Properties

Label 2-960-5.4-c3-0-69
Degree $2$
Conductor $960$
Sign $-0.688 - 0.724i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (8.10 − 7.70i)5-s − 22.2i·7-s − 9·9-s − 1.79·11-s + 58.2i·13-s + (−23.1 − 24.3i)15-s + 18.9i·17-s − 104.·19-s − 66.6·21-s − 49.6i·23-s + (6.37 − 124. i)25-s + 27i·27-s − 293.·29-s − 64.4·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.724 − 0.688i)5-s − 1.19i·7-s − 0.333·9-s − 0.0490·11-s + 1.24i·13-s + (−0.397 − 0.418i)15-s + 0.270i·17-s − 1.26·19-s − 0.692·21-s − 0.449i·23-s + (0.0509 − 0.998i)25-s + 0.192i·27-s − 1.87·29-s − 0.373·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.688 - 0.724i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.688 - 0.724i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4531987337\)
\(L(\frac12)\) \(\approx\) \(0.4531987337\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 + (-8.10 + 7.70i)T \)
good7 \( 1 + 22.2iT - 343T^{2} \)
11 \( 1 + 1.79T + 1.33e3T^{2} \)
13 \( 1 - 58.2iT - 2.19e3T^{2} \)
17 \( 1 - 18.9iT - 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + 49.6iT - 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 + 64.4T + 2.97e4T^{2} \)
37 \( 1 - 19.8iT - 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 247. iT - 7.95e4T^{2} \)
47 \( 1 + 384. iT - 1.03e5T^{2} \)
53 \( 1 - 463. iT - 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 - 137.T + 2.26e5T^{2} \)
67 \( 1 - 173. iT - 3.00e5T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 + 320. iT - 3.89e5T^{2} \)
79 \( 1 + 770.T + 4.93e5T^{2} \)
83 \( 1 + 173. iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 384. iT - 9.12e5T^{2} \)
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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

−9.061399834267445620381828865369, −8.349699690920884724558978473285, −7.30067402666960912461670581748, −6.62612940830560114252244471088, −5.77135411216349948249839241662, −4.59883035664238473308175217030, −3.85553622034528379319277810257, −2.17651159235255094976936830490, −1.39653221423124725637504785611, −0.10466918239197473057441049062, 1.95616523775568600498664745670, 2.80073767316836507443678076121, 3.77553096644257448858535320743, 5.30778269588247831751305868736, 5.63703641658160895088353320896, 6.60169732000237952391599215220, 7.73023831391309734830218666382, 8.696048629718463407870908186203, 9.389641359705812230403928354379, 10.13163772418681391627659010430

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