# Properties

 Degree $2$ Conductor $1386$ Sign $0.761 + 0.648i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.34 − 1.35i)5-s + (−0.222 + 2.63i)7-s + 0.999·8-s + (−2.34 − 1.35i)10-s + (−2.96 − 1.48i)11-s + 2.18i·13-s + (2.39 − 1.12i)14-s + (−0.5 − 0.866i)16-s + (0.0281 − 0.0488i)17-s + (2.38 − 1.37i)19-s + 2.71i·20-s + (0.192 + 3.31i)22-s + (6.02 − 3.47i)23-s + ⋯
 L(s)  = 1 + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (1.05 − 0.606i)5-s + (−0.0839 + 0.996i)7-s + 0.353·8-s + (−0.742 − 0.428i)10-s + (−0.893 − 0.449i)11-s + 0.604i·13-s + (0.639 − 0.300i)14-s + (−0.125 − 0.216i)16-s + (0.00683 − 0.0118i)17-s + (0.547 − 0.315i)19-s + 0.606i·20-s + (0.0409 + 0.705i)22-s + (1.25 − 0.725i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $0.761 + 0.648i$ Motivic weight: $$1$$ Character: $\chi_{1386} (1187, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ 0.761 + 0.648i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.610164094$$ $$L(\frac12)$$ $$\approx$$ $$1.610164094$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 + 0.866i)T$$
3 $$1$$
7 $$1 + (0.222 - 2.63i)T$$
11 $$1 + (2.96 + 1.48i)T$$
good5 $$1 + (-2.34 + 1.35i)T + (2.5 - 4.33i)T^{2}$$
13 $$1 - 2.18iT - 13T^{2}$$
17 $$1 + (-0.0281 + 0.0488i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2.38 + 1.37i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-6.02 + 3.47i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 - 6.43T + 29T^{2}$$
31 $$1 + (-3.65 + 6.32i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1.62 + 2.80i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 2.71T + 41T^{2}$$
43 $$1 - 10.6iT - 43T^{2}$$
47 $$1 + (2.58 - 1.49i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-10.1 - 5.88i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 + (-12.2 - 7.07i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (-10.2 + 5.93i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-0.749 + 1.29i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 1.82iT - 71T^{2}$$
73 $$1 + (-6.82 - 3.93i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (7.82 - 4.51i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + 17.2T + 83T^{2}$$
89 $$1 + (-7.43 + 4.29i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 - 6.25T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$