# Properties

 Label 2-260-20.7-c1-0-11 Degree $2$ Conductor $260$ Sign $0.677 - 0.735i$ Analytic cond. $2.07611$ Root an. cond. $1.44087$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.18 − 0.774i)2-s + (2.05 + 2.05i)3-s + (0.800 + 1.83i)4-s + (1.34 + 1.78i)5-s + (−0.841 − 4.03i)6-s + (0.845 − 0.845i)7-s + (0.472 − 2.78i)8-s + 5.48i·9-s + (−0.211 − 3.15i)10-s − 4.19i·11-s + (−2.12 + 5.42i)12-s + (−0.707 + 0.707i)13-s + (−1.65 + 0.345i)14-s + (−0.901 + 6.44i)15-s + (−2.71 + 2.93i)16-s + (−0.206 − 0.206i)17-s + ⋯
 L(s)  = 1 + (−0.836 − 0.547i)2-s + (1.18 + 1.18i)3-s + (0.400 + 0.916i)4-s + (0.602 + 0.798i)5-s + (−0.343 − 1.64i)6-s + (0.319 − 0.319i)7-s + (0.167 − 0.985i)8-s + 1.82i·9-s + (−0.0668 − 0.997i)10-s − 1.26i·11-s + (−0.613 + 1.56i)12-s + (−0.196 + 0.196i)13-s + (−0.442 + 0.0923i)14-s + (−0.232 + 1.66i)15-s + (−0.679 + 0.733i)16-s + (−0.0499 − 0.0499i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $0.677 - 0.735i$ Analytic conductor: $$2.07611$$ Root analytic conductor: $$1.44087$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{260} (27, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 260,\ (\ :1/2),\ 0.677 - 0.735i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.24186 + 0.544454i$$ $$L(\frac12)$$ $$\approx$$ $$1.24186 + 0.544454i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.18 + 0.774i)T$$
5 $$1 + (-1.34 - 1.78i)T$$
13 $$1 + (0.707 - 0.707i)T$$
good3 $$1 + (-2.05 - 2.05i)T + 3iT^{2}$$
7 $$1 + (-0.845 + 0.845i)T - 7iT^{2}$$
11 $$1 + 4.19iT - 11T^{2}$$
17 $$1 + (0.206 + 0.206i)T + 17iT^{2}$$
19 $$1 + 4.84T + 19T^{2}$$
23 $$1 + (-0.389 - 0.389i)T + 23iT^{2}$$
29 $$1 + 6.55iT - 29T^{2}$$
31 $$1 + 5.06iT - 31T^{2}$$
37 $$1 + (-8.49 - 8.49i)T + 37iT^{2}$$
41 $$1 + 5.61T + 41T^{2}$$
43 $$1 + (3.25 + 3.25i)T + 43iT^{2}$$
47 $$1 + (-8.91 + 8.91i)T - 47iT^{2}$$
53 $$1 + (-0.0714 + 0.0714i)T - 53iT^{2}$$
59 $$1 + 10.9T + 59T^{2}$$
61 $$1 + 3.57T + 61T^{2}$$
67 $$1 + (6.95 - 6.95i)T - 67iT^{2}$$
71 $$1 + 4.61iT - 71T^{2}$$
73 $$1 + (-2.57 + 2.57i)T - 73iT^{2}$$
79 $$1 - 2.19T + 79T^{2}$$
83 $$1 + (-11.1 - 11.1i)T + 83iT^{2}$$
89 $$1 + 15.0iT - 89T^{2}$$
97 $$1 + (5.51 + 5.51i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$