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Properties

Label 2-2880-180.139-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  + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s + 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + 1.73·35-s + (−0.5 + 0.866i)41-s − 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s + 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s + (−0.5 − 0.866i)29-s + 1.73·35-s + (−0.5 + 0.866i)41-s − 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-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: $\chi_{2880} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3100720927\)
\(L(\frac12)\) \(\approx\) \(0.3100720927\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.425843085150631896531568435039, −7.85931206515604018868889645177, −7.63033155480521738155611672403, −6.85842501375188650577943140702, −6.33438963666772706290111264723, −5.53277296425893229032705217626, −4.23659397101326177822823679224, −3.79362776807097909848194366055, −2.66862849851516393678702060796, −1.22300475139456811455549182232, 0.23598449708717053306657858830, 1.97488064813586656689865332117, 3.24754689870159944293772371394, 4.09940668331711756428119696595, 5.01900748094432207229325186479, 5.58718804853802737488813131175, 6.27479980721362033906995617869, 7.05130437208842874164083906136, 8.247441379412397282611070111415, 8.890234072129308472608749590411

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