Properties

Label 2-2001-1.1-c1-0-86
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  + 1.51·2-s − 3-s + 0.295·4-s − 1.86·5-s − 1.51·6-s + 1.57·7-s − 2.58·8-s + 9-s − 2.82·10-s + 0.419·11-s − 0.295·12-s + 4.28·13-s + 2.38·14-s + 1.86·15-s − 4.50·16-s + 1.70·17-s + 1.51·18-s − 5.83·19-s − 0.551·20-s − 1.57·21-s + 0.636·22-s + 23-s + 2.58·24-s − 1.51·25-s + 6.49·26-s − 27-s + 0.464·28-s + ⋯
L(s)  = 1  + 1.07·2-s − 0.577·3-s + 0.147·4-s − 0.834·5-s − 0.618·6-s + 0.593·7-s − 0.912·8-s + 0.333·9-s − 0.894·10-s + 0.126·11-s − 0.0853·12-s + 1.18·13-s + 0.636·14-s + 0.481·15-s − 1.12·16-s + 0.413·17-s + 0.357·18-s − 1.33·19-s − 0.123·20-s − 0.342·21-s + 0.135·22-s + 0.208·23-s + 0.527·24-s − 0.303·25-s + 1.27·26-s − 0.192·27-s + 0.0877·28-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 \( 1 - 1.51T + 2T^{2} \)
5 \( 1 + 1.86T + 5T^{2} \)
7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 - 0.419T + 11T^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
31 \( 1 + 2.45T + 31T^{2} \)
37 \( 1 + 3.07T + 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 + 3.20T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 5.61T + 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 - 5.23T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 - 15.9T + 73T^{2} \)
79 \( 1 - 5.67T + 79T^{2} \)
83 \( 1 + 3.89T + 83T^{2} \)
89 \( 1 + 6.42T + 89T^{2} \)
97 \( 1 - 2.73T + 97T^{2} \)
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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

−8.481671813607352521115324527151, −8.145102562087394868048100581441, −6.86842490218401650254584003369, −6.26164353872457605442089670578, −5.38115552431414949648989777807, −4.65866204721430688681813672214, −3.93202900147511801802400254663, −3.26971223588676075656076109747, −1.66180895046833995294914728141, 0, 1.66180895046833995294914728141, 3.26971223588676075656076109747, 3.93202900147511801802400254663, 4.65866204721430688681813672214, 5.38115552431414949648989777807, 6.26164353872457605442089670578, 6.86842490218401650254584003369, 8.145102562087394868048100581441, 8.481671813607352521115324527151

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