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Properties

Label 2-2001-2001.827-c0-0-4
Degree $2$
Conductor $2001$
Sign $0.129 - 0.991i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
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  + (−0.222 − 1.97i)2-s + (−0.781 + 0.623i)3-s + (−2.87 + 0.656i)4-s + (1.40 + 1.40i)6-s + (1.27 + 3.65i)8-s + (0.222 − 0.974i)9-s + (1.83 − 2.30i)12-s + (−0.541 − 1.12i)13-s + (4.28 − 2.06i)16-s + (−1.97 − 0.222i)18-s + (−0.781 + 0.623i)23-s + (−3.28 − 2.06i)24-s + (−0.222 − 0.974i)25-s + (−2.09 + 1.31i)26-s + (0.433 + 0.900i)27-s + ⋯
L(s)  = 1  + (−0.222 − 1.97i)2-s + (−0.781 + 0.623i)3-s + (−2.87 + 0.656i)4-s + (1.40 + 1.40i)6-s + (1.27 + 3.65i)8-s + (0.222 − 0.974i)9-s + (1.83 − 2.30i)12-s + (−0.541 − 1.12i)13-s + (4.28 − 2.06i)16-s + (−1.97 − 0.222i)18-s + (−0.781 + 0.623i)23-s + (−3.28 − 2.06i)24-s + (−0.222 − 0.974i)25-s + (−2.09 + 1.31i)26-s + (0.433 + 0.900i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07254216286\)
\(L(\frac12)\) \(\approx\) \(0.07254216286\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.781 - 0.623i)T \)
23 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (0.974 - 0.222i)T \)
good2 \( 1 + (0.222 + 1.97i)T + (-0.974 + 0.222i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.781 + 0.623i)T^{2} \)
13 \( 1 + (0.541 + 1.12i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (1.40 - 0.158i)T + (0.974 - 0.222i)T^{2} \)
37 \( 1 + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (-1.33 - 1.33i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - 0.445iT - T^{2} \)
61 \( 1 + (-0.433 + 0.900i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (1.40 - 0.678i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.05 + 0.119i)T + (0.974 + 0.222i)T^{2} \)
79 \( 1 + (0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (-0.433 - 0.900i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.269931707639151176072981675017, −8.310826685862906139762921665710, −7.51554069170609761282849952588, −5.86603154321734295860768053900, −5.22215500622446814190615074504, −4.37583685934389915180740728600, −3.62569044940580168365064957500, −2.78416199230669708858833641516, −1.52119896990927100255868071971, −0.06571674185132468088912579087, 1.70016150924670841522745612004, 3.93050692808936121945618242043, 4.69488846684200129450786947549, 5.53644793070980103875779341188, 6.08698252631095271571913544276, 6.89148131378077312453956314021, 7.42420164982516809870223952241, 7.980435609454632049874524128282, 9.042769609489460814904233117314, 9.500053068387660889049927531137

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