Properties

Label 2-2001-2001.827-c0-0-0
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  + (0.132 + 1.18i)2-s + (−0.149 − 0.988i)3-s + (−0.400 + 0.0913i)4-s + (1.14 − 0.307i)6-s + (0.231 + 0.660i)8-s + (−0.955 + 0.294i)9-s + (0.149 + 0.382i)12-s + (0.858 + 1.78i)13-s + (−1.11 + 0.538i)16-s + (−0.474 − 1.08i)18-s + (−0.781 + 0.623i)23-s + (0.618 − 0.327i)24-s + (−0.222 − 0.974i)25-s + (−1.98 + 1.24i)26-s + (0.433 + 0.900i)27-s + ⋯
L(s)  = 1  + (0.132 + 1.18i)2-s + (−0.149 − 0.988i)3-s + (−0.400 + 0.0913i)4-s + (1.14 − 0.307i)6-s + (0.231 + 0.660i)8-s + (−0.955 + 0.294i)9-s + (0.149 + 0.382i)12-s + (0.858 + 1.78i)13-s + (−1.11 + 0.538i)16-s + (−0.474 − 1.08i)18-s + (−0.781 + 0.623i)23-s + (0.618 − 0.327i)24-s + (−0.222 − 0.974i)25-s + (−1.98 + 1.24i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.246588598\)
\(L(\frac12)\) \(\approx\) \(1.246588598\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.149 + 0.988i)T \)
23 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-0.680 - 0.733i)T \)
good2 \( 1 + (-0.132 - 1.18i)T + (-0.974 + 0.222i)T^{2} \)
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 \( 1 + (-0.858 - 1.78i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
31 \( 1 + (-1.91 + 0.216i)T + (0.974 - 0.222i)T^{2} \)
37 \( 1 + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (1.07 + 1.07i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (-1.88 - 0.660i)T + (0.781 + 0.623i)T^{2} \)
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 \( 1 + (-1.67 + 0.807i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.928 + 0.104i)T + (0.974 + 0.222i)T^{2} \)
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

−9.060236696936489608060325569294, −8.468925337189437158229690800242, −7.84181981345360428785839166903, −6.95813472160342508874226113837, −6.46979552607933794847286232718, −5.96467042253913316480720601904, −4.95362081931146763461303166014, −4.02024084187807419616445222368, −2.49563576963726585309394126603, −1.54216384815411438833400623084, 0.952225748818871946914551364555, 2.56790671082701799040698567479, 3.23998683811978415896402883500, 4.00229158220619013283047043888, 4.86869146413211221546671715271, 5.81160975641838210171951444572, 6.58820291494089822093419631675, 7.968862616422662028834314829484, 8.493578593163773402708034466194, 9.587841765892769262813717392521

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