Properties

Label 2-2160-1.1-c3-0-27
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 − 9.46·7-s + 49.0·11-s + 55.4·13-s + 27.9·17-s − 26.3·19-s + 9.53·23-s + 25·25-s − 218.·29-s + 158.·31-s + 47.3·35-s + 189.·37-s + 246.·41-s − 40.2·43-s − 213.·47-s − 253.·49-s + 283.·53-s − 245.·55-s − 92·59-s + 370.·61-s − 277.·65-s − 1.08e3·67-s + 333.·71-s − 606.·73-s − 464.·77-s + 44.3·79-s − 107.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.511·7-s + 1.34·11-s + 1.18·13-s + 0.399·17-s − 0.318·19-s + 0.0864·23-s + 0.200·25-s − 1.39·29-s + 0.920·31-s + 0.228·35-s + 0.842·37-s + 0.937·41-s − 0.142·43-s − 0.664·47-s − 0.738·49-s + 0.734·53-s − 0.601·55-s − 0.203·59-s + 0.776·61-s − 0.528·65-s − 1.98·67-s + 0.557·71-s − 0.972·73-s − 0.687·77-s + 0.0631·79-s − 0.142·83-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

\(L(2)\) \(\approx\) \(2.153205989\)
\(L(\frac12)\) \(\approx\) \(2.153205989\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.46T + 343T^{2} \)
11 \( 1 - 49.0T + 1.33e3T^{2} \)
13 \( 1 - 55.4T + 2.19e3T^{2} \)
17 \( 1 - 27.9T + 4.91e3T^{2} \)
19 \( 1 + 26.3T + 6.85e3T^{2} \)
23 \( 1 - 9.53T + 1.21e4T^{2} \)
29 \( 1 + 218.T + 2.43e4T^{2} \)
31 \( 1 - 158.T + 2.97e4T^{2} \)
37 \( 1 - 189.T + 5.06e4T^{2} \)
41 \( 1 - 246.T + 6.89e4T^{2} \)
43 \( 1 + 40.2T + 7.95e4T^{2} \)
47 \( 1 + 213.T + 1.03e5T^{2} \)
53 \( 1 - 283.T + 1.48e5T^{2} \)
59 \( 1 + 92T + 2.05e5T^{2} \)
61 \( 1 - 370.T + 2.26e5T^{2} \)
67 \( 1 + 1.08e3T + 3.00e5T^{2} \)
71 \( 1 - 333.T + 3.57e5T^{2} \)
73 \( 1 + 606.T + 3.89e5T^{2} \)
79 \( 1 - 44.3T + 4.93e5T^{2} \)
83 \( 1 + 107.T + 5.71e5T^{2} \)
89 \( 1 + 257.T + 7.04e5T^{2} \)
97 \( 1 - 1.31e3T + 9.12e5T^{2} \)
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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

−8.782936960327829760281668685780, −7.995293022816128128620996962316, −7.11628665538161078952480085531, −6.32682588842465689425818093719, −5.79449024619700863114740012463, −4.46595466996278286007341673799, −3.81194870470042003794609257339, −3.06807169338575925706661682475, −1.65872956166561162131142991731, −0.69715916994554157241350064901, 0.69715916994554157241350064901, 1.65872956166561162131142991731, 3.06807169338575925706661682475, 3.81194870470042003794609257339, 4.46595466996278286007341673799, 5.79449024619700863114740012463, 6.32682588842465689425818093719, 7.11628665538161078952480085531, 7.995293022816128128620996962316, 8.782936960327829760281668685780

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