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Properties

Label 1-2001-2001.1085-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.952 - 0.303i$
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  + (0.755 + 0.654i)2-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.540 + 0.841i)8-s + (−0.281 + 0.959i)10-s + (−0.755 + 0.654i)11-s + (0.959 + 0.281i)13-s + (−0.909 − 0.415i)14-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + (0.989 − 0.142i)19-s + (−0.841 + 0.540i)20-s − 22-s + (−0.654 + 0.755i)25-s + (0.540 + 0.841i)26-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (0.142 + 0.989i)4-s + (0.415 + 0.909i)5-s + (−0.959 + 0.281i)7-s + (−0.540 + 0.841i)8-s + (−0.281 + 0.959i)10-s + (−0.755 + 0.654i)11-s + (0.959 + 0.281i)13-s + (−0.909 − 0.415i)14-s + (−0.959 + 0.281i)16-s + (0.989 + 0.142i)17-s + (0.989 − 0.142i)19-s + (−0.841 + 0.540i)20-s − 22-s + (−0.654 + 0.755i)25-s + (0.540 + 0.841i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.952 - 0.303i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1085, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.952 - 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3096676596 + 1.994950429i\)
\(L(\frac12)\) \(\approx\) \(-0.3096676596 + 1.994950429i\)
\(L(1)\) \(\approx\) \(0.9696523899 + 1.091599672i\)
\(L(1)\) \(\approx\) \(0.9696523899 + 1.091599672i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.755 + 0.654i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (0.989 - 0.142i)T \)
31 \( 1 + (-0.540 + 0.841i)T \)
37 \( 1 + (-0.909 - 0.415i)T \)
41 \( 1 + (-0.909 + 0.415i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.540 + 0.841i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (-0.540 - 0.841i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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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

−19.81717322312811169337818267696, −18.73030137153581168037327045149, −18.564458268044135026990364039833, −17.312269459679639196048046133, −16.30214393494880338535329046330, −16.037470059519724803281620616896, −15.199875102139787721499297056294, −13.96563854512772312045924096734, −13.55454308342484609770641969505, −13.05225567390357296321026397518, −12.27446769143935897878614016988, −11.6179424555108756697628581335, −10.51887539171232231308943977528, −10.079964768057950544845723343361, −9.26687743024647604196760621871, −8.47924141293620009774519219609, −7.38378487754088277462020322924, −6.310218224423803133418115234694, −5.52073568291232185632560240378, −5.24204573101105754878907640832, −3.84860633572576855882501056362, −3.432914415764914176464842780164, −2.448769647596272352665666615439, −1.30300365422527816956595368793, −0.52877545696387857610182710048, 1.68580176419832210745030570111, 2.89905202912324847699845405041, 3.20497487574546691964751654302, 4.17867736103657261491628113739, 5.475176464267415208321025728312, 5.75959823843311187005604376524, 6.80693713871055362567008884660, 7.18952682432323049733443760442, 8.13665238175379415892231847349, 9.1431367667121570764450007266, 9.945052914323571775930405318388, 10.72017692525290390705121377507, 11.67630087136178116702939047480, 12.44064755669088416393247448148, 13.18176151111279703843254687742, 13.77141186910709791339501052753, 14.506340473363059098301007815368, 15.218179729047118437877037217906, 15.985132637480246753765070826546, 16.34284804264448049896208376736, 17.472268930298346325766047309905, 18.13640749692665278419662778389, 18.663529362340686545741604413524, 19.6157061900138179006588313009, 20.65290982065829551524373560374

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