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Properties

Label 2-2001-2001.896-c0-0-3
Degree $2$
Conductor $2001$
Sign $0.238 - 0.971i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.791 + 0.497i)2-s + (0.365 + 0.930i)3-s + (−0.0549 − 0.114i)4-s + (−0.173 + 0.918i)6-s + (0.117 − 1.04i)8-s + (−0.733 + 0.680i)9-s + (0.0861 − 0.0928i)12-s + (1.49 + 1.19i)13-s + (0.534 − 0.670i)16-s + (−0.918 + 0.173i)18-s + (0.974 + 0.222i)23-s + (1.01 − 0.272i)24-s + (−0.900 + 0.433i)25-s + (0.589 + 1.68i)26-s + (−0.900 − 0.433i)27-s + ⋯
L(s)  = 1  + (0.791 + 0.497i)2-s + (0.365 + 0.930i)3-s + (−0.0549 − 0.114i)4-s + (−0.173 + 0.918i)6-s + (0.117 − 1.04i)8-s + (−0.733 + 0.680i)9-s + (0.0861 − 0.0928i)12-s + (1.49 + 1.19i)13-s + (0.534 − 0.670i)16-s + (−0.918 + 0.173i)18-s + (0.974 + 0.222i)23-s + (1.01 − 0.272i)24-s + (−0.900 + 0.433i)25-s + (0.589 + 1.68i)26-s + (−0.900 − 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (896, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923692505\)
\(L(\frac12)\) \(\approx\) \(1.923692505\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.365 - 0.930i)T \)
23 \( 1 + (-0.974 - 0.222i)T \)
29 \( 1 + (-0.997 - 0.0747i)T \)
good2 \( 1 + (-0.791 - 0.497i)T + (0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.974 - 0.222i)T^{2} \)
13 \( 1 + (-1.49 - 1.19i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.781 - 0.623i)T^{2} \)
31 \( 1 + (0.275 - 0.438i)T + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (-0.974 - 0.222i)T^{2} \)
41 \( 1 + (0.839 + 0.839i)T + iT^{2} \)
43 \( 1 + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (1.50 - 0.169i)T + (0.974 - 0.222i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + 1.80iT - T^{2} \)
61 \( 1 + (0.781 - 0.623i)T^{2} \)
67 \( 1 + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (-0.848 + 1.06i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.197 + 0.314i)T + (-0.433 + 0.900i)T^{2} \)
79 \( 1 + (0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.433 - 0.900i)T^{2} \)
97 \( 1 + (0.781 + 0.623i)T^{2} \)
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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

−9.381890021501511579361940647775, −8.899532518566600353445846355797, −8.022700914601131312913598940598, −6.87423214057207944592621816506, −6.24070498254304326273146431567, −5.35715657712781248340577153596, −4.67218080684964490290722804189, −3.86180572741484170946016351045, −3.26171499667276643962191552165, −1.63664517257999090804666446787, 1.24253000687596509081482811039, 2.50666355198014357605595258002, 3.24607763075991372930367846297, 3.97703321149938455316914157493, 5.17876601990079263269488525696, 5.93203328759352493371688061281, 6.72905590791436486609770086105, 7.81975378339190298728951440070, 8.321675970660408747651684572838, 8.859144946730870379005898111556

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