# Properties

 Label 2-390-5.4-c1-0-5 Degree $2$ Conductor $390$ Sign $1$ 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 + i·2-s + i·3-s − 4-s − 2.23i·5-s − 6-s − 3.23i·7-s − i·8-s − 9-s + 2.23·10-s + 4.47·11-s − i·12-s + i·13-s + 3.23·14-s + 2.23·15-s + 16-s − 7.23i·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.999i·5-s − 0.408·6-s − 1.22i·7-s − 0.353i·8-s − 0.333·9-s + 0.707·10-s + 1.34·11-s − 0.288i·12-s + 0.277i·13-s + 0.864·14-s + 0.577·15-s + 0.250·16-s − 1.75i·17-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $1$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.30559$$ $$L(\frac12)$$ $$\approx$$ $$1.30559$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1 - iT$$
5 $$1 + 2.23iT$$
13 $$1 - iT$$
good7 $$1 + 3.23iT - 7T^{2}$$
11 $$1 - 4.47T + 11T^{2}$$
17 $$1 + 7.23iT - 17T^{2}$$
19 $$1 - 2.76T + 19T^{2}$$
23 $$1 - 2.76iT - 23T^{2}$$
29 $$1 - 3.70T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 - 10.9iT - 37T^{2}$$
41 $$1 - 3.52T + 41T^{2}$$
43 $$1 + 2.47iT - 43T^{2}$$
47 $$1 + 12.9iT - 47T^{2}$$
53 $$1 + 0.472iT - 53T^{2}$$
59 $$1 - 8.47T + 59T^{2}$$
61 $$1 + 10.9T + 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 2.47T + 71T^{2}$$
73 $$1 - 13.2iT - 73T^{2}$$
79 $$1 - 4T + 79T^{2}$$
83 $$1 + 4.94iT - 83T^{2}$$
89 $$1 + 0.472T + 89T^{2}$$
97 $$1 - 3.70iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$