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Properties

Label 2-1584-11.10-c2-0-49
Degree $2$
Conductor $1584$
Sign $-0.636 + 0.771i$
Analytic cond. $43.1608$
Root an. cond. $6.56969$
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  + 5-s − 8.48i·7-s + (7 − 8.48i)11-s − 8.48i·13-s + 25.4i·17-s − 25.4i·19-s + 17·23-s − 24·25-s − 33.9i·29-s − 17·31-s − 8.48i·35-s + 47·37-s + 8.48i·41-s + 16.9i·43-s − 58·47-s + ⋯
L(s)  = 1  + 0.200·5-s − 1.21i·7-s + (0.636 − 0.771i)11-s − 0.652i·13-s + 1.49i·17-s − 1.33i·19-s + 0.739·23-s − 0.959·25-s − 1.17i·29-s − 0.548·31-s − 0.242i·35-s + 1.27·37-s + 0.206i·41-s + 0.394i·43-s − 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(43.1608\)
Root analytic conductor: \(6.56969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.553983853\)
\(L(\frac12)\) \(\approx\) \(1.553983853\)
\(L(2)\) 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 \( 1 \)
11 \( 1 + (-7 + 8.48i)T \)
good5 \( 1 - T + 25T^{2} \)
7 \( 1 + 8.48iT - 49T^{2} \)
13 \( 1 + 8.48iT - 169T^{2} \)
17 \( 1 - 25.4iT - 289T^{2} \)
19 \( 1 + 25.4iT - 361T^{2} \)
23 \( 1 - 17T + 529T^{2} \)
29 \( 1 + 33.9iT - 841T^{2} \)
31 \( 1 + 17T + 961T^{2} \)
37 \( 1 - 47T + 1.36e3T^{2} \)
41 \( 1 - 8.48iT - 1.68e3T^{2} \)
43 \( 1 - 16.9iT - 1.84e3T^{2} \)
47 \( 1 + 58T + 2.20e3T^{2} \)
53 \( 1 + 2T + 2.80e3T^{2} \)
59 \( 1 + 55T + 3.48e3T^{2} \)
61 \( 1 - 84.8iT - 3.72e3T^{2} \)
67 \( 1 + 89T + 4.48e3T^{2} \)
71 \( 1 + 7T + 5.04e3T^{2} \)
73 \( 1 + 127. iT - 5.32e3T^{2} \)
79 \( 1 + 33.9iT - 6.24e3T^{2} \)
83 \( 1 - 33.9iT - 6.88e3T^{2} \)
89 \( 1 - 97T + 7.92e3T^{2} \)
97 \( 1 + 121T + 9.40e3T^{2} \)
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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

−8.978027349168034711418324654403, −8.065477311587006708875125347062, −7.43311318541093660427539067961, −6.41782059763667302291466270197, −5.87173948811580016281734015905, −4.60515471660113915267289532421, −3.88241226712961191506486061725, −2.94088610624630429210073661786, −1.47826905041380451011269717049, −0.42436822419199694024768643839, 1.50309730887763496279945380039, 2.39674050681395693351786583564, 3.50088627889398548661735381241, 4.64642489588882339956048831671, 5.41766456323386761354853572508, 6.26249986534499155951676437060, 7.08468498246878391366359560734, 7.916733558048458798108569768720, 9.012176318511809295921605741880, 9.348354552170993143748238506561

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