# Properties

 Label 1-920-920.909-r0-0-0 Degree $1$ Conductor $920$ Sign $-0.684 - 0.728i$ Analytic cond. $4.27246$ Root an. cond. $4.27246$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.959 − 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.654 − 0.755i)27-s + (0.654 − 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯
 L(s)  = 1 + (−0.959 − 0.281i)3-s + (0.142 − 0.989i)7-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (0.654 − 0.755i)17-s + (0.654 + 0.755i)19-s + (−0.415 + 0.909i)21-s + (−0.654 − 0.755i)27-s + (0.654 − 0.755i)29-s + (−0.959 + 0.281i)31-s + (0.142 + 0.989i)33-s + (0.841 + 0.540i)37-s + (−0.142 + 0.989i)39-s + (0.841 − 0.540i)41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$920$$    =    $$2^{3} \cdot 5 \cdot 23$$ Sign: $-0.684 - 0.728i$ Analytic conductor: $$4.27246$$ Root analytic conductor: $$4.27246$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{920} (909, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 920,\ (0:\ ),\ -0.684 - 0.728i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3402918906 - 0.7869081826i$$ $$L(\frac12)$$ $$\approx$$ $$0.3402918906 - 0.7869081826i$$ $$L(1)$$ $$\approx$$ $$0.7172701756 - 0.3161823947i$$ $$L(1)$$ $$\approx$$ $$0.7172701756 - 0.3161823947i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
23 $$1$$
good3 $$1 + (-0.959 - 0.281i)T$$
7 $$1 + (0.142 - 0.989i)T$$
11 $$1 + (-0.415 - 0.909i)T$$
13 $$1 + (-0.142 - 0.989i)T$$
17 $$1 + (0.654 - 0.755i)T$$
19 $$1 + (0.654 + 0.755i)T$$
29 $$1 + (0.654 - 0.755i)T$$
31 $$1 + (-0.959 + 0.281i)T$$
37 $$1 + (0.841 + 0.540i)T$$
41 $$1 + (0.841 - 0.540i)T$$
43 $$1 + (-0.959 - 0.281i)T$$
47 $$1 - T$$
53 $$1 + (-0.142 + 0.989i)T$$
59 $$1 + (0.142 + 0.989i)T$$
61 $$1 + (0.959 - 0.281i)T$$
67 $$1 + (0.415 - 0.909i)T$$
71 $$1 + (0.415 - 0.909i)T$$
73 $$1 + (0.654 + 0.755i)T$$
79 $$1 + (-0.142 - 0.989i)T$$
83 $$1 + (0.841 + 0.540i)T$$
89 $$1 + (-0.959 - 0.281i)T$$
97 $$1 + (-0.841 + 0.540i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$