Properties

Label 2-2001-1.1-c1-0-57
Degree $2$
Conductor $2001$
Sign $-1$
Analytic cond. $15.9780$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.31·2-s − 3-s − 0.260·4-s + 2.51·5-s + 1.31·6-s − 2.25·7-s + 2.98·8-s + 9-s − 3.31·10-s − 0.206·11-s + 0.260·12-s − 0.720·13-s + 2.97·14-s − 2.51·15-s − 3.41·16-s + 2.26·17-s − 1.31·18-s − 4.58·19-s − 0.655·20-s + 2.25·21-s + 0.272·22-s + 23-s − 2.98·24-s + 1.31·25-s + 0.950·26-s − 27-s + 0.587·28-s + ⋯
L(s)  = 1  − 0.932·2-s − 0.577·3-s − 0.130·4-s + 1.12·5-s + 0.538·6-s − 0.851·7-s + 1.05·8-s + 0.333·9-s − 1.04·10-s − 0.0623·11-s + 0.0753·12-s − 0.199·13-s + 0.793·14-s − 0.649·15-s − 0.852·16-s + 0.548·17-s − 0.310·18-s − 1.05·19-s − 0.146·20-s + 0.491·21-s + 0.0581·22-s + 0.208·23-s − 0.608·24-s + 0.263·25-s + 0.186·26-s − 0.192·27-s + 0.111·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-1$
Analytic conductor: \(15.9780\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 + 1.31T + 2T^{2} \)
5 \( 1 - 2.51T + 5T^{2} \)
7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 + 0.206T + 11T^{2} \)
13 \( 1 + 0.720T + 13T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + 4.58T + 19T^{2} \)
31 \( 1 + 0.615T + 31T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + 3.39T + 41T^{2} \)
43 \( 1 + 8.50T + 43T^{2} \)
47 \( 1 - 5.89T + 47T^{2} \)
53 \( 1 - 5.61T + 53T^{2} \)
59 \( 1 - 1.23T + 59T^{2} \)
61 \( 1 - 9.37T + 61T^{2} \)
67 \( 1 + 9.43T + 67T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 + 1.27T + 73T^{2} \)
79 \( 1 - 9.36T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 - 1.72T + 97T^{2} \)
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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

−8.947905781639100994904443537068, −8.178327029917462077016428818851, −7.14694364453241280357832487862, −6.47227628374609674729135585437, −5.67937985869290355351448833926, −4.88564914494715297849902505642, −3.80879045957945628773915433500, −2.43042810073001041573317296685, −1.34312554582318347956188047667, 0, 1.34312554582318347956188047667, 2.43042810073001041573317296685, 3.80879045957945628773915433500, 4.88564914494715297849902505642, 5.67937985869290355351448833926, 6.47227628374609674729135585437, 7.14694364453241280357832487862, 8.178327029917462077016428818851, 8.947905781639100994904443537068

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