.265i)15-s + 0.476·16-s + 0.993i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(39-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3$$ Sign: $0.735 + 0.677i$ Motivic weight: $$38$$ Character: $\chi_{3} (2, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3,\ (\ :19),\ 0.735 + 0.677i)$$

## Particular Values

 $$L(\frac{39}{2})$$ $$\approx$$ $$2.306801110$$ $$L(\frac12)$$ $$\approx$$ $$2.306801110$$ $$L(20)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (8.54e8 + 7.87e8i)T$$
good2 $$1 + 2.26e5iT - 2.74e11T^{2}$$
5 $$1 + 6.89e12iT - 3.63e26T^{2}$$
7 $$1 - 1.19e16T + 1.29e32T^{2}$$
11 $$1 - 1.21e20iT - 3.74e39T^{2}$$
13 $$1 - 1.57e21T + 2.13e42T^{2}$$
17 $$1 - 2.37e23iT - 5.71e46T^{2}$$
19 $$1 + 3.40e23T + 3.91e48T^{2}$$
23 $$1 + 1.60e25iT - 5.56e51T^{2}$$
29 $$1 + 5.04e27iT - 3.72e55T^{2}$$
31 $$1 - 1.48e28T + 4.69e56T^{2}$$
37 $$1 - 4.44e29T + 3.90e59T^{2}$$
41 $$1 - 2.53e30iT - 1.93e61T^{2}$$
43 $$1 + 1.36e31T + 1.17e62T^{2}$$
47 $$1 + 1.20e31iT - 3.46e63T^{2}$$
53 $$1 - 1.42e32iT - 3.33e65T^{2}$$
59 $$1 - 5.18e33iT - 1.96e67T^{2}$$
61 $$1 + 1.25e34T + 6.95e67T^{2}$$
67 $$1 - 7.45e34T + 2.45e69T^{2}$$
71 $$1 + 2.70e35iT - 2.22e70T^{2}$$
73 $$1 - 1.76e35T + 6.40e70T^{2}$$
79 $$1 - 4.16e34T + 1.28e72T^{2}$$
83 $$1 + 4.66e35iT - 8.41e72T^{2}$$
89 $$1 + 1.44e37iT - 1.19e74T^{2}$$
97 $$1 - 4.27e37T + 3.14e75T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$