Properties

Label 2-2160-1.1-c3-0-40
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 + 2.40·7-s − 17.3·11-s + 71.2·13-s + 111.·17-s + 86.2·19-s + 148.·23-s + 25·25-s + 71.1·29-s − 212.·31-s + 12.0·35-s − 132.·37-s − 29.6·41-s + 126.·43-s + 103.·47-s − 337.·49-s − 298.·53-s − 86.6·55-s − 893.·59-s + 752.·61-s + 356.·65-s − 477.·67-s + 58.7·71-s + 873.·73-s − 41.6·77-s − 819.·79-s + 959.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.129·7-s − 0.474·11-s + 1.52·13-s + 1.58·17-s + 1.04·19-s + 1.34·23-s + 0.200·25-s + 0.455·29-s − 1.22·31-s + 0.0580·35-s − 0.590·37-s − 0.113·41-s + 0.449·43-s + 0.320·47-s − 0.983·49-s − 0.773·53-s − 0.212·55-s − 1.97·59-s + 1.57·61-s + 0.680·65-s − 0.870·67-s + 0.0981·71-s + 1.40·73-s − 0.0616·77-s − 1.16·79-s + 1.26·83-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: \(0\)
Selberg data: \((2,\ 2160,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.112136435\)
\(L(\frac12)\) \(\approx\) \(3.112136435\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.40T + 343T^{2} \)
11 \( 1 + 17.3T + 1.33e3T^{2} \)
13 \( 1 - 71.2T + 2.19e3T^{2} \)
17 \( 1 - 111.T + 4.91e3T^{2} \)
19 \( 1 - 86.2T + 6.85e3T^{2} \)
23 \( 1 - 148.T + 1.21e4T^{2} \)
29 \( 1 - 71.1T + 2.43e4T^{2} \)
31 \( 1 + 212.T + 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 + 29.6T + 6.89e4T^{2} \)
43 \( 1 - 126.T + 7.95e4T^{2} \)
47 \( 1 - 103.T + 1.03e5T^{2} \)
53 \( 1 + 298.T + 1.48e5T^{2} \)
59 \( 1 + 893.T + 2.05e5T^{2} \)
61 \( 1 - 752.T + 2.26e5T^{2} \)
67 \( 1 + 477.T + 3.00e5T^{2} \)
71 \( 1 - 58.7T + 3.57e5T^{2} \)
73 \( 1 - 873.T + 3.89e5T^{2} \)
79 \( 1 + 819.T + 4.93e5T^{2} \)
83 \( 1 - 959.T + 5.71e5T^{2} \)
89 \( 1 + 406.T + 7.04e5T^{2} \)
97 \( 1 - 623.T + 9.12e5T^{2} \)
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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

−8.777708122612520474203453569589, −7.919758929320974824256741318109, −7.25928859738734914303865168407, −6.25738080994842819233822972141, −5.53660561749462898576327555542, −4.92098860759809652570291217137, −3.56416570950987080052451531837, −3.04784389921331130888531039044, −1.62779928347131947355939831492, −0.871075873126293243858349532662, 0.871075873126293243858349532662, 1.62779928347131947355939831492, 3.04784389921331130888531039044, 3.56416570950987080052451531837, 4.92098860759809652570291217137, 5.53660561749462898576327555542, 6.25738080994842819233822972141, 7.25928859738734914303865168407, 7.919758929320974824256741318109, 8.777708122612520474203453569589

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