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Properties

Label 1-2001-2001.20-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.433 - 0.901i$
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  + (0.377 − 0.925i)2-s + (−0.714 − 0.699i)4-s + (−0.862 − 0.505i)5-s + (−0.794 − 0.607i)7-s + (−0.917 + 0.396i)8-s + (−0.794 + 0.607i)10-s + (0.986 − 0.162i)11-s + (−0.768 − 0.639i)13-s + (−0.862 + 0.505i)14-s + (0.0203 + 0.999i)16-s + (0.841 + 0.540i)17-s + (0.714 + 0.699i)19-s + (0.262 + 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (−0.882 + 0.470i)26-s + ⋯
L(s)  = 1  + (0.377 − 0.925i)2-s + (−0.714 − 0.699i)4-s + (−0.862 − 0.505i)5-s + (−0.794 − 0.607i)7-s + (−0.917 + 0.396i)8-s + (−0.794 + 0.607i)10-s + (0.986 − 0.162i)11-s + (−0.768 − 0.639i)13-s + (−0.862 + 0.505i)14-s + (0.0203 + 0.999i)16-s + (0.841 + 0.540i)17-s + (0.714 + 0.699i)19-s + (0.262 + 0.965i)20-s + (0.222 − 0.974i)22-s + (0.488 + 0.872i)25-s + (−0.882 + 0.470i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.433 - 0.901i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.433 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9935342449 - 0.6244454939i\)
\(L(\frac12)\) \(\approx\) \(0.9935342449 - 0.6244454939i\)
\(L(1)\) \(\approx\) \(0.7883875660 - 0.5227721539i\)
\(L(1)\) \(\approx\) \(0.7883875660 - 0.5227721539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.377 - 0.925i)T \)
5 \( 1 + (-0.862 - 0.505i)T \)
7 \( 1 + (-0.794 - 0.607i)T \)
11 \( 1 + (0.986 - 0.162i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (0.841 + 0.540i)T \)
19 \( 1 + (0.714 + 0.699i)T \)
31 \( 1 + (0.182 + 0.983i)T \)
37 \( 1 + (-0.557 + 0.830i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.182 + 0.983i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.452 - 0.891i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.591 + 0.806i)T \)
67 \( 1 + (-0.986 - 0.162i)T \)
71 \( 1 + (-0.742 + 0.670i)T \)
73 \( 1 + (0.947 + 0.320i)T \)
79 \( 1 + (-0.0203 + 0.999i)T \)
83 \( 1 + (0.996 - 0.0815i)T \)
89 \( 1 + (-0.999 + 0.0407i)T \)
97 \( 1 + (0.933 - 0.359i)T \)
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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

−19.927793967223990983723908526803, −19.10345989401700430319781645831, −18.76106069831993576822428295732, −17.801587844412080389582157435, −16.936032495666140897533111125592, −16.34748764299571034986994756452, −15.594509047767532333996951827186, −15.12071649429823105143619909171, −14.271571090676061920225854317801, −13.80835793811612767755433901506, −12.62403220383568470611230785733, −11.99627754519923333504518191701, −11.663473643382515571537830029299, −10.28048097106606153816483084581, −9.17416241925411020446958197616, −9.078850711136358935356505790857, −7.66961439996100727709686567397, −7.25279508422736264494094931695, −6.54608193383456091567483575469, −5.73690793620669978216417208016, −4.79600939758742481252623668503, −3.936133561124274837658608700303, −3.26652951129615133571977603683, −2.39565299307506422005472452319, −0.51764674227066907115555710421, 0.82375899116095449817169540632, 1.48633794798489665828069879009, 3.07813232791599355337417948157, 3.44459274847343554871468641312, 4.25260938638249344446081052146, 5.06606589291561815684929093679, 5.96830622923298055383647909559, 6.944809502632911588877512911484, 7.91469433990817952413050047206, 8.68041327547067223498553574392, 9.71236412415787621690283978483, 10.060460229064168458792222568162, 11.04732632841395551627725787173, 11.88309911334691493235555602734, 12.38305374269600361868886279859, 12.937752046816454296106193403564, 13.8917531334456873740445120820, 14.55505015409219159889885941109, 15.2884649858426056140016860699, 16.26974797078747642038014692102, 16.867050393391987176974896712344, 17.67040263011004194190034477713, 18.760783536477383766977995398024, 19.38689450229395817281361420643, 19.80716687232255742444407368182

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