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Properties

Label 2-1584-33.32-c3-0-45
Degree $2$
Conductor $1584$
Sign $0.605 + 0.795i$
Analytic cond. $93.4590$
Root an. cond. $9.66742$
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  + 7.07i·5-s + 14.2i·7-s + (−36.4 + 1.29i)11-s − 78.5i·13-s + 15.8·17-s + 32.0i·19-s + 114. i·23-s + 75·25-s + 114.·29-s − 129.·31-s − 100.·35-s − 301.·37-s − 287.·41-s − 105. i·43-s − 240. i·47-s + ⋯
L(s)  = 1  + 0.632i·5-s + 0.768i·7-s + (−0.999 + 0.0354i)11-s − 1.67i·13-s + 0.226·17-s + 0.386i·19-s + 1.03i·23-s + 0.599·25-s + 0.732·29-s − 0.747·31-s − 0.485·35-s − 1.34·37-s − 1.09·41-s − 0.373i·43-s − 0.746i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(93.4590\)
Root analytic conductor: \(9.66742\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.281419452\)
\(L(\frac12)\) \(\approx\) \(1.281419452\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 + (36.4 - 1.29i)T \)
good5 \( 1 - 7.07iT - 125T^{2} \)
7 \( 1 - 14.2iT - 343T^{2} \)
13 \( 1 + 78.5iT - 2.19e3T^{2} \)
17 \( 1 - 15.8T + 4.91e3T^{2} \)
19 \( 1 - 32.0iT - 6.85e3T^{2} \)
23 \( 1 - 114. iT - 1.21e4T^{2} \)
29 \( 1 - 114.T + 2.43e4T^{2} \)
31 \( 1 + 129.T + 2.97e4T^{2} \)
37 \( 1 + 301.T + 5.06e4T^{2} \)
41 \( 1 + 287.T + 6.89e4T^{2} \)
43 \( 1 + 105. iT - 7.95e4T^{2} \)
47 \( 1 + 240. iT - 1.03e5T^{2} \)
53 \( 1 + 320. iT - 1.48e5T^{2} \)
59 \( 1 + 600. iT - 2.05e5T^{2} \)
61 \( 1 + 398. iT - 2.26e5T^{2} \)
67 \( 1 + 260T + 3.00e5T^{2} \)
71 \( 1 - 672. iT - 3.57e5T^{2} \)
73 \( 1 + 1.14e3iT - 3.89e5T^{2} \)
79 \( 1 - 899. iT - 4.93e5T^{2} \)
83 \( 1 + 531.T + 5.71e5T^{2} \)
89 \( 1 - 1.19e3iT - 7.04e5T^{2} \)
97 \( 1 - 404.T + 9.12e5T^{2} \)
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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

−8.803498467643617992951103141661, −8.123257375875529237973861839852, −7.43218114065963936438068526138, −6.51572407204397282118293538978, −5.34722126345025048119155340662, −5.27995730102269773515419821812, −3.49848608694700097849450065210, −2.96643102865031711709677636882, −1.90746687134987415547704344010, −0.33821957057162126981939201281, 0.868205271215803731438003474297, 1.98285334386829952390424308470, 3.16614058657274622918266973248, 4.41441298266278010317996526979, 4.76469844697835542992600190685, 5.91553393228579587117424130698, 6.93051739152085611459304140586, 7.43397404650735651616603502216, 8.637804850431080407123911048962, 8.909606149783335439463782765234

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