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Properties

Label 1-2001-2001.1112-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.698 + 0.715i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.670 − 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (−0.806 − 0.591i)8-s + (−0.242 − 0.970i)10-s + (−0.998 + 0.0611i)11-s + (−0.794 − 0.607i)13-s + (0.830 − 0.557i)14-s + (−0.979 + 0.202i)16-s + (−0.540 − 0.841i)17-s + (−0.994 + 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (−0.983 + 0.182i)26-s + ⋯
L(s)  = 1  + (0.670 − 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (−0.806 − 0.591i)8-s + (−0.242 − 0.970i)10-s + (−0.998 + 0.0611i)11-s + (−0.794 − 0.607i)13-s + (0.830 − 0.557i)14-s + (−0.979 + 0.202i)16-s + (−0.540 − 0.841i)17-s + (−0.994 + 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (−0.983 + 0.182i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4764726009 - 1.130893193i\)
\(L(\frac12)\) \(\approx\) \(-0.4764726009 - 1.130893193i\)
\(L(1)\) \(\approx\) \(0.9183579924 - 0.8894646177i\)
\(L(1)\) \(\approx\) \(0.9183579924 - 0.8894646177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.670 - 0.742i)T \)
5 \( 1 + (0.557 - 0.830i)T \)
7 \( 1 + (0.970 + 0.242i)T \)
11 \( 1 + (-0.998 + 0.0611i)T \)
13 \( 1 + (-0.794 - 0.607i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (-0.994 + 0.101i)T \)
31 \( 1 + (-0.965 + 0.262i)T \)
37 \( 1 + (0.359 + 0.933i)T \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (0.965 + 0.262i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (-0.0203 + 0.999i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.0407 - 0.999i)T \)
67 \( 1 + (0.0611 - 0.998i)T \)
71 \( 1 + (0.488 - 0.872i)T \)
73 \( 1 + (-0.122 + 0.992i)T \)
79 \( 1 + (0.202 - 0.979i)T \)
83 \( 1 + (-0.685 + 0.728i)T \)
89 \( 1 + (-0.396 - 0.917i)T \)
97 \( 1 + (0.505 + 0.862i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.79367009529621002847433093100, −19.698967878334673735631523777, −18.72420089807569068408607133630, −18.00393672644919116646516450037, −17.414647917894362391727344023744, −16.87608226344709172778828458158, −15.90446890363425399870045711843, −14.95096153860599435882020051394, −14.70702501531858724002576689991, −13.990198150843289262928964257, −13.172521535394306672275210092348, −12.62291983683800481477637436631, −11.45250287290442161721298041930, −10.934580871653253384166177964928, −10.10547029806877867334457359811, −9.005051649441568842610640445422, −8.18775846789603985562421701028, −7.44396018191305203318759635574, −6.81405741785205594309361959504, −5.96066460580825524387497206927, −5.20388653745338838136282742575, −4.44053985108894593709983980597, −3.60598956603306173000392805650, −2.391731619517312321369490770334, −1.99895819201430189271564536445, 0.28414275677895663658929897016, 1.545626629334574893472602251058, 2.23790624601950940578391498534, 2.95845097952419722084934175834, 4.34975803579166058703736705783, 4.93006827442657018446090786447, 5.36483115889768638070212009764, 6.26670130382002337821176059781, 7.473954265497940527784603140083, 8.38169896647877123130744336389, 9.13908561360101640331129746524, 9.97215113903015800554228528844, 10.663723660400595183768097939787, 11.396165561463540007783331659873, 12.31158263422723534109154066797, 12.77334894202422691744378951995, 13.518687414239394860320307477176, 14.18266133555041278230342490117, 15.058791408698715080433468030479, 15.55463487145962910148663952255, 16.61867988283049976979291436470, 17.489555238660483724414305643274, 18.141579347146126392820511375802, 18.72241429645476442140480256611, 19.924349095091315692757521394823

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