Properties

Label 2-2880-8.5-c1-0-27
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 − 1.26·7-s − 3.46i·11-s + 3.46i·13-s − 3.46·17-s + 2i·19-s + 8.19·23-s − 25-s + 9.46·31-s + 1.26i·35-s − 6i·37-s − 2.53·41-s − 10.1i·43-s − 8.19·47-s − 5.39·49-s + ⋯
L(s)  = 1  − 0.447i·5-s − 0.479·7-s − 1.04i·11-s + 0.960i·13-s − 0.840·17-s + 0.458i·19-s + 1.70·23-s − 0.200·25-s + 1.69·31-s + 0.214i·35-s − 0.986i·37-s − 0.396·41-s − 1.55i·43-s − 1.19·47-s − 0.770·49-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

\(L(1)\) \(\approx\) \(1.196397080\)
\(L(\frac12)\) \(\approx\) \(1.196397080\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.46T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + 10.1iT - 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4.73iT - 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−8.675633811973169694877733674988, −7.947349591902545305202353906579, −6.78479144257994616752568653028, −6.46967290842681220755621965876, −5.41762659699721921527286959763, −4.67567359779411195895236525783, −3.74248122743724503558144945626, −2.89838561759075302840972019915, −1.70508584190718889663602764046, −0.40610297949479450680656394433, 1.21742491526225121391326462421, 2.68061071083011225059821945717, 3.09328418056973568516646821151, 4.45805377874823238036654968758, 4.95009263289322932831623936628, 6.13353844082001422107185555589, 6.73830414974396240417962854000, 7.37294549808494223126421982299, 8.228419053375635418483009860739, 9.017730713935180070592652277678

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