# Properties

 Label 2-2160-1.1-c3-0-13 Degree $2$ Conductor $2160$ Sign $1$ Analytic cond. $127.444$ Root an. cond. $11.2891$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 5·5-s − 17·7-s + 30·11-s − 61·13-s − 120·17-s + 43·19-s − 90·23-s + 25·25-s − 90·29-s − 8·31-s − 85·35-s + 317·37-s − 30·41-s + 220·43-s + 180·47-s − 54·49-s − 630·53-s + 150·55-s + 840·59-s + 599·61-s − 305·65-s − 107·67-s + 210·71-s − 421·73-s − 510·77-s − 353·79-s + 1.35e3·83-s + ⋯
 L(s)  = 1 + 0.447·5-s − 0.917·7-s + 0.822·11-s − 1.30·13-s − 1.71·17-s + 0.519·19-s − 0.815·23-s + 1/5·25-s − 0.576·29-s − 0.0463·31-s − 0.410·35-s + 1.40·37-s − 0.114·41-s + 0.780·43-s + 0.558·47-s − 0.157·49-s − 1.63·53-s + 0.367·55-s + 1.85·59-s + 1.25·61-s − 0.582·65-s − 0.195·67-s + 0.351·71-s − 0.674·73-s − 0.754·77-s − 0.502·79-s + 1.78·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2160$$    =    $$2^{4} \cdot 3^{3} \cdot 5$$ Sign: $1$ Analytic conductor: $$127.444$$ Root analytic conductor: $$11.2891$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: $\chi_{2160} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2160,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.469750079$$ $$L(\frac12)$$ $$\approx$$ $$1.469750079$$ $$L(\frac{5}{2})$$ 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 - p T$$
good7 $$1 + 17 T + p^{3} T^{2}$$
11 $$1 - 30 T + p^{3} T^{2}$$
13 $$1 + 61 T + p^{3} T^{2}$$
17 $$1 + 120 T + p^{3} T^{2}$$
19 $$1 - 43 T + p^{3} T^{2}$$
23 $$1 + 90 T + p^{3} T^{2}$$
29 $$1 + 90 T + p^{3} T^{2}$$
31 $$1 + 8 T + p^{3} T^{2}$$
37 $$1 - 317 T + p^{3} T^{2}$$
41 $$1 + 30 T + p^{3} T^{2}$$
43 $$1 - 220 T + p^{3} T^{2}$$
47 $$1 - 180 T + p^{3} T^{2}$$
53 $$1 + 630 T + p^{3} T^{2}$$
59 $$1 - 840 T + p^{3} T^{2}$$
61 $$1 - 599 T + p^{3} T^{2}$$
67 $$1 + 107 T + p^{3} T^{2}$$
71 $$1 - 210 T + p^{3} T^{2}$$
73 $$1 + 421 T + p^{3} T^{2}$$
79 $$1 + 353 T + p^{3} T^{2}$$
83 $$1 - 1350 T + p^{3} T^{2}$$
89 $$1 - 1020 T + p^{3} T^{2}$$
97 $$1 + 997 T + p^{3} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$