Properties

Label 2-2880-8.5-c1-0-34
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 + 4.73·7-s − 3.46i·11-s − 3.46i·13-s + 3.46·17-s − 2i·19-s + 2.19·23-s − 25-s − 2.53·31-s − 4.73i·35-s − 6i·37-s − 9.46·41-s − 0.196i·43-s − 2.19·47-s + 15.3·49-s + ⋯
L(s)  = 1  − 0.447i·5-s + 1.78·7-s − 1.04i·11-s − 0.960i·13-s + 0.840·17-s − 0.458i·19-s + 0.457·23-s − 0.200·25-s − 0.455·31-s − 0.799i·35-s − 0.986i·37-s − 1.47·41-s − 0.0299i·43-s − 0.320·47-s + 2.19·49-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

\(L(1)\) \(\approx\) \(2.260701655\)
\(L(\frac12)\) \(\approx\) \(2.260701655\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 + 0.196iT - 43T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 0.928iT - 61T^{2} \)
67 \( 1 + 0.196iT - 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 1.26iT - 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−8.623627749572606350101518978206, −7.84227679801819283806266207912, −7.46478658287170150840018942439, −6.14029630052336634648607145751, −5.31710967860216297578110082162, −4.97020720747110824894902887856, −3.88200187039545453162701796307, −2.91253215631202599062806034442, −1.67537820467108124592958741136, −0.76104772233906374179756945826, 1.48743359418671449145646546010, 2.02130523769817301593823351542, 3.33549009450246415815202602284, 4.41992202938802483792358204239, 4.90440615162653628871812081313, 5.77509651081506931069624856806, 6.89669638307692981804733847675, 7.38353231088624702540139069927, 8.163828778310000269990804204666, 8.767142687878435866037367964349

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