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Properties

Label 2-2160-540.499-c0-0-1
Degree $2$
Conductor $2160$
Sign $-0.973 + 0.230i$
Analytic cond. $1.07798$
Root an. cond. $1.03825$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.766 − 0.642i)5-s + (−1.20 − 0.439i)7-s + (−0.173 − 0.984i)9-s + (−0.984 + 0.173i)15-s + (−1.11 + 0.642i)21-s + (−0.642 + 0.233i)23-s + (0.173 + 0.984i)25-s + (−0.866 − 0.500i)27-s + (−0.326 − 1.85i)29-s + (0.642 + 1.11i)35-s + (−0.0603 + 0.342i)41-s + (−1.32 + 1.11i)43-s + (−0.500 + 0.866i)45-s + (−1.85 − 0.673i)47-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.766 − 0.642i)5-s + (−1.20 − 0.439i)7-s + (−0.173 − 0.984i)9-s + (−0.984 + 0.173i)15-s + (−1.11 + 0.642i)21-s + (−0.642 + 0.233i)23-s + (0.173 + 0.984i)25-s + (−0.866 − 0.500i)27-s + (−0.326 − 1.85i)29-s + (0.642 + 1.11i)35-s + (−0.0603 + 0.342i)41-s + (−1.32 + 1.11i)43-s + (−0.500 + 0.866i)45-s + (−1.85 − 0.673i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(1.07798\)
Root analytic conductor: \(1.03825\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :0),\ -0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7147323381\)
\(L(\frac12)\) \(\approx\) \(0.7147323381\)
\(L(1)\) 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 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
good7 \( 1 + (1.20 + 0.439i)T + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (1.32 - 1.11i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (1.85 + 0.673i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.342 + 1.93i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.342 + 1.93i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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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

−8.767804169084286068671160115134, −8.050187472476022496996277557667, −7.51636291519293379190791613291, −6.61591409637632890417446289077, −6.04019104858860948772525046743, −4.72594130580897303569989872639, −3.72887849315726965649619613611, −3.19587756333193274202274279800, −1.87733194809971872323133515156, −0.43686126268849439251006784473, 2.21857379344155103690454931894, 3.28067757920597654576925886296, 3.55447707490323781090395231366, 4.66565515572216207237634660148, 5.62944896157707472761883854568, 6.66462084757268767230526536843, 7.25033726067646718001138286751, 8.324109483734185258369549478252, 8.744909607622095951839518315885, 9.774164182945927551601487886443

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