Properties

Label 2-2001-1.1-c1-0-71
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  − 2.22·2-s + 3-s + 2.97·4-s + 0.537·5-s − 2.22·6-s − 1.88·7-s − 2.16·8-s + 9-s − 1.19·10-s + 1.11·11-s + 2.97·12-s + 1.56·13-s + 4.19·14-s + 0.537·15-s − 1.11·16-s − 3.41·17-s − 2.22·18-s − 2.62·19-s + 1.59·20-s − 1.88·21-s − 2.49·22-s + 23-s − 2.16·24-s − 4.71·25-s − 3.49·26-s + 27-s − 5.59·28-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.577·3-s + 1.48·4-s + 0.240·5-s − 0.910·6-s − 0.711·7-s − 0.765·8-s + 0.333·9-s − 0.378·10-s + 0.336·11-s + 0.857·12-s + 0.434·13-s + 1.12·14-s + 0.138·15-s − 0.278·16-s − 0.828·17-s − 0.525·18-s − 0.602·19-s + 0.356·20-s − 0.410·21-s − 0.531·22-s + 0.208·23-s − 0.442·24-s − 0.942·25-s − 0.685·26-s + 0.192·27-s − 1.05·28-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 \( 1 + 2.22T + 2T^{2} \)
5 \( 1 - 0.537T + 5T^{2} \)
7 \( 1 + 1.88T + 7T^{2} \)
11 \( 1 - 1.11T + 11T^{2} \)
13 \( 1 - 1.56T + 13T^{2} \)
17 \( 1 + 3.41T + 17T^{2} \)
19 \( 1 + 2.62T + 19T^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 - 1.64T + 37T^{2} \)
41 \( 1 - 5.06T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 6.06T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 8.80T + 59T^{2} \)
61 \( 1 + 3.73T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 + 2.43T + 71T^{2} \)
73 \( 1 - 1.02T + 73T^{2} \)
79 \( 1 - 1.24T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 7.41T + 97T^{2} \)
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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

−8.833852153293619342815041258429, −8.257885135266819077463183479716, −7.41327097175532181819858586103, −6.66791525650015669515014085182, −6.03333691972207041370487311060, −4.55044108040538804185647944625, −3.50301651376845247222701889284, −2.38272842662696696594179343753, −1.50388471664017327210206339184, 0, 1.50388471664017327210206339184, 2.38272842662696696594179343753, 3.50301651376845247222701889284, 4.55044108040538804185647944625, 6.03333691972207041370487311060, 6.66791525650015669515014085182, 7.41327097175532181819858586103, 8.257885135266819077463183479716, 8.833852153293619342815041258429

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