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Properties

Label 1-2001-2001.608-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.985 + 0.167i$
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  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.415 + 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (0.415 + 0.909i)26-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (0.841 − 0.540i)10-s + (0.142 + 0.989i)11-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.415 + 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (0.415 + 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.985 + 0.167i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.985 + 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8995700020 + 0.07565133385i\)
\(L(\frac12)\) \(\approx\) \(0.8995700020 + 0.07565133385i\)
\(L(1)\) \(\approx\) \(0.7288311481 + 0.1844747270i\)
\(L(1)\) \(\approx\) \(0.7288311481 + 0.1844747270i\)

Euler product

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

−19.667130230498711079751914367099, −19.1032484742280019564903314026, −18.765270940686021996141971238117, −18.22357743041510117713590331837, −16.97533619822422996915096548744, −16.4228423668129998982873541152, −15.550974653170915487034031696991, −14.67302774414243100313811071532, −13.96206820607786393394544094359, −13.200782271431382317798984321784, −12.37289570164874161993285275542, −11.76595946196815411973633231986, −11.02467786834837398876571412031, −10.45653019429927456937738037054, −9.61350462729795458046867317647, −8.69605439053289551746211230081, −8.255652556421700986899557265752, −7.14109108088235308420927903050, −6.15742881156953315415487357977, −5.51857193841201576718995130366, −3.958417995931617113665717536338, −3.72911708399325625351475688427, −2.86568774981104244273286008030, −2.01775273857669154518459547145, −0.71117363258253554467832514444, 0.540564312247732810069511525380, 1.49240324446901994475786561673, 3.254009119230449470723497703466, 3.95309015293718816914569843200, 4.71374215729392109718935802133, 5.520393791635247442059070054263, 6.46132111280959084070940963318, 7.18023058790764310351376879643, 7.84653645116672232461004062731, 8.65819369496469793070169101956, 9.32252134830274219110367868274, 10.139297103265227951456074980109, 10.8485805531898874640370625121, 12.31758553449639779727464608098, 12.577917216685692057267400507380, 13.44955432719994546135822593027, 14.11909147658426822024130238041, 15.30712472731389203092424768962, 15.46545496314764292426416912844, 16.432234727426937027976904811309, 16.84881057429512737620600471132, 17.567298851768714883631193034388, 18.4700182514931182376326986737, 19.19585410898796509316446030511, 19.88035224112057586912646921327

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