Properties

Label 2-2160-1.1-c3-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s − 14.5·7-s − 49.2·11-s + 72.1·13-s − 118.·17-s − 123.·19-s − 91.4·23-s + 25·25-s − 174.·29-s + 46.2·31-s + 72.5·35-s + 154.·37-s + 364.·41-s − 125.·43-s − 221.·47-s − 132.·49-s + 13.6·53-s + 246.·55-s + 239.·59-s − 54.5·61-s − 360.·65-s + 76.0·67-s + 728.·71-s − 501.·73-s + 715.·77-s − 397.·79-s − 1.36e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.783·7-s − 1.35·11-s + 1.53·13-s − 1.68·17-s − 1.48·19-s − 0.829·23-s + 0.200·25-s − 1.11·29-s + 0.268·31-s + 0.350·35-s + 0.688·37-s + 1.38·41-s − 0.445·43-s − 0.687·47-s − 0.385·49-s + 0.0354·53-s + 0.604·55-s + 0.527·59-s − 0.114·61-s − 0.688·65-s + 0.138·67-s + 1.21·71-s − 0.804·73-s + 1.05·77-s − 0.566·79-s − 1.81·83-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

\(L(2)\) \(\approx\) \(0.6218994256\)
\(L(\frac12)\) \(\approx\) \(0.6218994256\)
\(L(\frac{5}{2})\) 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 \( 1 + 14.5T + 343T^{2} \)
11 \( 1 + 49.2T + 1.33e3T^{2} \)
13 \( 1 - 72.1T + 2.19e3T^{2} \)
17 \( 1 + 118.T + 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + 91.4T + 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 - 46.2T + 2.97e4T^{2} \)
37 \( 1 - 154.T + 5.06e4T^{2} \)
41 \( 1 - 364.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + 221.T + 1.03e5T^{2} \)
53 \( 1 - 13.6T + 1.48e5T^{2} \)
59 \( 1 - 239.T + 2.05e5T^{2} \)
61 \( 1 + 54.5T + 2.26e5T^{2} \)
67 \( 1 - 76.0T + 3.00e5T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 + 501.T + 3.89e5T^{2} \)
79 \( 1 + 397.T + 4.93e5T^{2} \)
83 \( 1 + 1.36e3T + 5.71e5T^{2} \)
89 \( 1 + 1.46e3T + 7.04e5T^{2} \)
97 \( 1 - 335.T + 9.12e5T^{2} \)
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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

−8.525200820273198151800212692527, −8.174719679086535478926035390060, −7.10716059641723724604890786719, −6.32070761248735785846845142379, −5.76217083934571988092499013672, −4.48742436322822095429510331479, −3.90429191007281031337615339753, −2.84883472603925296204039487719, −1.93904557679539961835098877897, −0.33686646246872973884210936249, 0.33686646246872973884210936249, 1.93904557679539961835098877897, 2.84883472603925296204039487719, 3.90429191007281031337615339753, 4.48742436322822095429510331479, 5.76217083934571988092499013672, 6.32070761248735785846845142379, 7.10716059641723724604890786719, 8.174719679086535478926035390060, 8.525200820273198151800212692527

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