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Properties

Label 2-2880-180.79-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.173 - 0.984i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)15-s + 0.999·21-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)29-s + 0.999·35-s + (0.5 + 0.866i)41-s + (1 − 1.73i)43-s − 0.999·45-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)15-s + 0.999·21-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)29-s + 0.999·35-s + (0.5 + 0.866i)41-s + (1 − 1.73i)43-s − 0.999·45-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.646004778\)
\(L(\frac12)\) \(\approx\) \(1.646004778\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.223826718305855265114034195060, −8.480369050468391839810714468529, −7.42534117913834486056216984344, −7.18621530404378794048179723054, −5.92320148961871584019092971743, −5.24341693315974065382397879082, −4.27769218216159212974723174349, −3.55658777993906907654344569534, −2.73326392370793457871694347760, −1.64509517683527146491139362783, 1.05835520682313524296989683089, 2.10711834991941699631872380659, 2.74022108219464296623192514342, 4.09654127510504351517244595889, 5.00557815200628854761433989997, 5.86720076966175889530050794394, 6.35293503791239846851923782925, 7.51327550696187853948440466282, 8.052520986540843387610259395733, 8.911404599159117196362578289378

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