Properties

Label 2-960-3.2-c2-0-6
Degree $2$
Conductor $960$
Sign $-0.995 + 0.0972i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.291 + 2.98i)3-s − 2.23i·5-s − 4.46·7-s + (−8.82 + 1.74i)9-s + 17.8i·11-s + 11.0·13-s + (6.67 − 0.652i)15-s − 0.794i·17-s + 26.5·19-s + (−1.30 − 13.3i)21-s + 14.9i·23-s − 5.00·25-s + (−7.77 − 25.8i)27-s + 5.58i·29-s − 53.1·31-s + ⋯
L(s)  = 1  + (0.0972 + 0.995i)3-s − 0.447i·5-s − 0.637·7-s + (−0.981 + 0.193i)9-s + 1.62i·11-s + 0.846·13-s + (0.445 − 0.0434i)15-s − 0.0467i·17-s + 1.39·19-s + (−0.0619 − 0.634i)21-s + 0.649i·23-s − 0.200·25-s + (−0.287 − 0.957i)27-s + 0.192i·29-s − 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.995 + 0.0972i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.995 + 0.0972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7906956891\)
\(L(\frac12)\) \(\approx\) \(0.7906956891\)
\(L(2)\) 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 \( 1 + (-0.291 - 2.98i)T \)
5 \( 1 + 2.23iT \)
good7 \( 1 + 4.46T + 49T^{2} \)
11 \( 1 - 17.8iT - 121T^{2} \)
13 \( 1 - 11.0T + 169T^{2} \)
17 \( 1 + 0.794iT - 289T^{2} \)
19 \( 1 - 26.5T + 361T^{2} \)
23 \( 1 - 14.9iT - 529T^{2} \)
29 \( 1 - 5.58iT - 841T^{2} \)
31 \( 1 + 53.1T + 961T^{2} \)
37 \( 1 + 51.7T + 1.36e3T^{2} \)
41 \( 1 + 67.8iT - 1.68e3T^{2} \)
43 \( 1 + 40.8T + 1.84e3T^{2} \)
47 \( 1 - 12.3iT - 2.20e3T^{2} \)
53 \( 1 - 37.0iT - 2.80e3T^{2} \)
59 \( 1 - 61.0iT - 3.48e3T^{2} \)
61 \( 1 + 97.8T + 3.72e3T^{2} \)
67 \( 1 - 3.02T + 4.48e3T^{2} \)
71 \( 1 + 57.0iT - 5.04e3T^{2} \)
73 \( 1 + 31.4T + 5.32e3T^{2} \)
79 \( 1 - 2.16T + 6.24e3T^{2} \)
83 \( 1 - 13.0iT - 6.88e3T^{2} \)
89 \( 1 - 173. iT - 7.92e3T^{2} \)
97 \( 1 + 91.6T + 9.40e3T^{2} \)
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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

−10.14724559556455592148873514616, −9.332592065075429465071405839232, −9.020780599094533896287136582223, −7.75879489839944908587736331302, −6.95080447921851817822264625650, −5.67025557639794909464512855775, −5.03940091934524033492258565094, −3.97382354035986665690668136251, −3.22226461469635449414046701055, −1.71008926201489548928966182916, 0.24527777706336231126112865406, 1.53023045162711131337625681143, 3.11034255837504595253132317020, 3.43787398599680204296039846405, 5.35981101349232979328977589551, 6.13734801805896619526393456648, 6.73972947372571278497907676274, 7.71382230711252723627486590627, 8.500994032685757298216708684972, 9.201872454425173998216305548043

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