Properties

Label 2-2001-1.1-c1-0-33
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  − 2.50·2-s − 3-s + 4.28·4-s − 2.73·5-s + 2.50·6-s − 1.54·7-s − 5.71·8-s + 9-s + 6.86·10-s − 1.30·11-s − 4.28·12-s + 3.43·13-s + 3.86·14-s + 2.73·15-s + 5.76·16-s − 2.28·17-s − 2.50·18-s − 2.39·19-s − 11.7·20-s + 1.54·21-s + 3.27·22-s + 23-s + 5.71·24-s + 2.50·25-s − 8.60·26-s − 27-s − 6.59·28-s + ⋯
L(s)  = 1  − 1.77·2-s − 0.577·3-s + 2.14·4-s − 1.22·5-s + 1.02·6-s − 0.582·7-s − 2.02·8-s + 0.333·9-s + 2.17·10-s − 0.393·11-s − 1.23·12-s + 0.952·13-s + 1.03·14-s + 0.707·15-s + 1.44·16-s − 0.553·17-s − 0.590·18-s − 0.548·19-s − 2.62·20-s + 0.336·21-s + 0.697·22-s + 0.208·23-s + 1.16·24-s + 0.501·25-s − 1.68·26-s − 0.192·27-s − 1.24·28-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 \( 1 + 2.50T + 2T^{2} \)
5 \( 1 + 2.73T + 5T^{2} \)
7 \( 1 + 1.54T + 7T^{2} \)
11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 - 3.43T + 13T^{2} \)
17 \( 1 + 2.28T + 17T^{2} \)
19 \( 1 + 2.39T + 19T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 3.81T + 41T^{2} \)
43 \( 1 - 4.29T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 0.703T + 53T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + 0.0657T + 61T^{2} \)
67 \( 1 + 0.500T + 67T^{2} \)
71 \( 1 + 0.422T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + 2.63T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 3.71T + 97T^{2} \)
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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

−8.806014209515192850562666857668, −7.954565283564172651526302788148, −7.56367475864801142323470437384, −6.55529144143285644056407892779, −6.12633247855376382650470621356, −4.62922247106009666986247257325, −3.61509556081520773657488788115, −2.45111046656329092446787127668, −0.994633543089141866606701045434, 0, 0.994633543089141866606701045434, 2.45111046656329092446787127668, 3.61509556081520773657488788115, 4.62922247106009666986247257325, 6.12633247855376382650470621356, 6.55529144143285644056407892779, 7.56367475864801142323470437384, 7.954565283564172651526302788148, 8.806014209515192850562666857668

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