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Properties

Label 1-2001-2001.632-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.684 + 0.729i$
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  + (0.979 + 0.202i)2-s + (0.917 + 0.396i)4-s + (−0.714 − 0.699i)5-s + (−0.557 + 0.830i)7-s + (0.818 + 0.574i)8-s + (−0.557 − 0.830i)10-s + (0.970 + 0.242i)11-s + (−0.862 − 0.505i)13-s + (−0.714 + 0.699i)14-s + (0.685 + 0.728i)16-s + (−0.654 + 0.755i)17-s + (−0.917 − 0.396i)19-s + (−0.377 − 0.925i)20-s + (0.900 + 0.433i)22-s + (0.0203 + 0.999i)25-s + (−0.742 − 0.670i)26-s + ⋯
L(s)  = 1  + (0.979 + 0.202i)2-s + (0.917 + 0.396i)4-s + (−0.714 − 0.699i)5-s + (−0.557 + 0.830i)7-s + (0.818 + 0.574i)8-s + (−0.557 − 0.830i)10-s + (0.970 + 0.242i)11-s + (−0.862 − 0.505i)13-s + (−0.714 + 0.699i)14-s + (0.685 + 0.728i)16-s + (−0.654 + 0.755i)17-s + (−0.917 − 0.396i)19-s + (−0.377 − 0.925i)20-s + (0.900 + 0.433i)22-s + (0.0203 + 0.999i)25-s + (−0.742 − 0.670i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.684 + 0.729i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.684 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5934111549 + 1.370176261i\)
\(L(\frac12)\) \(\approx\) \(0.5934111549 + 1.370176261i\)
\(L(1)\) \(\approx\) \(1.335968404 + 0.3997759136i\)
\(L(1)\) \(\approx\) \(1.335968404 + 0.3997759136i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.979 + 0.202i)T \)
5 \( 1 + (-0.714 - 0.699i)T \)
7 \( 1 + (-0.557 + 0.830i)T \)
11 \( 1 + (0.970 + 0.242i)T \)
13 \( 1 + (-0.862 - 0.505i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.917 - 0.396i)T \)
31 \( 1 + (0.488 + 0.872i)T \)
37 \( 1 + (-0.101 - 0.994i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.488 + 0.872i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.996 - 0.0815i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (-0.986 - 0.162i)T \)
67 \( 1 + (-0.970 + 0.242i)T \)
71 \( 1 + (0.452 + 0.891i)T \)
73 \( 1 + (0.882 - 0.470i)T \)
79 \( 1 + (-0.685 + 0.728i)T \)
83 \( 1 + (-0.992 - 0.122i)T \)
89 \( 1 + (-0.0611 + 0.998i)T \)
97 \( 1 + (0.523 - 0.852i)T \)
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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

−19.93037583817228597119901321650, −19.10908051377456782376968813143, −18.63960580994447899382174177199, −17.12663133514744959491803887497, −16.75629825588685433360845269413, −15.86469867455820201083619340412, −15.1048467637725426531375187810, −14.54450602045216978599754332946, −13.78392931756683340608520188265, −13.25528842849100100130815242735, −12.02944426570646281142487070503, −11.839480542452420403531686254143, −10.90147181240496873596874444371, −10.2529582804524255092404667610, −9.47069770839213344595486998070, −8.2318759266389998092341545818, −7.166796070381016873636432155654, −6.802289895181959680938330040516, −6.18518808375967308419182030455, −4.845444076211925890358775406284, −4.159075540571975708400256671614, −3.58498194587587540261819552826, −2.73143850858686216247659805635, −1.77640802677123109100773614898, −0.33202070799314535709074825075, 1.45436118276416567273720324686, 2.45786639183523648531483873601, 3.3292614190824869790230948504, 4.229522052548314899841203717569, 4.79077669009104148286953187521, 5.70543908336856859796385034030, 6.51798329030186834828884249949, 7.174859893889536554169901268217, 8.257419970700873234834359069876, 8.77948217192082835920455471845, 9.77448598130506929823835444654, 10.85240191996716994004424901431, 11.67709300943020338061561629892, 12.336066195280899812980643677090, 12.71124599125328254559670289549, 13.43485555280761621966194118225, 14.642392733831840042072191642428, 15.065309967212290003729736941408, 15.64501061178231330271918719879, 16.46104228854570473167664340385, 17.06167207632884651859309492053, 17.81926496299733110403993061108, 19.19967662986585239014313154526, 19.801942885842069950910626542439, 19.91232368081575637052032675433

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