# Properties

 Label 1-260-260.163-r1-0-0 Degree $1$ Conductor $260$ Sign $0.868 + 0.496i$ Analytic cond. $27.9408$ Root an. cond. $27.9408$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)41-s + (−0.866 − 0.5i)43-s + ⋯
 L(s)  = 1 + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + (0.866 − 0.5i)41-s + (−0.866 − 0.5i)43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $0.868 + 0.496i$ Analytic conductor: $$27.9408$$ Root analytic conductor: $$27.9408$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{260} (163, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 260,\ (1:\ ),\ 0.868 + 0.496i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.176588553 + 0.3126283321i$$ $$L(\frac12)$$ $$\approx$$ $$1.176588553 + 0.3126283321i$$ $$L(1)$$ $$\approx$$ $$0.8181400540 + 0.1392047768i$$ $$L(1)$$ $$\approx$$ $$0.8181400540 + 0.1392047768i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
13 $$1$$
good3 $$1 + (-0.866 + 0.5i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
11 $$1 + (0.866 - 0.5i)T$$
17 $$1 + (-0.866 - 0.5i)T$$
19 $$1 + (-0.866 - 0.5i)T$$
23 $$1 + (0.866 - 0.5i)T$$
29 $$1 + (0.5 + 0.866i)T$$
31 $$1 - iT$$
37 $$1 + (0.5 + 0.866i)T$$
41 $$1 + (0.866 - 0.5i)T$$
43 $$1 + (-0.866 - 0.5i)T$$
47 $$1 + T$$
53 $$1 + iT$$
59 $$1 + (0.866 + 0.5i)T$$
61 $$1 + (-0.5 + 0.866i)T$$
67 $$1 + (0.5 + 0.866i)T$$
71 $$1 + (0.866 + 0.5i)T$$
73 $$1 + T$$
79 $$1 + T$$
83 $$1 + T$$
89 $$1 + (0.866 - 0.5i)T$$
97 $$1 + (-0.5 + 0.866i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$