Properties

Label 1-2001-2001.311-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.981 + 0.191i$
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  + (0.891 − 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (0.162 − 0.986i)8-s + (−0.359 − 0.933i)10-s + (0.639 − 0.768i)11-s + (−0.557 − 0.830i)13-s + (−0.994 + 0.101i)14-s + (−0.301 − 0.953i)16-s + (−0.755 − 0.654i)17-s + (0.806 + 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.872 − 0.488i)26-s + ⋯
L(s)  = 1  + (0.891 − 0.452i)2-s + (0.591 − 0.806i)4-s + (0.101 − 0.994i)5-s + (−0.933 − 0.359i)7-s + (0.162 − 0.986i)8-s + (−0.359 − 0.933i)10-s + (0.639 − 0.768i)11-s + (−0.557 − 0.830i)13-s + (−0.994 + 0.101i)14-s + (−0.301 − 0.953i)16-s + (−0.755 − 0.654i)17-s + (0.806 + 0.591i)19-s + (−0.742 − 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.872 − 0.488i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.981 + 0.191i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.981 + 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2054510022 - 2.127502162i\)
\(L(\frac12)\) \(\approx\) \(-0.2054510022 - 2.127502162i\)
\(L(1)\) \(\approx\) \(1.106080132 - 1.087287621i\)
\(L(1)\) \(\approx\) \(1.106080132 - 1.087287621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.891 - 0.452i)T \)
5 \( 1 + (0.101 - 0.994i)T \)
7 \( 1 + (-0.933 - 0.359i)T \)
11 \( 1 + (0.639 - 0.768i)T \)
13 \( 1 + (-0.557 - 0.830i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.806 + 0.591i)T \)
31 \( 1 + (0.925 + 0.377i)T \)
37 \( 1 + (0.852 - 0.523i)T \)
41 \( 1 + (-0.540 - 0.841i)T \)
43 \( 1 + (-0.925 + 0.377i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (-0.685 - 0.728i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (-0.998 + 0.0611i)T \)
67 \( 1 + (0.768 - 0.639i)T \)
71 \( 1 + (0.0203 + 0.999i)T \)
73 \( 1 + (0.983 + 0.182i)T \)
79 \( 1 + (0.953 + 0.301i)T \)
83 \( 1 + (-0.339 + 0.940i)T \)
89 \( 1 + (-0.574 - 0.818i)T \)
97 \( 1 + (0.699 - 0.714i)T \)
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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

−20.227604971557640190308236918246, −19.76171345816840561903719509849, −18.931420315280728213354922082963, −18.080645863695168505357849160093, −17.23665509281043631112639220039, −16.67947899615332145013438954018, −15.63295086300259832349157173700, −15.20362450611242892690624013220, −14.593003180893413003961977320664, −13.70437670930319846797080012427, −13.23307806544357535812713477507, −12.157272280772794948115013457542, −11.76575774419342091423770170050, −10.88678588266666022931201117800, −9.83922490399827893902412504287, −9.267194751029734089183389340247, −8.10186604898958069302734281713, −7.125576145686682611845416369582, −6.60520946780906470408013344373, −6.20478099202816724810718581671, −5.02126821064102496357977214210, −4.20874080682940489428578418543, −3.37939774681421059354302660702, −2.597278668755954029485251416655, −1.87137921274429328444951993710, 0.51762547912801076762310296745, 1.301177498514494822969263686049, 2.56632721797449202138005794303, 3.33462388114367086070726294266, 4.11927470218361037947580467592, 4.96837998992276903823273106237, 5.71124647033209507263839069427, 6.4233926195501488480471433334, 7.307286834413205445177121310131, 8.349163978815704472809073385208, 9.43835279821770123712240533644, 9.79529076530907729981371505748, 10.774291303889940019737651461827, 11.64609262725206151498401166059, 12.33917915282571435657928170098, 12.89075212207956524893092890031, 13.70620361649006553538074399509, 14.03895238522623164474944039878, 15.23554036479046046121821105977, 15.911449554920915603983805296416, 16.457909507599746966677334155259, 17.18358441362788877654211748959, 18.23062322302855510898230605257, 19.216181867088840340217077331444, 19.84498879653018338662398212643

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