Properties

Label 2-2001-1.1-c1-0-90
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  + 0.586·2-s + 3-s − 1.65·4-s − 0.842·5-s + 0.586·6-s + 1.23·7-s − 2.14·8-s + 9-s − 0.494·10-s + 1.47·11-s − 1.65·12-s − 3.26·13-s + 0.725·14-s − 0.842·15-s + 2.05·16-s − 4.93·17-s + 0.586·18-s + 0.260·19-s + 1.39·20-s + 1.23·21-s + 0.861·22-s + 23-s − 2.14·24-s − 4.28·25-s − 1.91·26-s + 27-s − 2.05·28-s + ⋯
L(s)  = 1  + 0.414·2-s + 0.577·3-s − 0.828·4-s − 0.376·5-s + 0.239·6-s + 0.468·7-s − 0.757·8-s + 0.333·9-s − 0.156·10-s + 0.443·11-s − 0.478·12-s − 0.904·13-s + 0.194·14-s − 0.217·15-s + 0.514·16-s − 1.19·17-s + 0.138·18-s + 0.0598·19-s + 0.312·20-s + 0.270·21-s + 0.183·22-s + 0.208·23-s − 0.437·24-s − 0.857·25-s − 0.374·26-s + 0.192·27-s − 0.387·28-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: $\chi_{2001} (1, \cdot )$
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 \( 1 - 0.586T + 2T^{2} \)
5 \( 1 + 0.842T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + 3.26T + 13T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 - 0.260T + 19T^{2} \)
31 \( 1 + 0.101T + 31T^{2} \)
37 \( 1 + 3.33T + 37T^{2} \)
41 \( 1 + 0.732T + 41T^{2} \)
43 \( 1 + 5.01T + 43T^{2} \)
47 \( 1 + 8.71T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 - 8.76T + 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 + 3.21T + 67T^{2} \)
71 \( 1 + 1.72T + 71T^{2} \)
73 \( 1 + 2.64T + 73T^{2} \)
79 \( 1 - 0.465T + 79T^{2} \)
83 \( 1 + 8.34T + 83T^{2} \)
89 \( 1 + 5.13T + 89T^{2} \)
97 \( 1 + 6.87T + 97T^{2} \)
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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

−8.678466619146187577338266763149, −8.168361637360035223137861344465, −7.28198913324182202788897262180, −6.41882935193630449270722352345, −5.26609604873747387380326446572, −4.58235497223594517057033821793, −3.91422748192339750027640790953, −2.97158089731959578853628216902, −1.75791002852859904551242507440, 0, 1.75791002852859904551242507440, 2.97158089731959578853628216902, 3.91422748192339750027640790953, 4.58235497223594517057033821793, 5.26609604873747387380326446572, 6.41882935193630449270722352345, 7.28198913324182202788897262180, 8.168361637360035223137861344465, 8.678466619146187577338266763149

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