Properties

Label 2-2880-12.11-c1-0-10
Degree $2$
Conductor $2880$
Sign $0.816 - 0.577i$
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 − 1.63i·7-s + 3.95·11-s − 0.585·13-s + 4i·17-s + 7.91i·19-s − 5.59·23-s − 25-s + 7.65i·29-s + 5.59i·31-s − 1.63·35-s + 9.07·37-s + 1.41i·41-s − 7.91i·43-s + 3.27·47-s + ⋯
L(s)  = 1  − 0.447i·5-s − 0.619i·7-s + 1.19·11-s − 0.162·13-s + 0.970i·17-s + 1.81i·19-s − 1.16·23-s − 0.200·25-s + 1.42i·29-s + 1.00i·31-s − 0.277·35-s + 1.49·37-s + 0.220i·41-s − 1.20i·43-s + 0.478·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760332107\)
\(L(\frac12)\) \(\approx\) \(1.760332107\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.63iT - 7T^{2} \)
11 \( 1 - 3.95T + 11T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 7.91iT - 19T^{2} \)
23 \( 1 + 5.59T + 23T^{2} \)
29 \( 1 - 7.65iT - 29T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 7.91iT - 43T^{2} \)
47 \( 1 - 3.27T + 47T^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 - 0.828T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 4.63T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 - 3.27T + 83T^{2} \)
89 \( 1 - 5.41iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−8.869202605903624043856446708479, −8.055180136658735949279571223515, −7.47275669297353105654942339396, −6.41622840118481888233655382187, −5.96144565435835448009776597054, −4.88487661890120544920768645039, −3.93910255020595304086284638360, −3.56390601710761963128041247114, −1.91561841288903355510489933925, −1.13849376961797709324512791744, 0.63575014393193569196652788980, 2.20848205121969443505050239236, 2.82093596657131920067120015442, 4.03894203887196505012707908061, 4.67699444635761927528198211698, 5.82387824911023386308755757974, 6.36414841646036761581620134905, 7.16773547473073688790229512129, 7.892345135368314978513005755132, 8.808015352539101119239057379826

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