Properties

Label 2-2001-1.1-c1-0-101
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  + 1.98·2-s + 3-s + 1.95·4-s − 1.78·5-s + 1.98·6-s − 0.840·7-s − 0.0814·8-s + 9-s − 3.55·10-s − 6.06·11-s + 1.95·12-s − 4.54·13-s − 1.67·14-s − 1.78·15-s − 4.08·16-s + 5.04·17-s + 1.98·18-s − 0.331·19-s − 3.50·20-s − 0.840·21-s − 12.0·22-s + 23-s − 0.0814·24-s − 1.80·25-s − 9.03·26-s + 27-s − 1.64·28-s + ⋯
L(s)  = 1  + 1.40·2-s + 0.577·3-s + 0.979·4-s − 0.799·5-s + 0.812·6-s − 0.317·7-s − 0.0288·8-s + 0.333·9-s − 1.12·10-s − 1.83·11-s + 0.565·12-s − 1.25·13-s − 0.447·14-s − 0.461·15-s − 1.02·16-s + 1.22·17-s + 0.468·18-s − 0.0759·19-s − 0.782·20-s − 0.183·21-s − 2.57·22-s + 0.208·23-s − 0.0166·24-s − 0.361·25-s − 1.77·26-s + 0.192·27-s − 0.311·28-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 \( 1 - 1.98T + 2T^{2} \)
5 \( 1 + 1.78T + 5T^{2} \)
7 \( 1 + 0.840T + 7T^{2} \)
11 \( 1 + 6.06T + 11T^{2} \)
13 \( 1 + 4.54T + 13T^{2} \)
17 \( 1 - 5.04T + 17T^{2} \)
19 \( 1 + 0.331T + 19T^{2} \)
31 \( 1 - 0.306T + 31T^{2} \)
37 \( 1 - 8.76T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 8.93T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 9.45T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 7.91T + 71T^{2} \)
73 \( 1 - 2.31T + 73T^{2} \)
79 \( 1 + 3.85T + 79T^{2} \)
83 \( 1 - 9.11T + 83T^{2} \)
89 \( 1 + 0.431T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−8.527412025595522429933423348187, −7.76325140606858243100741284892, −7.30748947167281556248848245981, −6.20778454784127851435522182577, −5.15263622121855022852395431148, −4.83069051743904971821740615058, −3.66984452968375463229237287443, −3.06911542463815757260558197807, −2.29396793374380473688411339290, 0, 2.29396793374380473688411339290, 3.06911542463815757260558197807, 3.66984452968375463229237287443, 4.83069051743904971821740615058, 5.15263622121855022852395431148, 6.20778454784127851435522182577, 7.30748947167281556248848245981, 7.76325140606858243100741284892, 8.527412025595522429933423348187

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