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Properties

Label 2-2880-16.5-c1-0-8
Degree $2$
Conductor $2880$
Sign $0.488 - 0.872i$
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 − 4.06i·7-s + (1.15 + 1.15i)11-s + (−2.48 + 2.48i)13-s − 3.58·17-s + (−4.19 + 4.19i)19-s + 4.42i·23-s + 1.00i·25-s + (−2.76 + 2.76i)29-s + 10.7·31-s + (−2.87 + 2.87i)35-s + (−4.11 − 4.11i)37-s + 10.9i·41-s + (−0.217 − 0.217i)43-s + 7.21·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 1.53i·7-s + (0.349 + 0.349i)11-s + (−0.689 + 0.689i)13-s − 0.869·17-s + (−0.963 + 0.963i)19-s + 0.923i·23-s + 0.200i·25-s + (−0.513 + 0.513i)29-s + 1.92·31-s + (−0.486 + 0.486i)35-s + (−0.675 − 0.675i)37-s + 1.71i·41-s + (−0.0331 − 0.0331i)43-s + 1.05·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070064403\)
\(L(\frac12)\) \(\approx\) \(1.070064403\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.06iT - 7T^{2} \)
11 \( 1 + (-1.15 - 1.15i)T + 11iT^{2} \)
13 \( 1 + (2.48 - 2.48i)T - 13iT^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + (4.19 - 4.19i)T - 19iT^{2} \)
23 \( 1 - 4.42iT - 23T^{2} \)
29 \( 1 + (2.76 - 2.76i)T - 29iT^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (4.11 + 4.11i)T + 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (0.217 + 0.217i)T + 43iT^{2} \)
47 \( 1 - 7.21T + 47T^{2} \)
53 \( 1 + (-4.93 - 4.93i)T + 53iT^{2} \)
59 \( 1 + (-9.20 - 9.20i)T + 59iT^{2} \)
61 \( 1 + (-1.03 + 1.03i)T - 61iT^{2} \)
67 \( 1 + (-0.201 + 0.201i)T - 67iT^{2} \)
71 \( 1 + 6.73iT - 71T^{2} \)
73 \( 1 + 1.83iT - 73T^{2} \)
79 \( 1 - 6.37T + 79T^{2} \)
83 \( 1 + (-3.17 + 3.17i)T - 83iT^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.881057265136040648721440475427, −8.074062547263688066390297332582, −7.29904245397116346200838533234, −6.84513442193268316453143466326, −5.94768136449219197531961526170, −4.63788460593391506234608954452, −4.31628087597780368292557594841, −3.51584144951210313119782060939, −2.12073014095625245239426452789, −1.05756269733627486848054549863, 0.38106908099300328693124633950, 2.36299670065088933163077192561, 2.58321250571791126697639241913, 3.88106457837580827785685999700, 4.82753317121593312902802970870, 5.55492947104226921257115111598, 6.46341904915531537380873389849, 6.93355918260543235614727538268, 8.185336409356097107660858597774, 8.563137593013615226209515214207

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