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Properties

Label 2-2160-12.11-c3-0-52
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 + 15.0i·7-s + 42.5·11-s − 43.6·13-s + 39.0i·17-s − 43.1i·19-s + 151.·23-s − 25·25-s − 118. i·29-s − 270. i·31-s − 75.1·35-s + 202.·37-s + 95.4i·41-s − 47.6i·43-s + 236.·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.811i·7-s + 1.16·11-s − 0.931·13-s + 0.557i·17-s − 0.520i·19-s + 1.37·23-s − 0.200·25-s − 0.760i·29-s − 1.56i·31-s − 0.362·35-s + 0.901·37-s + 0.363i·41-s − 0.169i·43-s + 0.735·47-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

\(L(2)\) \(\approx\) \(2.356037300\)
\(L(\frac12)\) \(\approx\) \(2.356037300\)
\(L(\frac{5}{2})\) 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 \( 1 - 15.0iT - 343T^{2} \)
11 \( 1 - 42.5T + 1.33e3T^{2} \)
13 \( 1 + 43.6T + 2.19e3T^{2} \)
17 \( 1 - 39.0iT - 4.91e3T^{2} \)
19 \( 1 + 43.1iT - 6.85e3T^{2} \)
23 \( 1 - 151.T + 1.21e4T^{2} \)
29 \( 1 + 118. iT - 2.43e4T^{2} \)
31 \( 1 + 270. iT - 2.97e4T^{2} \)
37 \( 1 - 202.T + 5.06e4T^{2} \)
41 \( 1 - 95.4iT - 6.89e4T^{2} \)
43 \( 1 + 47.6iT - 7.95e4T^{2} \)
47 \( 1 - 236.T + 1.03e5T^{2} \)
53 \( 1 + 227. iT - 1.48e5T^{2} \)
59 \( 1 + 335.T + 2.05e5T^{2} \)
61 \( 1 - 324.T + 2.26e5T^{2} \)
67 \( 1 + 161. iT - 3.00e5T^{2} \)
71 \( 1 + 451.T + 3.57e5T^{2} \)
73 \( 1 - 315.T + 3.89e5T^{2} \)
79 \( 1 - 753. iT - 4.93e5T^{2} \)
83 \( 1 - 554.T + 5.71e5T^{2} \)
89 \( 1 + 739. iT - 7.04e5T^{2} \)
97 \( 1 - 1.71e3T + 9.12e5T^{2} \)
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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

−8.976169519210912406646627145765, −7.978388519699883442649797033112, −7.19188069824985285194892143977, −6.42021113806954393541067829363, −5.74850268701289563157359131084, −4.74515171740814341058404574765, −3.89813246604426083697872362960, −2.80324344900558349119279936374, −2.07820439060660618932795897906, −0.72692362387743227038536098835, 0.72139503586435457736128591431, 1.48479360903128857355474273489, 2.85126959339786004345399664929, 3.81123644768563459403677756278, 4.64376718692318303721968502244, 5.31597024285308790883776875534, 6.48255170082578288064100118945, 7.12823003612717933154439537181, 7.73470738010756740351692475418, 8.929478006699708268516990194021

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