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Properties

Label 2-2880-48.11-c1-0-29
Degree $2$
Conductor $2880$
Sign $-0.861 + 0.506i$
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 − 2.45·7-s + (−0.168 − 0.168i)11-s + (2.45 − 2.45i)13-s + 1.49i·17-s + (−1.78 − 1.78i)19-s − 1.51i·23-s − 1.00i·25-s + (−4.35 − 4.35i)29-s + 3.96i·31-s + (−1.73 + 1.73i)35-s + (1.35 + 1.35i)37-s + 2.13·41-s + (−5.24 + 5.24i)43-s − 7.46·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s − 0.927·7-s + (−0.0506 − 0.0506i)11-s + (0.679 − 0.679i)13-s + 0.363i·17-s + (−0.410 − 0.410i)19-s − 0.315i·23-s − 0.200i·25-s + (−0.808 − 0.808i)29-s + 0.711i·31-s + (−0.293 + 0.293i)35-s + (0.222 + 0.222i)37-s + 0.333·41-s + (−0.799 + 0.799i)43-s − 1.08·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.861 + 0.506i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.861 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6869515907\)
\(L(\frac12)\) \(\approx\) \(0.6869515907\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.45T + 7T^{2} \)
11 \( 1 + (0.168 + 0.168i)T + 11iT^{2} \)
13 \( 1 + (-2.45 + 2.45i)T - 13iT^{2} \)
17 \( 1 - 1.49iT - 17T^{2} \)
19 \( 1 + (1.78 + 1.78i)T + 19iT^{2} \)
23 \( 1 + 1.51iT - 23T^{2} \)
29 \( 1 + (4.35 + 4.35i)T + 29iT^{2} \)
31 \( 1 - 3.96iT - 31T^{2} \)
37 \( 1 + (-1.35 - 1.35i)T + 37iT^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 + (5.24 - 5.24i)T - 43iT^{2} \)
47 \( 1 + 7.46T + 47T^{2} \)
53 \( 1 + (-8.51 + 8.51i)T - 53iT^{2} \)
59 \( 1 + (-0.862 - 0.862i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 - 3.75iT - 73T^{2} \)
79 \( 1 + 8.09iT - 79T^{2} \)
83 \( 1 + (5.60 - 5.60i)T - 83iT^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 + 15.1T + 97T^{2} \)
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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

−8.479028946462318852314091409708, −7.83818125225156125492400157037, −6.75096507715953365658460150754, −6.21562044287850887509413462945, −5.48427396606671840618335454390, −4.53190597422385539426053446412, −3.56612245087103784447823182727, −2.79925931540566648373199875802, −1.57695894663993372656161408741, −0.21243682473257702446694053076, 1.47353931232013203482553057972, 2.57246672697402884557099428619, 3.50523683731385849972695397804, 4.22196607855315583578980425439, 5.41959351994038402722358805588, 6.08687017442300661214013653440, 6.79176842692759561863226326740, 7.41441020978620508622027163615, 8.454561848539868560852965437069, 9.160670285292011837801537582402

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