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Properties

Label 2-2160-12.11-c3-0-28
Degree $2$
Conductor $2160$
Sign $0.866 - 0.5i$
Analytic cond. $127.444$
Root an. cond. $11.2891$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5i·5-s − 7.56i·7-s − 32.1·11-s − 70.8·13-s − 78.3i·17-s − 58.7i·19-s − 126.·23-s − 25·25-s + 131. i·29-s + 294. i·31-s + 37.8·35-s − 242.·37-s + 32.2i·41-s − 178. i·43-s + 202.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 0.408i·7-s − 0.880·11-s − 1.51·13-s − 1.11i·17-s − 0.709i·19-s − 1.14·23-s − 0.200·25-s + 0.839i·29-s + 1.70i·31-s + 0.182·35-s − 1.07·37-s + 0.122i·41-s − 0.631i·43-s + 0.628·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(127.444\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 0.866 - 0.5i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.118634004\)
\(L(\frac12)\) \(\approx\) \(1.118634004\)
\(L(\frac{5}{2})\) 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 \( 1 - 5iT \)
good7 \( 1 + 7.56iT - 343T^{2} \)
11 \( 1 + 32.1T + 1.33e3T^{2} \)
13 \( 1 + 70.8T + 2.19e3T^{2} \)
17 \( 1 + 78.3iT - 4.91e3T^{2} \)
19 \( 1 + 58.7iT - 6.85e3T^{2} \)
23 \( 1 + 126.T + 1.21e4T^{2} \)
29 \( 1 - 131. iT - 2.43e4T^{2} \)
31 \( 1 - 294. iT - 2.97e4T^{2} \)
37 \( 1 + 242.T + 5.06e4T^{2} \)
41 \( 1 - 32.2iT - 6.89e4T^{2} \)
43 \( 1 + 178. iT - 7.95e4T^{2} \)
47 \( 1 - 202.T + 1.03e5T^{2} \)
53 \( 1 + 754. iT - 1.48e5T^{2} \)
59 \( 1 - 674.T + 2.05e5T^{2} \)
61 \( 1 - 228.T + 2.26e5T^{2} \)
67 \( 1 - 150. iT - 3.00e5T^{2} \)
71 \( 1 + 902.T + 3.57e5T^{2} \)
73 \( 1 - 1.05e3T + 3.89e5T^{2} \)
79 \( 1 - 679. iT - 4.93e5T^{2} \)
83 \( 1 - 898.T + 5.71e5T^{2} \)
89 \( 1 - 546. iT - 7.04e5T^{2} \)
97 \( 1 + 1.02e3T + 9.12e5T^{2} \)
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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

−8.776296821406692063932112943755, −7.897306270299042634237695462994, −7.04818978920931689133964296952, −6.84965375006629256149167279659, −5.26502388675434448597077758377, −5.07143416744071421993835823647, −3.82067985406166227803252087703, −2.82577056268190733349071932722, −2.12450992579963673627380532811, −0.54811594685245985671863912062, 0.37651573932827174931537494248, 1.92073242485360773981143109555, 2.54364286421483609637259548572, 3.87304288304531667384842601304, 4.61129901649719536727495884376, 5.62772467454750540961962992991, 6.01990158836469343924502384503, 7.33545377794308405019354711260, 7.900313779896570517610332017656, 8.533781152168941728937774488429

Graph of the $Z$-function along the critical line