Properties

Label 2-2001-1.1-c1-0-83
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  − 0.865·2-s + 3-s − 1.25·4-s + 1.66·5-s − 0.865·6-s − 0.715·7-s + 2.81·8-s + 9-s − 1.44·10-s − 0.656·11-s − 1.25·12-s − 4.37·13-s + 0.619·14-s + 1.66·15-s + 0.0635·16-s − 7.15·17-s − 0.865·18-s + 6.56·19-s − 2.08·20-s − 0.715·21-s + 0.568·22-s − 23-s + 2.81·24-s − 2.22·25-s + 3.78·26-s + 27-s + 0.895·28-s + ⋯
L(s)  = 1  − 0.612·2-s + 0.577·3-s − 0.625·4-s + 0.744·5-s − 0.353·6-s − 0.270·7-s + 0.995·8-s + 0.333·9-s − 0.456·10-s − 0.198·11-s − 0.360·12-s − 1.21·13-s + 0.165·14-s + 0.430·15-s + 0.0158·16-s − 1.73·17-s − 0.204·18-s + 1.50·19-s − 0.465·20-s − 0.156·21-s + 0.121·22-s − 0.208·23-s + 0.574·24-s − 0.445·25-s + 0.742·26-s + 0.192·27-s + 0.169·28-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 \( 1 + 0.865T + 2T^{2} \)
5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 0.715T + 7T^{2} \)
11 \( 1 + 0.656T + 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 + 7.15T + 17T^{2} \)
19 \( 1 - 6.56T + 19T^{2} \)
31 \( 1 - 0.705T + 31T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + 12.6T + 41T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 - 8.50T + 47T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 + 4.81T + 59T^{2} \)
61 \( 1 - 2.51T + 61T^{2} \)
67 \( 1 + 2.81T + 67T^{2} \)
71 \( 1 - 0.253T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 0.611T + 79T^{2} \)
83 \( 1 - 2.06T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−8.904704381263216884158021627486, −8.173414260767610638602659962862, −7.32349218803630308320014115936, −6.66688470030188812727642744517, −5.36974594319357538081360019351, −4.79619832863514215636766824474, −3.72574072111506657014122739012, −2.56902408649765190830487222917, −1.63836418820017179196332876904, 0, 1.63836418820017179196332876904, 2.56902408649765190830487222917, 3.72574072111506657014122739012, 4.79619832863514215636766824474, 5.36974594319357538081360019351, 6.66688470030188812727642744517, 7.32349218803630308320014115936, 8.173414260767610638602659962862, 8.904704381263216884158021627486

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