Properties

Label 2-2880-8.5-c1-0-30
Degree $2$
Conductor $2880$
Sign $-0.258 + 0.965i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 3.46i·11-s + 3.46i·13-s − 3.46·17-s − 4i·19-s − 25-s − 6i·29-s − 3.46·31-s − 3.46i·37-s − 6.92·41-s + 4i·43-s + 12·47-s − 7·49-s − 6i·53-s + 3.46·55-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.04i·11-s + 0.960i·13-s − 0.840·17-s − 0.917i·19-s − 0.200·25-s − 1.11i·29-s − 0.622·31-s − 0.569i·37-s − 1.08·41-s + 0.609i·43-s + 1.75·47-s − 49-s − 0.824i·53-s + 0.467·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9595613684\)
\(L(\frac12)\) \(\approx\) \(0.9595613684\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 3.46iT - 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.660028135054982375262293720611, −7.80094038044776034619138615307, −6.90212988544887289641533699685, −6.40482649389017380677193510373, −5.54737349489928589122694773746, −4.56212449011991369590298358773, −3.76762034189593931564614921777, −2.80144468149461499075924721802, −1.87663066030947128011637230269, −0.30443296453344264688207702028, 1.29787060371798793505263825713, 2.31174958229351208147513721045, 3.44171575832765357576671137490, 4.34224938271149257922366313263, 5.13095000685091091668022225604, 5.82399342219003607646171727105, 6.83833214215272057996577430250, 7.47528367014606436729730219680, 8.286979462035644621372744009763, 8.940748345366480721326000951085

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