# Properties

 Label 2-2160-1.1-c3-0-5 Degree $2$ Conductor $2160$ Sign $1$ Analytic cond. $127.444$ Root an. cond. $11.2891$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 5·5-s − 14.5·7-s − 49.2·11-s + 72.1·13-s − 118.·17-s − 123.·19-s − 91.4·23-s + 25·25-s − 174.·29-s + 46.2·31-s + 72.5·35-s + 154.·37-s + 364.·41-s − 125.·43-s − 221.·47-s − 132.·49-s + 13.6·53-s + 246.·55-s + 239.·59-s − 54.5·61-s − 360.·65-s + 76.0·67-s + 728.·71-s − 501.·73-s + 715.·77-s − 397.·79-s − 1.36e3·83-s + ⋯
 L(s)  = 1 − 0.447·5-s − 0.783·7-s − 1.35·11-s + 1.53·13-s − 1.68·17-s − 1.48·19-s − 0.829·23-s + 0.200·25-s − 1.11·29-s + 0.268·31-s + 0.350·35-s + 0.688·37-s + 1.38·41-s − 0.445·43-s − 0.687·47-s − 0.385·49-s + 0.0354·53-s + 0.604·55-s + 0.527·59-s − 0.114·61-s − 0.688·65-s + 0.138·67-s + 1.21·71-s − 0.804·73-s + 1.05·77-s − 0.566·79-s − 1.81·83-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: no Arithmetic: yes Character: $\chi_{2160} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2160,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6218994256$$ $$L(\frac12)$$ $$\approx$$ $$0.6218994256$$ $$L(\frac{5}{2})$$ 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 + 14.5T + 343T^{2}$$
11 $$1 + 49.2T + 1.33e3T^{2}$$
13 $$1 - 72.1T + 2.19e3T^{2}$$
17 $$1 + 118.T + 4.91e3T^{2}$$
19 $$1 + 123.T + 6.85e3T^{2}$$
23 $$1 + 91.4T + 1.21e4T^{2}$$
29 $$1 + 174.T + 2.43e4T^{2}$$
31 $$1 - 46.2T + 2.97e4T^{2}$$
37 $$1 - 154.T + 5.06e4T^{2}$$
41 $$1 - 364.T + 6.89e4T^{2}$$
43 $$1 + 125.T + 7.95e4T^{2}$$
47 $$1 + 221.T + 1.03e5T^{2}$$
53 $$1 - 13.6T + 1.48e5T^{2}$$
59 $$1 - 239.T + 2.05e5T^{2}$$
61 $$1 + 54.5T + 2.26e5T^{2}$$
67 $$1 - 76.0T + 3.00e5T^{2}$$
71 $$1 - 728.T + 3.57e5T^{2}$$
73 $$1 + 501.T + 3.89e5T^{2}$$
79 $$1 + 397.T + 4.93e5T^{2}$$
83 $$1 + 1.36e3T + 5.71e5T^{2}$$
89 $$1 + 1.46e3T + 7.04e5T^{2}$$
97 $$1 - 335.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$