Properties

Label 2-2160-1.1-c3-0-72
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 $1$

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: $\chi_{2160} (1, \cdot )$
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 \( 1 - 2.40T + 343T^{2} \)
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

−8.492393500992495822148633075504, −7.55752122098615708508870629840, −6.77452763908664872656796770314, −6.02754366729099058801556211595, −5.15748677329458019868069892075, −3.98172740002470697049815080978, −3.66188762588000779099928579359, −2.25668718417237057877709237878, −1.25009991303512396055323738024, 0, 1.25009991303512396055323738024, 2.25668718417237057877709237878, 3.66188762588000779099928579359, 3.98172740002470697049815080978, 5.15748677329458019868069892075, 6.02754366729099058801556211595, 6.77452763908664872656796770314, 7.55752122098615708508870629840, 8.492393500992495822148633075504

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