Properties

Label 2-2001-1.1-c1-0-53
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  − 0.743·2-s + 3-s − 1.44·4-s − 3.36·5-s − 0.743·6-s − 2.98·7-s + 2.56·8-s + 9-s + 2.49·10-s + 1.73·11-s − 1.44·12-s + 0.418·13-s + 2.21·14-s − 3.36·15-s + 0.989·16-s + 3.04·17-s − 0.743·18-s + 6.79·19-s + 4.86·20-s − 2.98·21-s − 1.28·22-s + 23-s + 2.56·24-s + 6.30·25-s − 0.311·26-s + 27-s + 4.31·28-s + ⋯
L(s)  = 1  − 0.525·2-s + 0.577·3-s − 0.723·4-s − 1.50·5-s − 0.303·6-s − 1.12·7-s + 0.906·8-s + 0.333·9-s + 0.790·10-s + 0.522·11-s − 0.417·12-s + 0.116·13-s + 0.592·14-s − 0.868·15-s + 0.247·16-s + 0.737·17-s − 0.175·18-s + 1.55·19-s + 1.08·20-s − 0.651·21-s − 0.274·22-s + 0.208·23-s + 0.523·24-s + 1.26·25-s − 0.0610·26-s + 0.192·27-s + 0.816·28-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 \( 1 + 0.743T + 2T^{2} \)
5 \( 1 + 3.36T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 0.418T + 13T^{2} \)
17 \( 1 - 3.04T + 17T^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 + 8.40T + 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + 1.01T + 47T^{2} \)
53 \( 1 - 1.08T + 53T^{2} \)
59 \( 1 + 0.434T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 4.29T + 67T^{2} \)
71 \( 1 + 7.01T + 71T^{2} \)
73 \( 1 + 9.28T + 73T^{2} \)
79 \( 1 - 2.43T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 4.37T + 97T^{2} \)
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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

−8.886894866680530317670249195322, −7.964407955418548135731818354437, −7.50185496541243731499986219207, −6.76735393239233091629900919114, −5.44965017917238054918728589553, −4.45387690681618710874905565352, −3.50023417849304708803727768226, −3.27851684412075141378758473367, −1.24522085264184878085266314942, 0, 1.24522085264184878085266314942, 3.27851684412075141378758473367, 3.50023417849304708803727768226, 4.45387690681618710874905565352, 5.44965017917238054918728589553, 6.76735393239233091629900919114, 7.50185496541243731499986219207, 7.964407955418548135731818354437, 8.886894866680530317670249195322

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