# Properties

 Label 2-260-260.43-c1-0-10 Degree $2$ Conductor $260$ Sign $0.872 - 0.488i$ 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 + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (1.86 + 1.23i)5-s + (−1.99 + 2i)8-s + (2.59 − 1.5i)9-s + (−3 − 0.999i)10-s + (−3.59 + 0.232i)13-s + (1.99 − 3.46i)16-s + (6.96 + 1.86i)17-s + (−3 + 3i)18-s + (4.46 + 0.267i)20-s + (1.96 + 4.59i)25-s + (4.83 − 1.63i)26-s + (5.76 + 3.33i)29-s + (−1.46 + 5.46i)32-s + ⋯
 L(s)  = 1 + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (0.834 + 0.550i)5-s + (−0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.948 − 0.316i)10-s + (−0.997 + 0.0643i)13-s + (0.499 − 0.866i)16-s + (1.68 + 0.452i)17-s + (−0.707 + 0.707i)18-s + (0.998 + 0.0599i)20-s + (0.392 + 0.919i)25-s + (0.947 − 0.320i)26-s + (1.07 + 0.618i)29-s + (−0.258 + 0.965i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $0.872 - 0.488i$ Analytic conductor: $$2.07611$$ Root analytic conductor: $$1.44087$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{260} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 260,\ (\ :1/2),\ 0.872 - 0.488i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.968320 + 0.252881i$$ $$L(\frac12)$$ $$\approx$$ $$0.968320 + 0.252881i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.36 - 0.366i)T$$
5 $$1 + (-1.86 - 1.23i)T$$
13 $$1 + (3.59 - 0.232i)T$$
good3 $$1 + (-2.59 + 1.5i)T^{2}$$
7 $$1 + (6.06 + 3.5i)T^{2}$$
11 $$1 + (-5.5 - 9.52i)T^{2}$$
17 $$1 + (-6.96 - 1.86i)T + (14.7 + 8.5i)T^{2}$$
19 $$1 + (9.5 - 16.4i)T^{2}$$
23 $$1 + (-19.9 + 11.5i)T^{2}$$
29 $$1 + (-5.76 - 3.33i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + (1.13 - 0.303i)T + (32.0 - 18.5i)T^{2}$$
41 $$1 + (-1.66 - 0.964i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (37.2 + 21.5i)T^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 + (10.2 + 10.2i)T + 53iT^{2}$$
59 $$1 + (29.5 - 51.0i)T^{2}$$
61 $$1 + (7.33 + 12.6i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (9.83 - 9.83i)T - 73iT^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-4.75 + 17.7i)T + (-84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$