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Properties

Label 2-2880-180.79-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8047620346\)
\(L(\frac12)\) \(\approx\) \(0.8047620346\)
\(L(1)\) 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

−9.124341262513931921036419522028, −8.261204561306139041109713835443, −7.49448396396202385775543767190, −6.67207579762922035564784758164, −6.17534803587148751106937407049, −5.61262907558489642550975910784, −4.61513174007322232574137771784, −3.15083391219480897328328518960, −2.54665829285715161419014266577, −1.55355513168815157663624875263, 0.52939089770347656307058012856, 1.98609132456151855772709693258, 3.50661512155154079706557656836, 4.02378035417871130917379258860, 4.96050209383363365413286642559, 5.59461820268370552783368270706, 6.33758571730686593590579872675, 7.19762023609525562275726827369, 8.186223973329353668423572242360, 8.980976686057987926543437577601

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