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Properties

Label 1-2001-2001.404-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.0673 + 0.997i$
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  + (−0.359 + 0.933i)2-s + (−0.742 − 0.670i)4-s + (0.882 − 0.470i)5-s + (−0.992 − 0.122i)7-s + (0.891 − 0.452i)8-s + (0.122 + 0.992i)10-s + (−0.728 − 0.685i)11-s + (−0.947 − 0.320i)13-s + (0.470 − 0.882i)14-s + (0.101 + 0.994i)16-s + (0.281 + 0.959i)17-s + (−0.670 + 0.742i)19-s + (−0.970 − 0.242i)20-s + (0.900 − 0.433i)22-s + (0.557 − 0.830i)25-s + (0.639 − 0.768i)26-s + ⋯
L(s)  = 1  + (−0.359 + 0.933i)2-s + (−0.742 − 0.670i)4-s + (0.882 − 0.470i)5-s + (−0.992 − 0.122i)7-s + (0.891 − 0.452i)8-s + (0.122 + 0.992i)10-s + (−0.728 − 0.685i)11-s + (−0.947 − 0.320i)13-s + (0.470 − 0.882i)14-s + (0.101 + 0.994i)16-s + (0.281 + 0.959i)17-s + (−0.670 + 0.742i)19-s + (−0.970 − 0.242i)20-s + (0.900 − 0.433i)22-s + (0.557 − 0.830i)25-s + (0.639 − 0.768i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0673 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0673 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.0673 + 0.997i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.0673 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6238739049 + 0.5831515737i\)
\(L(\frac12)\) \(\approx\) \(0.6238739049 + 0.5831515737i\)
\(L(1)\) \(\approx\) \(0.7149624201 + 0.2470942417i\)
\(L(1)\) \(\approx\) \(0.7149624201 + 0.2470942417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.359 + 0.933i)T \)
5 \( 1 + (0.882 - 0.470i)T \)
7 \( 1 + (-0.992 - 0.122i)T \)
11 \( 1 + (-0.728 - 0.685i)T \)
13 \( 1 + (-0.947 - 0.320i)T \)
17 \( 1 + (0.281 + 0.959i)T \)
19 \( 1 + (-0.670 + 0.742i)T \)
31 \( 1 + (-0.607 + 0.794i)T \)
37 \( 1 + (0.983 - 0.182i)T \)
41 \( 1 + (0.755 - 0.654i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (0.714 - 0.699i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (-0.999 + 0.0203i)T \)
67 \( 1 + (-0.685 - 0.728i)T \)
71 \( 1 + (-0.862 + 0.505i)T \)
73 \( 1 + (0.998 + 0.0611i)T \)
79 \( 1 + (0.994 + 0.101i)T \)
83 \( 1 + (-0.917 + 0.396i)T \)
89 \( 1 + (0.202 + 0.979i)T \)
97 \( 1 + (0.965 - 0.262i)T \)
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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

−19.82991549070341273506313766456, −18.97753074943759874427347251885, −18.42392175938600521158353166810, −17.81770725658204578169848665674, −16.99569331811202674363218680504, −16.45031502550254736233941081531, −15.325796163780157363246862475982, −14.53335057571565828027212571767, −13.61104915503489162683100064127, −13.07042946730297248721015545998, −12.46830408610450643944874990206, −11.60548818795556848546166114143, −10.69612919935564249825981942846, −10.05301351214167706212254683766, −9.467775009449568293686077172872, −9.03553399024988927021173765800, −7.624770986783693516771807730224, −7.124388505988106327172999794311, −6.0877957051366035779212379126, −5.0940511258920618880000079385, −4.344218711476165099647069759129, −3.082367106413439392706400499603, −2.56196071125447678955696141040, −1.943849147797473011447649603719, −0.44042263171646686328586987376, 0.78579321781004218011485436462, 1.94985701009380805716065333013, 3.05343326018544466104589373811, 4.18007725454183913232057880715, 5.12200818796120354177483863484, 5.95059708267496627988411448904, 6.22066653172498728451802351489, 7.36090208450895676456241277733, 8.08986808356538538975010633069, 8.90520393792690622694830141731, 9.60195956063674544267787827518, 10.2706273575666711538467671119, 10.77830525588574563662024598339, 12.598419326305527639542193965135, 12.732087895478793780446372375361, 13.60750182841147744641049428403, 14.355642971476160033698733965932, 15.0383714056850233353399696406, 15.991010217907873721839946560404, 16.59098079971708462866201922382, 16.98473428101821420562467275529, 17.86526116222957853737328320029, 18.478558302480345254106956845272, 19.45573032465939555273632059607, 19.70626342277811415296248435015

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