# Properties

 Label 2-390-65.37-c1-0-11 Degree $2$ Conductor $390$ Sign $0.469 + 0.882i$ Analytic cond. $3.11416$ Root an. cond. $1.76469$ 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 + (0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (−0.900 − 2.04i)5-s + (0.965 − 0.258i)6-s + (0.0298 − 0.0517i)7-s − 0.999i·8-s + (0.866 + 0.499i)9-s + (−1.80 − 1.32i)10-s + (2.65 + 0.710i)11-s + (0.707 − 0.707i)12-s + (0.914 − 3.48i)13-s − 0.0597i·14-s + (−0.340 − 2.20i)15-s + (−0.5 − 0.866i)16-s + (1.22 + 4.56i)17-s + ⋯
 L(s)  = 1 + (0.612 − 0.353i)2-s + (0.557 + 0.149i)3-s + (0.249 − 0.433i)4-s + (−0.402 − 0.915i)5-s + (0.394 − 0.105i)6-s + (0.0112 − 0.0195i)7-s − 0.353i·8-s + (0.288 + 0.166i)9-s + (−0.570 − 0.418i)10-s + (0.799 + 0.214i)11-s + (0.204 − 0.204i)12-s + (0.253 − 0.967i)13-s − 0.0159i·14-s + (−0.0879 − 0.570i)15-s + (−0.125 − 0.216i)16-s + (0.296 + 1.10i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $0.469 + 0.882i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 0.469 + 0.882i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.86824 - 1.12250i$$ $$L(\frac12)$$ $$\approx$$ $$1.86824 - 1.12250i$$ $$L(\frac{3}{2})$$ 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 + (-0.965 - 0.258i)T$$
5 $$1 + (0.900 + 2.04i)T$$
13 $$1 + (-0.914 + 3.48i)T$$
good7 $$1 + (-0.0298 + 0.0517i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (-2.65 - 0.710i)T + (9.52 + 5.5i)T^{2}$$
17 $$1 + (-1.22 - 4.56i)T + (-14.7 + 8.5i)T^{2}$$
19 $$1 + (2.19 + 8.18i)T + (-16.4 + 9.5i)T^{2}$$
23 $$1 + (0.837 - 3.12i)T + (-19.9 - 11.5i)T^{2}$$
29 $$1 + (3.06 - 1.76i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (-5.45 - 5.45i)T + 31iT^{2}$$
37 $$1 + (-0.224 - 0.388i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (2.07 - 7.76i)T + (-35.5 - 20.5i)T^{2}$$
43 $$1 + (3.94 - 1.05i)T + (37.2 - 21.5i)T^{2}$$
47 $$1 - 8.10T + 47T^{2}$$
53 $$1 + (3.62 - 3.62i)T - 53iT^{2}$$
59 $$1 + (8.24 - 2.20i)T + (51.0 - 29.5i)T^{2}$$
61 $$1 + (-5.07 + 8.79i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (8.49 - 4.90i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (1.76 - 0.474i)T + (61.4 - 35.5i)T^{2}$$
73 $$1 - 11.0iT - 73T^{2}$$
79 $$1 + 12.6iT - 79T^{2}$$
83 $$1 - 0.810T + 83T^{2}$$
89 $$1 + (-3.49 + 13.0i)T + (-77.0 - 44.5i)T^{2}$$
97 $$1 + (-9.22 - 5.32i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$