# Properties

 Degree $2$ Conductor $1386$ Sign $0.958 + 0.286i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.25 − 2.18i)5-s + (1.48 − 2.19i)7-s + 0.999i·8-s + (2.18 − 1.25i)10-s + (0.866 − 0.5i)11-s + 2.44i·13-s + (2.38 − 1.15i)14-s + (−0.5 + 0.866i)16-s + (2.64 + 4.58i)17-s + (−4.52 − 2.61i)19-s + 2.51·20-s + 0.999·22-s + (4.28 + 2.47i)23-s + ⋯
 L(s)  = 1 + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.562 − 0.975i)5-s + (0.560 − 0.827i)7-s + 0.353i·8-s + (0.689 − 0.398i)10-s + (0.261 − 0.150i)11-s + 0.679i·13-s + (0.636 − 0.308i)14-s + (−0.125 + 0.216i)16-s + (0.641 + 1.11i)17-s + (−1.03 − 0.599i)19-s + 0.562·20-s + 0.213·22-s + (0.893 + 0.515i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $0.958 + 0.286i$ Motivic weight: $$1$$ Character: $\chi_{1386} (89, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ 0.958 + 0.286i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.892509079$$ $$L(\frac12)$$ $$\approx$$ $$2.892509079$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.866 - 0.5i)T$$
3 $$1$$
7 $$1 + (-1.48 + 2.19i)T$$
11 $$1 + (-0.866 + 0.5i)T$$
good5 $$1 + (-1.25 + 2.18i)T + (-2.5 - 4.33i)T^{2}$$
13 $$1 - 2.44iT - 13T^{2}$$
17 $$1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (4.52 + 2.61i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-4.28 - 2.47i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + 8.05iT - 29T^{2}$$
31 $$1 + (-7.44 + 4.29i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (-5.91 + 10.2i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + 1.92T + 41T^{2}$$
43 $$1 + 11.9T + 43T^{2}$$
47 $$1 + (-0.637 + 1.10i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-0.414 + 0.239i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (-3.02 - 5.23i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-10.7 - 6.20i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (6.11 + 10.5i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 14.7iT - 71T^{2}$$
73 $$1 + (8.52 - 4.92i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (-3.52 + 6.10i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 9.35T + 83T^{2}$$
89 $$1 + (7.92 - 13.7i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 13.6iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$