# Properties

 Label 2-390-65.8-c1-0-4 Degree $2$ Conductor $390$ Sign $0.432 - 0.901i$ 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 + (0.707 + 0.707i)3-s + 4-s + (−2.22 + 0.227i)5-s + (0.707 + 0.707i)6-s + 4.05i·7-s + 8-s + 1.00i·9-s + (−2.22 + 0.227i)10-s + (1.63 − 1.63i)11-s + (0.707 + 0.707i)12-s + (3.18 + 1.68i)13-s + 4.05i·14-s + (−1.73 − 1.41i)15-s + 16-s + (0.932 + 0.932i)17-s + ⋯
 L(s)  = 1 + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.994 + 0.101i)5-s + (0.288 + 0.288i)6-s + 1.53i·7-s + 0.353·8-s + 0.333i·9-s + (−0.703 + 0.0720i)10-s + (0.493 − 0.493i)11-s + (0.204 + 0.204i)12-s + (0.883 + 0.467i)13-s + 1.08i·14-s + (−0.447 − 0.364i)15-s + 0.250·16-s + (0.226 + 0.226i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $0.432 - 0.901i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (73, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ 0.432 - 0.901i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.73060 + 1.08975i$$ $$L(\frac12)$$ $$\approx$$ $$1.73060 + 1.08975i$$ $$L(\frac{3}{2})$$ 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 + (-0.707 - 0.707i)T$$
5 $$1 + (2.22 - 0.227i)T$$
13 $$1 + (-3.18 - 1.68i)T$$
good7 $$1 - 4.05iT - 7T^{2}$$
11 $$1 + (-1.63 + 1.63i)T - 11iT^{2}$$
17 $$1 + (-0.932 - 0.932i)T + 17iT^{2}$$
19 $$1 + (4.66 - 4.66i)T - 19iT^{2}$$
23 $$1 + (-3.93 + 3.93i)T - 23iT^{2}$$
29 $$1 + 6.86iT - 29T^{2}$$
31 $$1 + (6.67 + 6.67i)T + 31iT^{2}$$
37 $$1 + 8.04iT - 37T^{2}$$
41 $$1 + (-6.69 - 6.69i)T + 41iT^{2}$$
43 $$1 + (2.58 - 2.58i)T - 43iT^{2}$$
47 $$1 - 0.559iT - 47T^{2}$$
53 $$1 + (6.34 + 6.34i)T + 53iT^{2}$$
59 $$1 + (2.29 + 2.29i)T + 59iT^{2}$$
61 $$1 - 6.48T + 61T^{2}$$
67 $$1 - 7.13T + 67T^{2}$$
71 $$1 + (-5.56 - 5.56i)T + 71iT^{2}$$
73 $$1 - 7.36T + 73T^{2}$$
79 $$1 + 1.91iT - 79T^{2}$$
83 $$1 + 0.718iT - 83T^{2}$$
89 $$1 + (4.02 + 4.02i)T + 89iT^{2}$$
97 $$1 + 6.60T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$