# Properties

 Label 1-260-260.83-r1-0-0 Degree $1$ Conductor $260$ Sign $0.256 + 0.966i$ 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 − i·3-s − 7-s − 9-s − i·11-s + i·17-s − i·19-s + i·21-s + i·23-s + i·27-s − 29-s + i·31-s − 33-s + 37-s − i·41-s + i·43-s + ⋯
 L(s)  = 1 − i·3-s − 7-s − 9-s − i·11-s + i·17-s − i·19-s + i·21-s + i·23-s + i·27-s − 29-s + i·31-s − 33-s + 37-s − i·41-s + i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $0.256 + 0.966i$ Analytic conductor: $$27.9408$$ Root analytic conductor: $$27.9408$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{260} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 260,\ (1:\ ),\ 0.256 + 0.966i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4054398590 + 0.3118226147i$$ $$L(\frac12)$$ $$\approx$$ $$0.4054398590 + 0.3118226147i$$ $$L(1)$$ $$\approx$$ $$0.7532252852 - 0.2000298488i$$ $$L(1)$$ $$\approx$$ $$0.7532252852 - 0.2000298488i$$

## Euler product

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