# Properties

 Label 2-1080-1.1-c1-0-7 Degree $2$ Conductor $1080$ Sign $1$ Analytic cond. $8.62384$ Root an. cond. $2.93663$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s + 2·7-s + 11-s + 13-s − 17-s + 4·19-s + 23-s + 25-s − 5·29-s + 31-s + 2·35-s + 6·37-s + 7·43-s + 7·47-s − 3·49-s − 12·53-s + 55-s − 4·59-s + 10·61-s + 65-s − 4·67-s + 12·71-s + 6·73-s + 2·77-s + 15·79-s + 2·83-s − 85-s + ⋯
 L(s)  = 1 + 0.447·5-s + 0.755·7-s + 0.301·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s + 0.179·31-s + 0.338·35-s + 0.986·37-s + 1.06·43-s + 1.02·47-s − 3/7·49-s − 1.64·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s + 0.124·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.227·77-s + 1.68·79-s + 0.219·83-s − 0.108·85-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1080$$    =    $$2^{3} \cdot 3^{3} \cdot 5$$ Sign: $1$ Analytic conductor: $$8.62384$$ Root analytic conductor: $$2.93663$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1080,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.998267417$$ $$L(\frac12)$$ $$\approx$$ $$1.998267417$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1 - T$$
good7 $$1 - 2 T + p T^{2}$$
11 $$1 - T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 + T + p T^{2}$$
19 $$1 - 4 T + p T^{2}$$
23 $$1 - T + p T^{2}$$
29 $$1 + 5 T + p T^{2}$$
31 $$1 - T + p T^{2}$$
37 $$1 - 6 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 7 T + p T^{2}$$
47 $$1 - 7 T + p T^{2}$$
53 $$1 + 12 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 15 T + p T^{2}$$
83 $$1 - 2 T + p T^{2}$$
89 $$1 + 12 T + p T^{2}$$
97 $$1 - 10 T + p T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$