# Properties

 Label 2-580-1.1-c1-0-7 Degree $2$ Conductor $580$ Sign $-1$ Analytic cond. $4.63132$ Root an. cond. $2.15205$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s − 2·7-s − 3·9-s − 4·11-s − 6·13-s − 4·17-s + 4·19-s + 6·23-s + 25-s − 29-s − 2·35-s − 8·37-s − 2·41-s + 4·43-s − 3·45-s − 4·47-s − 3·49-s − 2·53-s − 4·55-s + 8·59-s + 10·61-s + 6·63-s − 6·65-s − 10·67-s − 8·71-s + 8·77-s + 8·79-s + ⋯
 L(s)  = 1 + 0.447·5-s − 0.755·7-s − 9-s − 1.20·11-s − 1.66·13-s − 0.970·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.185·29-s − 0.338·35-s − 1.31·37-s − 0.312·41-s + 0.609·43-s − 0.447·45-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 0.539·55-s + 1.04·59-s + 1.28·61-s + 0.755·63-s − 0.744·65-s − 1.22·67-s − 0.949·71-s + 0.911·77-s + 0.900·79-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$580$$    =    $$2^{2} \cdot 5 \cdot 29$$ Sign: $-1$ Analytic conductor: $$4.63132$$ Root analytic conductor: $$2.15205$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 580,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - T$$
29 $$1 + T$$
good3 $$1 + p T^{2}$$
7 $$1 + 2 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 + 4 T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 - 4 T + p T^{2}$$
47 $$1 + 4 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 - 8 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 + 10 T + p T^{2}$$
71 $$1 + 8 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 - 8 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 + 12 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$