# Properties

 Label 2-1560-1560.389-c0-0-5 Degree $2$ Conductor $1560$ Sign $-1$ Analytic cond. $0.778541$ Root an. cond. $0.882349$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s − 2i·11-s + 12-s + 13-s + i·15-s + 16-s − i·18-s + i·20-s + ⋯
 L(s)  = 1 − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s − 2i·11-s + 12-s + 13-s + i·15-s + 16-s − i·18-s + i·20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1560$$    =    $$2^{3} \cdot 3 \cdot 5 \cdot 13$$ Sign: $-1$ Analytic conductor: $$0.778541$$ Root analytic conductor: $$0.882349$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1560} (389, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1560,\ (\ :0),\ -1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6371074505$$ $$L(\frac12)$$ $$\approx$$ $$0.6371074505$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + T$$
5 $$1 + iT$$
13 $$1 - T$$
good7 $$1 + T^{2}$$
11 $$1 + 2iT - T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + T^{2}$$
23 $$1 + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + 2T + T^{2}$$
47 $$1 + 2iT - T^{2}$$
53 $$1 - T^{2}$$
59 $$1 - 2iT - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + T^{2}$$
79 $$1 - 2T + T^{2}$$
83 $$1 + T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$