Properties

Label 2-8001-1.1-c1-0-157
Degree $2$
Conductor $8001$
Sign $-1$
Analytic cond. $63.8883$
Root an. cond. $7.99301$
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  − 1.63·2-s + 0.672·4-s + 0.107·5-s − 7-s + 2.16·8-s − 0.176·10-s − 1.23·11-s − 2.67·13-s + 1.63·14-s − 4.89·16-s − 6.14·17-s + 1.57·19-s + 0.0725·20-s + 2.01·22-s − 5.97·23-s − 4.98·25-s + 4.37·26-s − 0.672·28-s + 9.60·29-s + 1.81·31-s + 3.65·32-s + 10.0·34-s − 0.107·35-s + 7.63·37-s − 2.57·38-s + 0.233·40-s + 4.73·41-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.336·4-s + 0.0482·5-s − 0.377·7-s + 0.767·8-s − 0.0557·10-s − 0.372·11-s − 0.742·13-s + 0.436·14-s − 1.22·16-s − 1.48·17-s + 0.361·19-s + 0.0162·20-s + 0.430·22-s − 1.24·23-s − 0.997·25-s + 0.858·26-s − 0.127·28-s + 1.78·29-s + 0.325·31-s + 0.646·32-s + 1.72·34-s − 0.0182·35-s + 1.25·37-s − 0.418·38-s + 0.0369·40-s + 0.739·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
Sign: $-1$
Analytic conductor: \(63.8883\)
Root analytic conductor: \(7.99301\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
127 \( 1 + T \)
good2 \( 1 + 1.63T + 2T^{2} \)
5 \( 1 - 0.107T + 5T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + 2.67T + 13T^{2} \)
17 \( 1 + 6.14T + 17T^{2} \)
19 \( 1 - 1.57T + 19T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 - 9.60T + 29T^{2} \)
31 \( 1 - 1.81T + 31T^{2} \)
37 \( 1 - 7.63T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 + 5.68T + 59T^{2} \)
61 \( 1 + 8.36T + 61T^{2} \)
67 \( 1 + 4.40T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 8.13T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.65772498415484632211544554456, −7.03930685283610596795824713396, −6.26324576139180944218132118451, −5.54307586849786280729439995123, −4.36839892091370915708217976244, −4.22808986878229730635879436886, −2.65226315046539572374974831300, −2.23587235333006437902012892666, −0.946001048035913700547777939165, 0, 0.946001048035913700547777939165, 2.23587235333006437902012892666, 2.65226315046539572374974831300, 4.22808986878229730635879436886, 4.36839892091370915708217976244, 5.54307586849786280729439995123, 6.26324576139180944218132118451, 7.03930685283610596795824713396, 7.65772498415484632211544554456

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