# Properties

 Degree $2$ Conductor $1386$ Sign $-0.820 - 0.572i$ 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 + (1.37 + 2.37i)5-s + (−1.37 + 2.26i)7-s − 0.999·8-s + (−1.37 + 2.37i)10-s + (0.5 − 0.866i)11-s + 5.49·13-s + (−2.64 − 0.0585i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (4.01 + 6.96i)19-s − 2.74·20-s + 0.999·22-s + (−0.645 − 1.11i)23-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.614 + 1.06i)5-s + (−0.519 + 0.854i)7-s − 0.353·8-s + (−0.434 + 0.752i)10-s + (0.150 − 0.261i)11-s + 1.52·13-s + (−0.706 − 0.0156i)14-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + (0.921 + 1.59i)19-s − 0.614·20-s + 0.213·22-s + (−0.134 − 0.232i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $-0.820 - 0.572i$ Motivic weight: $$1$$ Character: $\chi_{1386} (991, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ -0.820 - 0.572i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.058532104$$ $$L(\frac12)$$ $$\approx$$ $$2.058532104$$ $$L(\frac{3}{2})$$ 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 + (1.37 - 2.26i)T$$
11 $$1 + (-0.5 + 0.866i)T$$
good5 $$1 + (-1.37 - 2.37i)T + (-2.5 + 4.33i)T^{2}$$
13 $$1 - 5.49T + 13T^{2}$$
17 $$1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-4.01 - 6.96i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (0.645 + 1.11i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 4.54T + 29T^{2}$$
31 $$1 + (2.54 - 4.40i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2.01 - 3.49i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + 5.54T + 41T^{2}$$
43 $$1 + 4.03T + 43T^{2}$$
47 $$1 + (4.64 + 8.04i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-2.74 + 4.75i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (4.76 - 8.25i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-0.829 - 1.43i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (1.77 - 3.06i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 2.54T + 71T^{2}$$
73 $$1 + (-4.29 + 7.43i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (6.11 + 10.5i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 14.5T + 83T^{2}$$
89 $$1 + (3.25 + 5.63i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 0.0872T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$