# Properties

 Label 2-260-260.203-c0-0-0 Degree $2$ Conductor $260$ Sign $0.749 - 0.661i$ Analytic cond. $0.129756$ Root an. cond. $0.360217$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + i·5-s − 8-s + i·9-s − i·10-s − i·13-s + 16-s + (1 + i)17-s − i·18-s + i·20-s − 25-s + i·26-s − 2i·29-s − 32-s + ⋯
 L(s)  = 1 − 2-s + 4-s + i·5-s − 8-s + i·9-s − i·10-s − i·13-s + 16-s + (1 + i)17-s − i·18-s + i·20-s − 25-s + i·26-s − 2i·29-s − 32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $0.749 - 0.661i$ Analytic conductor: $$0.129756$$ Root analytic conductor: $$0.360217$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{260} (203, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 260,\ (\ :0),\ 0.749 - 0.661i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5059126929$$ $$L(\frac12)$$ $$\approx$$ $$0.5059126929$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
5 $$1 - iT$$
13 $$1 + iT$$
good3 $$1 - iT^{2}$$
7 $$1 + T^{2}$$
11 $$1 + iT^{2}$$
17 $$1 + (-1 - i)T + iT^{2}$$
19 $$1 + iT^{2}$$
23 $$1 + iT^{2}$$
29 $$1 + 2iT - T^{2}$$
31 $$1 - iT^{2}$$
37 $$1 - T^{2}$$
41 $$1 + (1 + i)T + iT^{2}$$
43 $$1 + iT^{2}$$
47 $$1 + T^{2}$$
53 $$1 + (1 + i)T + iT^{2}$$
59 $$1 - iT^{2}$$
61 $$1 + T^{2}$$
67 $$1 - T^{2}$$
71 $$1 - iT^{2}$$
73 $$1 - 2T + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + (-1 - i)T + iT^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$