# Properties

 Label 2-630-5.4-c1-0-10 Degree $2$ Conductor $630$ Sign $-0.447 + 0.894i$ Analytic cond. $5.03057$ Root an. cond. $2.24289$ 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 − 4-s + (−1 + 2i)5-s − i·7-s + i·8-s + (2 + i)10-s − 2·11-s − 6i·13-s − 14-s + 16-s − 4i·17-s + 6·19-s + (1 − 2i)20-s + 2i·22-s − 8i·23-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.377i·7-s + 0.353i·8-s + (0.632 + 0.316i)10-s − 0.603·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s − 0.970i·17-s + 1.37·19-s + (0.223 − 0.447i)20-s + 0.426i·22-s − 1.66i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$630$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$5.03057$$ Root analytic conductor: $$2.24289$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{630} (379, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 630,\ (\ :1/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.540391 - 0.874372i$$ $$L(\frac12)$$ $$\approx$$ $$0.540391 - 0.874372i$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 + (1 - 2i)T$$
7 $$1 + iT$$
good11 $$1 + 2T + 11T^{2}$$
13 $$1 + 6iT - 13T^{2}$$
17 $$1 + 4iT - 17T^{2}$$
19 $$1 - 6T + 19T^{2}$$
23 $$1 + 8iT - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + 2T + 31T^{2}$$
37 $$1 - 4iT - 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 - 6iT - 53T^{2}$$
59 $$1 + 8T + 59T^{2}$$
61 $$1 + 10T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 - 14iT - 73T^{2}$$
79 $$1 - 12T + 79T^{2}$$
83 $$1 + 8iT - 83T^{2}$$
89 $$1 + 10T + 89T^{2}$$
97 $$1 + 10iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$