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Properties

Label 1-2001-2001.443-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.137 + 0.990i$
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  + (0.0407 + 0.999i)2-s + (−0.996 + 0.0815i)4-s + (−0.452 − 0.891i)5-s + (0.488 − 0.872i)7-s + (−0.122 − 0.992i)8-s + (0.872 − 0.488i)10-s + (−0.965 − 0.262i)11-s + (−0.742 + 0.670i)13-s + (0.891 + 0.452i)14-s + (0.986 − 0.162i)16-s + (−0.989 + 0.142i)17-s + (0.0815 + 0.996i)19-s + (0.523 + 0.852i)20-s + (0.222 − 0.974i)22-s + (−0.591 + 0.806i)25-s + (−0.699 − 0.714i)26-s + ⋯
L(s)  = 1  + (0.0407 + 0.999i)2-s + (−0.996 + 0.0815i)4-s + (−0.452 − 0.891i)5-s + (0.488 − 0.872i)7-s + (−0.122 − 0.992i)8-s + (0.872 − 0.488i)10-s + (−0.965 − 0.262i)11-s + (−0.742 + 0.670i)13-s + (0.891 + 0.452i)14-s + (0.986 − 0.162i)16-s + (−0.989 + 0.142i)17-s + (0.0815 + 0.996i)19-s + (0.523 + 0.852i)20-s + (0.222 − 0.974i)22-s + (−0.591 + 0.806i)25-s + (−0.699 − 0.714i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.137 + 0.990i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.137 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4932707532 + 0.5666203663i\)
\(L(\frac12)\) \(\approx\) \(0.4932707532 + 0.5666203663i\)
\(L(1)\) \(\approx\) \(0.7248157090 + 0.2405984682i\)
\(L(1)\) \(\approx\) \(0.7248157090 + 0.2405984682i\)

Euler product

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

−19.63281906677631974784395004655, −19.1776826574510252496722842636, −18.25508331453084379760099057382, −17.91484154696305712929292782007, −17.30067898666401065451535279338, −15.72299088344317116330833153490, −15.32399825342979994284616040108, −14.652016297484186986009207985901, −13.760917809611743535021306524742, −13.01638408061155613186722345109, −12.20451142399918007152431569323, −11.600409376200640070038801616339, −10.87094666050613819055956157392, −10.32773351662058413240717834448, −9.449773368598143349433732112644, −8.564715862339254691437215685246, −7.87891003452071900457094006388, −7.02684539279988487857046211383, −5.848734878304337196300048873246, −4.96928950524968739930856643687, −4.41349385825277446390243807527, −3.095775619924613439772567841393, −2.63666869130194777112678173847, −1.97970868872371521125689482219, −0.35249306437775965396558941094, 0.77171590187179029737858870828, 1.95305088393624110244503640687, 3.48661249768696825626979137174, 4.31678417692073627382775794704, 4.847997367279825977967695370675, 5.55848698316888731515493892646, 6.70089029534739575438734705013, 7.3578685921314213529135238522, 8.2120001435924852788435464634, 8.49765959848463803826585339442, 9.63313747024273231195159080792, 10.27725916849668646219181189365, 11.33896001476732524705026317235, 12.21803098085898655279823387186, 13.01550543881600006623817029055, 13.63071706653672959817983266454, 14.297716684009099945950322185699, 15.19107227074123461923258036619, 15.830674775995959264745526041563, 16.596830234839582216982736024091, 16.982323209238968147765601176540, 17.77497071192325297739617539797, 18.52107244968795210689203811523, 19.45067763701718663146801825415, 20.021951673032993205011242622480

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