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Properties

Label 2-2880-16.5-c1-0-0
Degree $2$
Conductor $2880$
Sign $-0.870 + 0.492i$
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 + 2.76i·7-s + (−3.51 − 3.51i)11-s + (−4.55 + 4.55i)13-s + 5.00·17-s + (0.812 − 0.812i)19-s − 7.48i·23-s + 1.00i·25-s + (−6.03 + 6.03i)29-s − 7.58·31-s + (−1.95 + 1.95i)35-s + (1.08 + 1.08i)37-s − 3.15i·41-s + (3.10 + 3.10i)43-s − 2.76·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.04i·7-s + (−1.05 − 1.05i)11-s + (−1.26 + 1.26i)13-s + 1.21·17-s + (0.186 − 0.186i)19-s − 1.56i·23-s + 0.200i·25-s + (−1.12 + 1.12i)29-s − 1.36·31-s + (−0.330 + 0.330i)35-s + (0.178 + 0.178i)37-s − 0.492i·41-s + (0.474 + 0.474i)43-s − 0.402·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.870 + 0.492i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.870 + 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.009370904304\)
\(L(\frac12)\) \(\approx\) \(0.009370904304\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.76iT - 7T^{2} \)
11 \( 1 + (3.51 + 3.51i)T + 11iT^{2} \)
13 \( 1 + (4.55 - 4.55i)T - 13iT^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
19 \( 1 + (-0.812 + 0.812i)T - 19iT^{2} \)
23 \( 1 + 7.48iT - 23T^{2} \)
29 \( 1 + (6.03 - 6.03i)T - 29iT^{2} \)
31 \( 1 + 7.58T + 31T^{2} \)
37 \( 1 + (-1.08 - 1.08i)T + 37iT^{2} \)
41 \( 1 + 3.15iT - 41T^{2} \)
43 \( 1 + (-3.10 - 3.10i)T + 43iT^{2} \)
47 \( 1 + 2.76T + 47T^{2} \)
53 \( 1 + (6.41 + 6.41i)T + 53iT^{2} \)
59 \( 1 + (5.13 + 5.13i)T + 59iT^{2} \)
61 \( 1 + (2.49 - 2.49i)T - 61iT^{2} \)
67 \( 1 + (-3.14 + 3.14i)T - 67iT^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + 8.95T + 79T^{2} \)
83 \( 1 + (2.86 - 2.86i)T - 83iT^{2} \)
89 \( 1 - 7.23iT - 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.245881144525429735446473454805, −8.524669531164985188072496286042, −7.69664743111446217215803416987, −6.97370700641562796257234784383, −6.04479997186121499288225169524, −5.40471814691128756331617808853, −4.77307056927600548432412151961, −3.41227480338707096565503179564, −2.66388809536771912426209545104, −1.85347616920253123532808655794, 0.00280737492608246471877342335, 1.37865670786154213136285872687, 2.51867161487262286661846655306, 3.50309693072860433287413645101, 4.45480531714813692153045889335, 5.41889862292461642625260569047, 5.65325617732343487814312242978, 7.27213318940041203350560680509, 7.52348517143093564415935976116, 7.961613861229654492420054963126

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