# Properties

 Label 2-1386-21.17-c1-0-5 Degree $2$ Conductor $1386$ Sign $0.345 - 0.938i$ Analytic cond. $11.0672$ Root an. cond. $3.32675$ Motivic weight $1$ Arithmetic yes Rational no 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 + (1.22 + 2.12i)5-s + (−1 + 2.44i)7-s − 0.999i·8-s + (2.12 + 1.22i)10-s + (0.866 + 0.5i)11-s + 2.44i·13-s + (0.358 + 2.62i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + (−1.5 + 0.866i)19-s + 2.44·20-s + 0.999·22-s + (−6.27 + 3.62i)23-s + ⋯
 L(s)  = 1 + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.547 + 0.948i)5-s + (−0.377 + 0.925i)7-s − 0.353i·8-s + (0.670 + 0.387i)10-s + (0.261 + 0.150i)11-s + 0.679i·13-s + (0.0958 + 0.700i)14-s + (−0.125 − 0.216i)16-s + (−0.210 + 0.363i)17-s + (−0.344 + 0.198i)19-s + 0.547·20-s + 0.213·22-s + (−1.30 + 0.755i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1386$$    =    $$2 \cdot 3^{2} \cdot 7 \cdot 11$$ Sign: $0.345 - 0.938i$ Analytic conductor: $$11.0672$$ Root analytic conductor: $$3.32675$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1386} (1277, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1386,\ (\ :1/2),\ 0.345 - 0.938i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.225924978$$ $$L(\frac12)$$ $$\approx$$ $$2.225924978$$ $$L(\frac{3}{2})$$ 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 - 2.44i)T$$
11 $$1 + (-0.866 - 0.5i)T$$
good5 $$1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2}$$
13 $$1 - 2.44iT - 13T^{2}$$
17 $$1 + (0.866 - 1.5i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 + (6.27 - 3.62i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + 1.24iT - 29T^{2}$$
31 $$1 + (-1.24 - 0.717i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 1.43T + 41T^{2}$$
43 $$1 - T + 43T^{2}$$
47 $$1 + (-3.82 - 6.62i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (26.5 + 45.8i)T^{2}$$
59 $$1 + (7.49 - 12.9i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-6.24 + 10.8i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 - 7.24iT - 71T^{2}$$
73 $$1 + (-4.24 - 2.44i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 12.8T + 83T^{2}$$
89 $$1 + (0.507 + 0.878i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 3.16iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$