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Properties

Label 2-2880-16.13-c1-0-4
Degree $2$
Conductor $2880$
Sign $-0.914 + 0.403i$
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  + (−0.707 + 0.707i)5-s + 4.30i·7-s + (−4.45 + 4.45i)11-s + (−0.918 − 0.918i)13-s − 2.39·17-s + (5.30 + 5.30i)19-s − 2.19i·23-s − 1.00i·25-s + (4.30 + 4.30i)29-s − 5.39·31-s + (−3.04 − 3.04i)35-s + (0.841 − 0.841i)37-s − 6.87i·41-s + (6.63 − 6.63i)43-s − 11.0·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + 1.62i·7-s + (−1.34 + 1.34i)11-s + (−0.254 − 0.254i)13-s − 0.580·17-s + (1.21 + 1.21i)19-s − 0.458i·23-s − 0.200i·25-s + (0.798 + 0.798i)29-s − 0.969·31-s + (−0.515 − 0.515i)35-s + (0.138 − 0.138i)37-s − 1.07i·41-s + (1.01 − 1.01i)43-s − 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.914 + 0.403i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6310750681\)
\(L(\frac12)\) \(\approx\) \(0.6310750681\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 4.30iT - 7T^{2} \)
11 \( 1 + (4.45 - 4.45i)T - 11iT^{2} \)
13 \( 1 + (0.918 + 0.918i)T + 13iT^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 + (-5.30 - 5.30i)T + 19iT^{2} \)
23 \( 1 + 2.19iT - 23T^{2} \)
29 \( 1 + (-4.30 - 4.30i)T + 29iT^{2} \)
31 \( 1 + 5.39T + 31T^{2} \)
37 \( 1 + (-0.841 + 0.841i)T - 37iT^{2} \)
41 \( 1 + 6.87iT - 41T^{2} \)
43 \( 1 + (-6.63 + 6.63i)T - 43iT^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + (0.600 - 0.600i)T - 53iT^{2} \)
59 \( 1 + (0.850 - 0.850i)T - 59iT^{2} \)
61 \( 1 + (7.50 + 7.50i)T + 61iT^{2} \)
67 \( 1 + (-8.00 - 8.00i)T + 67iT^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 + 1.27iT - 73T^{2} \)
79 \( 1 + 8.05T + 79T^{2} \)
83 \( 1 + (-0.110 - 0.110i)T + 83iT^{2} \)
89 \( 1 + 5.90iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−9.163600199960440931613865506138, −8.423603150165588654044666492804, −7.67259888664300547292017542501, −7.09884745940151250801938153114, −6.01897455441500051916381761951, −5.30497340132069300428005258536, −4.76933273397578518947201662257, −3.46150159972892571514282164178, −2.57224049354461536488264501631, −1.91667727520990162631651054823, 0.21562868142463652224746249274, 1.11667045721437470851106018858, 2.75958297428665262350500194264, 3.47498942093589258008007939288, 4.50569716070832531437373247159, 5.04755189128515079387911392336, 6.08632441296978313990671663319, 6.98912096050505109140679942356, 7.69793340556670588779057675668, 8.104205546440663720798466946577

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