Properties

Label 2-2001-2001.1034-c0-0-0
Degree $2$
Conductor $2001$
Sign $-0.960 - 0.278i$
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  + (−1.69 + 1.06i)2-s + (−0.988 + 0.149i)3-s + (1.29 − 2.69i)4-s + (1.51 − 1.30i)6-s + (0.446 + 3.96i)8-s + (0.955 − 0.294i)9-s + (−0.882 + 2.86i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (−1.30 + 1.51i)18-s + (0.974 − 0.222i)23-s + (−1.03 − 3.85i)24-s + (−0.900 − 0.433i)25-s + (0.967 − 2.76i)26-s + (−0.900 + 0.433i)27-s + ⋯
L(s)  = 1  + (−1.69 + 1.06i)2-s + (−0.988 + 0.149i)3-s + (1.29 − 2.69i)4-s + (1.51 − 1.30i)6-s + (0.446 + 3.96i)8-s + (0.955 − 0.294i)9-s + (−0.882 + 2.86i)12-s + (−1.14 + 0.914i)13-s + (−3.10 − 3.88i)16-s + (−1.30 + 1.51i)18-s + (0.974 − 0.222i)23-s + (−1.03 − 3.85i)24-s + (−0.900 − 0.433i)25-s + (0.967 − 2.76i)26-s + (−0.900 + 0.433i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.960 - 0.278i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.960 - 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2164402762\)
\(L(\frac12)\) \(\approx\) \(0.2164402762\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 + (0.563 + 0.826i)T \)
good2 \( 1 + (1.69 - 1.06i)T + (0.433 - 0.900i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (1.14 - 0.914i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.781 + 0.623i)T^{2} \)
31 \( 1 + (-1.02 - 1.63i)T + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.974 + 0.222i)T^{2} \)
41 \( 1 + (0.565 - 0.565i)T - iT^{2} \)
43 \( 1 + (-0.433 - 0.900i)T^{2} \)
47 \( 1 + (0.369 + 0.0416i)T + (0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - 1.80iT - T^{2} \)
61 \( 1 + (0.781 + 0.623i)T^{2} \)
67 \( 1 + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.806 - 1.28i)T + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (0.974 - 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.781 - 0.623i)T^{2} \)
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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

−9.708504015032349667060713406820, −8.953178828848571389316941626675, −8.133747827860928452696829063839, −7.16199174573195953465760086309, −6.86358623346853727540105508349, −6.03787540786941866114154661344, −5.22738225717061837306654565234, −4.52437411788101704768510644434, −2.36796812779140371690488613519, −1.19991699225866404565140561636, 0.34728250528033734823058044462, 1.61786924811916406230213698516, 2.64695073738592227748773922737, 3.69517669325971802560484180572, 4.85136164817021656938782752080, 6.05976318549451767531801125051, 7.08971638363295089464546542278, 7.55412132327584438993029463980, 8.237305172015228004653160055933, 9.382214201885955562093812823350

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