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Properties

Label 2-2001-2001.137-c0-0-3
Degree $2$
Conductor $2001$
Sign $-0.413 - 0.910i$
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  + (1.85 + 0.649i)2-s + (−0.733 + 0.680i)3-s + (2.23 + 1.78i)4-s + (−1.80 + 0.785i)6-s + (1.94 + 3.09i)8-s + (0.0747 − 0.997i)9-s + (−2.85 + 0.213i)12-s + (−1.61 + 0.367i)13-s + (0.963 + 4.21i)16-s + (0.785 − 1.80i)18-s + (0.433 − 0.900i)23-s + (−3.53 − 0.946i)24-s + (0.623 − 0.781i)25-s + (−3.22 − 0.363i)26-s + (0.623 + 0.781i)27-s + ⋯
L(s)  = 1  + (1.85 + 0.649i)2-s + (−0.733 + 0.680i)3-s + (2.23 + 1.78i)4-s + (−1.80 + 0.785i)6-s + (1.94 + 3.09i)8-s + (0.0747 − 0.997i)9-s + (−2.85 + 0.213i)12-s + (−1.61 + 0.367i)13-s + (0.963 + 4.21i)16-s + (0.785 − 1.80i)18-s + (0.433 − 0.900i)23-s + (−3.53 − 0.946i)24-s + (0.623 − 0.781i)25-s + (−3.22 − 0.363i)26-s + (0.623 + 0.781i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.706222789\)
\(L(\frac12)\) \(\approx\) \(2.706222789\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.733 - 0.680i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (0.149 - 0.988i)T \)
good2 \( 1 + (-1.85 - 0.649i)T + (0.781 + 0.623i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.433 + 0.900i)T^{2} \)
13 \( 1 + (1.61 - 0.367i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.974 - 0.222i)T^{2} \)
31 \( 1 + (-0.170 + 0.488i)T + (-0.781 - 0.623i)T^{2} \)
37 \( 1 + (-0.433 + 0.900i)T^{2} \)
41 \( 1 + (-1.25 + 1.25i)T - iT^{2} \)
43 \( 1 + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (1.36 + 0.856i)T + (0.433 + 0.900i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + 1.24iT - T^{2} \)
61 \( 1 + (-0.974 - 0.222i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.443 + 1.94i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.605 - 1.73i)T + (-0.781 + 0.623i)T^{2} \)
79 \( 1 + (0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.781 - 0.623i)T^{2} \)
97 \( 1 + (-0.974 + 0.222i)T^{2} \)
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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

−9.741820947198884286257891471530, −8.648786309742392337272325238676, −7.56723481446320720979301539908, −6.85032456987443902032215049467, −6.30758670904083205838499131766, −5.33663710238938423551268426102, −4.81594484239358513901587873991, −4.24366159485796438312396845416, −3.22916949280416365402717261232, −2.30350169317568086001848993554, 1.29090988496769093430522470024, 2.38547438805976720059076577390, 3.13798172799044205942396945916, 4.42838375757615264039420597281, 5.03988864872704445320427016157, 5.64505380794757261239671976788, 6.46592440561171525423863064340, 7.20782901414428749860414955921, 7.77206271694374519043856726507, 9.563037527937794182149873175403

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