Properties

 Label 2-2160-1.1-c3-0-60 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 + 3.85·7-s − 42.2·11-s + 4.96·13-s − 25.8·17-s − 28.9·19-s + 191.·23-s + 25·25-s + 287.·29-s − 52.6·31-s − 19.2·35-s − 225.·37-s + 73.9·41-s + 275.·43-s + 192.·47-s − 328.·49-s + 275.·53-s + 211.·55-s − 497.·59-s + 44.0·61-s − 24.8·65-s − 761.·67-s − 264.·71-s + 728.·73-s − 163.·77-s − 664.·79-s − 1.49e3·83-s + ⋯
 L(s)  = 1 − 0.447·5-s + 0.208·7-s − 1.15·11-s + 0.105·13-s − 0.369·17-s − 0.350·19-s + 1.73·23-s + 0.200·25-s + 1.84·29-s − 0.305·31-s − 0.0931·35-s − 1.00·37-s + 0.281·41-s + 0.976·43-s + 0.596·47-s − 0.956·49-s + 0.713·53-s + 0.518·55-s − 1.09·59-s + 0.0924·61-s − 0.0473·65-s − 1.38·67-s − 0.441·71-s + 1.16·73-s − 0.241·77-s − 0.946·79-s − 1.97·83-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 - 3.85T + 343T^{2}$$
11 $$1 + 42.2T + 1.33e3T^{2}$$
13 $$1 - 4.96T + 2.19e3T^{2}$$
17 $$1 + 25.8T + 4.91e3T^{2}$$
19 $$1 + 28.9T + 6.85e3T^{2}$$
23 $$1 - 191.T + 1.21e4T^{2}$$
29 $$1 - 287.T + 2.43e4T^{2}$$
31 $$1 + 52.6T + 2.97e4T^{2}$$
37 $$1 + 225.T + 5.06e4T^{2}$$
41 $$1 - 73.9T + 6.89e4T^{2}$$
43 $$1 - 275.T + 7.95e4T^{2}$$
47 $$1 - 192.T + 1.03e5T^{2}$$
53 $$1 - 275.T + 1.48e5T^{2}$$
59 $$1 + 497.T + 2.05e5T^{2}$$
61 $$1 - 44.0T + 2.26e5T^{2}$$
67 $$1 + 761.T + 3.00e5T^{2}$$
71 $$1 + 264.T + 3.57e5T^{2}$$
73 $$1 - 728.T + 3.89e5T^{2}$$
79 $$1 + 664.T + 4.93e5T^{2}$$
83 $$1 + 1.49e3T + 5.71e5T^{2}$$
89 $$1 - 106.T + 7.04e5T^{2}$$
97 $$1 + 924.T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$