# Properties

 Label 1-260-260.207-r0-0-0 Degree $1$ Conductor $260$ Sign $-0.525 + 0.850i$ Analytic cond. $1.20743$ Root an. cond. $1.20743$ 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 + i·7-s − 9-s + 11-s + i·17-s − 19-s − 21-s + i·23-s − i·27-s − 29-s + 31-s + i·33-s − i·37-s − 41-s + i·43-s + ⋯
 L(s)  = 1 + i·3-s + i·7-s − 9-s + 11-s + i·17-s − 19-s − 21-s + i·23-s − i·27-s − 29-s + 31-s + i·33-s − i·37-s − 41-s + i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $-0.525 + 0.850i$ Analytic conductor: $$1.20743$$ Root analytic conductor: $$1.20743$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{260} (207, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 260,\ (0:\ ),\ -0.525 + 0.850i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5240298130 + 0.9399022273i$$ $$L(\frac12)$$ $$\approx$$ $$0.5240298130 + 0.9399022273i$$ $$L(1)$$ $$\approx$$ $$0.8616406587 + 0.5224811559i$$ $$L(1)$$ $$\approx$$ $$0.8616406587 + 0.5224811559i$$

## 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 + iT$$
23 $$1$$
29 $$1 - T$$
31 $$1$$
37 $$1 + T$$
41 $$1$$
43 $$1$$
47 $$1$$
53 $$1$$
59 $$1$$
61 $$1 + iT$$
67 $$1$$
71 $$1 - T$$
73 $$1$$
79 $$1 - T$$
83 $$1$$
89 $$1 + iT$$
97 $$1$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$