# Properties

 Label 2-2160-1.1-c3-0-74 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 $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 5·5-s − 23.9·7-s + 57.9·11-s − 8.16·13-s + 50.0·17-s − 69.7·19-s + 4.92·23-s + 25·25-s − 79.4·29-s − 260.·31-s − 119.·35-s − 223.·37-s + 337.·41-s − 326.·43-s + 89.6·47-s + 229.·49-s + 543.·53-s + 289.·55-s + 92·59-s + 159.·61-s − 40.8·65-s + 910.·67-s − 293.·71-s + 142.·73-s − 1.38e3·77-s − 1.10e3·79-s + 813.·83-s + ⋯
 L(s)  = 1 + 0.447·5-s − 1.29·7-s + 1.58·11-s − 0.174·13-s + 0.714·17-s − 0.842·19-s + 0.0446·23-s + 0.200·25-s − 0.508·29-s − 1.50·31-s − 0.577·35-s − 0.994·37-s + 1.28·41-s − 1.15·43-s + 0.278·47-s + 0.668·49-s + 1.40·53-s + 0.710·55-s + 0.203·59-s + 0.334·61-s − 0.0778·65-s + 1.66·67-s − 0.490·71-s + 0.227·73-s − 2.05·77-s − 1.57·79-s + 1.07·83-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: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2160,\ (\ :3/2),\ -1)$$

## 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 + 23.9T + 343T^{2}$$
11 $$1 - 57.9T + 1.33e3T^{2}$$
13 $$1 + 8.16T + 2.19e3T^{2}$$
17 $$1 - 50.0T + 4.91e3T^{2}$$
19 $$1 + 69.7T + 6.85e3T^{2}$$
23 $$1 - 4.92T + 1.21e4T^{2}$$
29 $$1 + 79.4T + 2.43e4T^{2}$$
31 $$1 + 260.T + 2.97e4T^{2}$$
37 $$1 + 223.T + 5.06e4T^{2}$$
41 $$1 - 337.T + 6.89e4T^{2}$$
43 $$1 + 326.T + 7.95e4T^{2}$$
47 $$1 - 89.6T + 1.03e5T^{2}$$
53 $$1 - 543.T + 1.48e5T^{2}$$
59 $$1 - 92T + 2.05e5T^{2}$$
61 $$1 - 159.T + 2.26e5T^{2}$$
67 $$1 - 910.T + 3.00e5T^{2}$$
71 $$1 + 293.T + 3.57e5T^{2}$$
73 $$1 - 142.T + 3.89e5T^{2}$$
79 $$1 + 1.10e3T + 4.93e5T^{2}$$
83 $$1 - 813.T + 5.71e5T^{2}$$
89 $$1 + 956.T + 7.04e5T^{2}$$
97 $$1 - 106.T + 9.12e5T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−8.586073635541684526636636617397, −7.31858737809018870030372597318, −6.72296317108167638162863543011, −6.07736387890552511899348628238, −5.30887165113565709222079893066, −3.97956198606435513412392390498, −3.51203911652305220053848828474, −2.32885282283661426561726151817, −1.25664347571108240023114863116, 0, 1.25664347571108240023114863116, 2.32885282283661426561726151817, 3.51203911652305220053848828474, 3.97956198606435513412392390498, 5.30887165113565709222079893066, 6.07736387890552511899348628238, 6.72296317108167638162863543011, 7.31858737809018870030372597318, 8.586073635541684526636636617397