# Properties

 Label 2-1080-1.1-c3-0-33 Degree $2$ Conductor $1080$ Sign $-1$ Analytic cond. $63.7220$ Root an. cond. $7.98261$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5·5-s − 30.2·7-s + 67.3·11-s − 60.7·13-s + 61.1·17-s − 27.8·19-s − 40.4·23-s + 25·25-s + 212.·29-s + 167.·31-s − 151.·35-s − 366.·37-s − 363.·41-s + 153.·43-s − 434.·47-s + 570.·49-s + 79.6·53-s + 336.·55-s + 339.·59-s − 525.·61-s − 303.·65-s + 131.·67-s − 296.·71-s − 1.23e3·73-s − 2.03e3·77-s − 621.·79-s − 76.3·83-s + ⋯
 L(s)  = 1 + 0.447·5-s − 1.63·7-s + 1.84·11-s − 1.29·13-s + 0.872·17-s − 0.336·19-s − 0.366·23-s + 0.200·25-s + 1.35·29-s + 0.969·31-s − 0.729·35-s − 1.62·37-s − 1.38·41-s + 0.545·43-s − 1.34·47-s + 1.66·49-s + 0.206·53-s + 0.825·55-s + 0.748·59-s − 1.10·61-s − 0.580·65-s + 0.240·67-s − 0.495·71-s − 1.97·73-s − 3.01·77-s − 0.885·79-s − 0.100·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: $$63.7220$$ Root analytic conductor: $$7.98261$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1080,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 5T$$
good7 $$1 + 30.2T + 343T^{2}$$
11 $$1 - 67.3T + 1.33e3T^{2}$$
13 $$1 + 60.7T + 2.19e3T^{2}$$
17 $$1 - 61.1T + 4.91e3T^{2}$$
19 $$1 + 27.8T + 6.85e3T^{2}$$
23 $$1 + 40.4T + 1.21e4T^{2}$$
29 $$1 - 212.T + 2.43e4T^{2}$$
31 $$1 - 167.T + 2.97e4T^{2}$$
37 $$1 + 366.T + 5.06e4T^{2}$$
41 $$1 + 363.T + 6.89e4T^{2}$$
43 $$1 - 153.T + 7.95e4T^{2}$$
47 $$1 + 434.T + 1.03e5T^{2}$$
53 $$1 - 79.6T + 1.48e5T^{2}$$
59 $$1 - 339.T + 2.05e5T^{2}$$
61 $$1 + 525.T + 2.26e5T^{2}$$
67 $$1 - 131.T + 3.00e5T^{2}$$
71 $$1 + 296.T + 3.57e5T^{2}$$
73 $$1 + 1.23e3T + 3.89e5T^{2}$$
79 $$1 + 621.T + 4.93e5T^{2}$$
83 $$1 + 76.3T + 5.71e5T^{2}$$
89 $$1 + 192.T + 7.04e5T^{2}$$
97 $$1 + 874.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$