# Properties

 Label 2-390-65.64-c1-0-3 Degree $2$ Conductor $390$ Sign $0.955 - 0.296i$ 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 − 2-s − i·3-s + 4-s + (1.48 + 1.67i)5-s + i·6-s + 4.15·7-s − 8-s − 9-s + (−1.48 − 1.67i)10-s + 5.35i·11-s − i·12-s + (−3.28 − 1.48i)13-s − 4.15·14-s + (1.67 − 1.48i)15-s + 16-s + 1.19i·17-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.662 + 0.749i)5-s + 0.408i·6-s + 1.57·7-s − 0.353·8-s − 0.333·9-s + (−0.468 − 0.529i)10-s + 1.61i·11-s − 0.288i·12-s + (−0.911 − 0.410i)13-s − 1.11·14-s + (0.432 − 0.382i)15-s + 0.250·16-s + 0.289i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $0.955 - 0.296i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (259, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 0.955 - 0.296i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.22078 + 0.184933i$$ $$L(\frac12)$$ $$\approx$$ $$1.22078 + 0.184933i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + iT$$
5 $$1 + (-1.48 - 1.67i)T$$
13 $$1 + (3.28 + 1.48i)T$$
good7 $$1 - 4.15T + 7T^{2}$$
11 $$1 - 5.35iT - 11T^{2}$$
17 $$1 - 1.19iT - 17T^{2}$$
19 $$1 - 2.80iT - 19T^{2}$$
23 $$1 + 0.806iT - 23T^{2}$$
29 $$1 - 7.50T + 29T^{2}$$
31 $$1 + 7.92iT - 31T^{2}$$
37 $$1 - 7.35T + 37T^{2}$$
41 $$1 + 3.61iT - 41T^{2}$$
43 $$1 + 6.57iT - 43T^{2}$$
47 $$1 + 12.3T + 47T^{2}$$
53 $$1 + 5.53iT - 53T^{2}$$
59 $$1 - 12.8iT - 59T^{2}$$
61 $$1 - 6.31T + 61T^{2}$$
67 $$1 - 4.57T + 67T^{2}$$
71 $$1 + 8.96iT - 71T^{2}$$
73 $$1 + 4.08T + 73T^{2}$$
79 $$1 + 12.4T + 79T^{2}$$
83 $$1 + 10.0T + 83T^{2}$$
89 $$1 + 5.03iT - 89T^{2}$$
97 $$1 + 2.93T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$