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Properties

Label 1-2001-2001.1031-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.149 + 0.988i$
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  + (0.999 + 0.0407i)2-s + (0.996 + 0.0815i)4-s + (−0.452 + 0.891i)5-s + (−0.488 − 0.872i)7-s + (0.992 + 0.122i)8-s + (−0.488 + 0.872i)10-s + (0.262 + 0.965i)11-s + (0.742 + 0.670i)13-s + (−0.452 − 0.891i)14-s + (0.986 + 0.162i)16-s + (−0.142 + 0.989i)17-s + (−0.996 − 0.0815i)19-s + (−0.523 + 0.852i)20-s + (0.222 + 0.974i)22-s + (−0.591 − 0.806i)25-s + (0.714 + 0.699i)26-s + ⋯
L(s)  = 1  + (0.999 + 0.0407i)2-s + (0.996 + 0.0815i)4-s + (−0.452 + 0.891i)5-s + (−0.488 − 0.872i)7-s + (0.992 + 0.122i)8-s + (−0.488 + 0.872i)10-s + (0.262 + 0.965i)11-s + (0.742 + 0.670i)13-s + (−0.452 − 0.891i)14-s + (0.986 + 0.162i)16-s + (−0.142 + 0.989i)17-s + (−0.996 − 0.0815i)19-s + (−0.523 + 0.852i)20-s + (0.222 + 0.974i)22-s + (−0.591 − 0.806i)25-s + (0.714 + 0.699i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.149 + 0.988i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.149 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.652431126 + 1.920216060i\)
\(L(\frac12)\) \(\approx\) \(1.652431126 + 1.920216060i\)
\(L(1)\) \(\approx\) \(1.642213211 + 0.5389240752i\)
\(L(1)\) \(\approx\) \(1.642213211 + 0.5389240752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.999 + 0.0407i)T \)
5 \( 1 + (-0.452 + 0.891i)T \)
7 \( 1 + (-0.488 - 0.872i)T \)
11 \( 1 + (0.262 + 0.965i)T \)
13 \( 1 + (0.742 + 0.670i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.996 - 0.0815i)T \)
31 \( 1 + (0.101 + 0.994i)T \)
37 \( 1 + (-0.0203 - 0.999i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.101 + 0.994i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.818 - 0.574i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (-0.339 + 0.940i)T \)
67 \( 1 + (-0.262 + 0.965i)T \)
71 \( 1 + (-0.917 + 0.396i)T \)
73 \( 1 + (-0.862 - 0.505i)T \)
79 \( 1 + (-0.986 + 0.162i)T \)
83 \( 1 + (0.794 + 0.607i)T \)
89 \( 1 + (0.947 + 0.320i)T \)
97 \( 1 + (0.979 - 0.202i)T \)
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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

−20.0761041911142599589186757339, −18.97077672059524189240623481152, −18.72924006458824157607555504268, −17.2874871915920769280935999624, −16.56989546113839397226637313011, −15.91976929095784256382434192319, −15.45338203739398159232146920239, −14.72425848897255051831470194129, −13.53744673497320094899651042694, −13.263062857910547564675540723775, −12.42393551600570861480207651658, −11.758213592349446551126595658934, −11.22359093504985424881491313273, −10.228192987482663516077331636619, −9.1414337305094072144215404159, −8.47179630496895162118099088077, −7.73345613056376589094485026580, −6.53378761125397564653533190399, −5.93453949522764905044945052440, −5.24183570642295497858893579959, −4.40466211223182314788325872306, −3.50665956134576910232399246190, −2.86332493190895490650955342897, −1.7652272884757503905099925502, −0.58311935047075178987587736983, 1.44107319897239750576174548084, 2.32830629567901968075377587667, 3.363529488656058004914412897064, 4.07246237787241526458395073886, 4.43210283643346541969904556184, 5.86098577518830340203765681567, 6.6067280323279211327379547265, 7.00628798791061914028101908723, 7.791058359375786898653514083716, 8.8646882744376411507591174076, 10.1901816092503918837747756636, 10.58348591601962544405563677172, 11.29152820081362419704694441318, 12.16033853724927676476893837352, 12.86668672508735921959385427886, 13.55730760517234732589094868745, 14.46642527751486686238653472467, 14.76052170001907138122790840031, 15.72645397305511119197950736620, 16.2214189452145571150687140004, 17.18738519162382184363771294559, 17.80573033087150391526929350442, 19.19435991085100093660128248945, 19.3365478410271113658531291121, 20.21100179735653665769853801505

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