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Properties

Label 2-960-15.14-c2-0-21
Degree $2$
Conductor $960$
Sign $-0.290 - 0.956i$
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  + (2.72 − 1.26i)3-s + (−0.689 + 4.95i)5-s + 0.735i·7-s + (5.82 − 6.86i)9-s + 10.9i·11-s + 21.1i·13-s + (4.36 + 14.3i)15-s + 7.03·17-s − 23.1·19-s + (0.927 + 2.00i)21-s − 24.7·23-s + (−24.0 − 6.82i)25-s + (7.21 − 26.0i)27-s − 32.3i·29-s − 34.9·31-s + ⋯
L(s)  = 1  + (0.907 − 0.420i)3-s + (−0.137 + 0.990i)5-s + 0.105i·7-s + (0.647 − 0.762i)9-s + 0.995i·11-s + 1.63i·13-s + (0.290 + 0.956i)15-s + 0.413·17-s − 1.21·19-s + (0.0441 + 0.0953i)21-s − 1.07·23-s + (−0.962 − 0.272i)25-s + (0.267 − 0.963i)27-s − 1.11i·29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.290 - 0.956i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.290 - 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.880629619\)
\(L(\frac12)\) \(\approx\) \(1.880629619\)
\(L(2)\) 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 \( 1 + (-2.72 + 1.26i)T \)
5 \( 1 + (0.689 - 4.95i)T \)
good7 \( 1 - 0.735iT - 49T^{2} \)
11 \( 1 - 10.9iT - 121T^{2} \)
13 \( 1 - 21.1iT - 169T^{2} \)
17 \( 1 - 7.03T + 289T^{2} \)
19 \( 1 + 23.1T + 361T^{2} \)
23 \( 1 + 24.7T + 529T^{2} \)
29 \( 1 + 32.3iT - 841T^{2} \)
31 \( 1 + 34.9T + 961T^{2} \)
37 \( 1 - 37.7iT - 1.36e3T^{2} \)
41 \( 1 - 39.0iT - 1.68e3T^{2} \)
43 \( 1 - 22.6iT - 1.84e3T^{2} \)
47 \( 1 - 39.1T + 2.20e3T^{2} \)
53 \( 1 + 60.9T + 2.80e3T^{2} \)
59 \( 1 - 7.79iT - 3.48e3T^{2} \)
61 \( 1 - 11.1T + 3.72e3T^{2} \)
67 \( 1 - 33.3iT - 4.48e3T^{2} \)
71 \( 1 + 96.9iT - 5.04e3T^{2} \)
73 \( 1 - 134. iT - 5.32e3T^{2} \)
79 \( 1 - 121.T + 6.24e3T^{2} \)
83 \( 1 - 90.2T + 6.88e3T^{2} \)
89 \( 1 - 53.1iT - 7.92e3T^{2} \)
97 \( 1 - 115. iT - 9.40e3T^{2} \)
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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

−9.838330820332665683338159375850, −9.381600416485342962705289086546, −8.276557385979093014752587065792, −7.55834717732232830972866821386, −6.75247618821425211695200579404, −6.19879275314546592763168294963, −4.42365531752858865747630880562, −3.79191244247804123893609064300, −2.45475921884043351937876890429, −1.82858198534042788965404352427, 0.49801111762467393571392888499, 1.99679206519668328220624525649, 3.33614570913892093726926243837, 4.01769598235165789944338529454, 5.20280963570190498232544472041, 5.86058328658843058657046453599, 7.42083680293073685215854707477, 8.119676741650616502324899901611, 8.700732701125921501185411351965, 9.370119738659477109333456939158

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