# Properties

 Label 2-2160-1.1-c3-0-26 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 + 24.3·7-s + 26.8·11-s − 68.2·13-s − 61.8·17-s + 75.1·19-s + 43.3·23-s + 25·25-s + 174.·29-s − 222.·31-s − 121.·35-s + 67.1·37-s − 22.2·41-s + 84.7·43-s + 585.·47-s + 251.·49-s + 38.3·53-s − 134.·55-s − 92·59-s − 226.·61-s + 341.·65-s − 858.·67-s − 116.·71-s + 911.·73-s + 655.·77-s + 285.·79-s + 999.·83-s + ⋯
 L(s)  = 1 − 0.447·5-s + 1.31·7-s + 0.737·11-s − 1.45·13-s − 0.882·17-s + 0.907·19-s + 0.393·23-s + 0.200·25-s + 1.12·29-s − 1.29·31-s − 0.588·35-s + 0.298·37-s − 0.0845·41-s + 0.300·43-s + 1.81·47-s + 0.734·49-s + 0.0994·53-s − 0.329·55-s − 0.203·59-s − 0.474·61-s + 0.651·65-s − 1.56·67-s − 0.195·71-s + 1.46·73-s + 0.970·77-s + 0.406·79-s + 1.32·83-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

 $$L(2)$$ $$\approx$$ $$2.281596396$$ $$L(\frac12)$$ $$\approx$$ $$2.281596396$$ $$L(\frac{5}{2})$$ 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 - 24.3T + 343T^{2}$$
11 $$1 - 26.8T + 1.33e3T^{2}$$
13 $$1 + 68.2T + 2.19e3T^{2}$$
17 $$1 + 61.8T + 4.91e3T^{2}$$
19 $$1 - 75.1T + 6.85e3T^{2}$$
23 $$1 - 43.3T + 1.21e4T^{2}$$
29 $$1 - 174.T + 2.43e4T^{2}$$
31 $$1 + 222.T + 2.97e4T^{2}$$
37 $$1 - 67.1T + 5.06e4T^{2}$$
41 $$1 + 22.2T + 6.89e4T^{2}$$
43 $$1 - 84.7T + 7.95e4T^{2}$$
47 $$1 - 585.T + 1.03e5T^{2}$$
53 $$1 - 38.3T + 1.48e5T^{2}$$
59 $$1 + 92T + 2.05e5T^{2}$$
61 $$1 + 226.T + 2.26e5T^{2}$$
67 $$1 + 858.T + 3.00e5T^{2}$$
71 $$1 + 116.T + 3.57e5T^{2}$$
73 $$1 - 911.T + 3.89e5T^{2}$$
79 $$1 - 285.T + 4.93e5T^{2}$$
83 $$1 - 999.T + 5.71e5T^{2}$$
89 $$1 + 374.T + 7.04e5T^{2}$$
97 $$1 + 227.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$