Properties

Label 2-8001-1.1-c1-0-93
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.191·2-s − 1.96·4-s − 3.92·5-s − 7-s + 0.759·8-s + 0.751·10-s + 5.26·11-s + 6.07·13-s + 0.191·14-s + 3.78·16-s + 4.12·17-s + 0.0374·19-s + 7.69·20-s − 1.00·22-s + 3.57·23-s + 10.3·25-s − 1.16·26-s + 1.96·28-s + 2.19·29-s + 4.14·31-s − 2.24·32-s − 0.790·34-s + 3.92·35-s − 9.31·37-s − 0.00716·38-s − 2.97·40-s + 9.48·41-s + ⋯
L(s)  = 1  − 0.135·2-s − 0.981·4-s − 1.75·5-s − 0.377·7-s + 0.268·8-s + 0.237·10-s + 1.58·11-s + 1.68·13-s + 0.0512·14-s + 0.945·16-s + 1.00·17-s + 0.00858·19-s + 1.72·20-s − 0.215·22-s + 0.745·23-s + 2.07·25-s − 0.228·26-s + 0.371·28-s + 0.408·29-s + 0.745·31-s − 0.396·32-s − 0.135·34-s + 0.662·35-s − 1.53·37-s − 0.00116·38-s − 0.470·40-s + 1.48·41-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: \(0\)
Selberg data: \((2,\ 8001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.327936885\)
\(L(\frac12)\) \(\approx\) \(1.327936885\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.191T + 2T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
13 \( 1 - 6.07T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 0.0374T + 19T^{2} \)
23 \( 1 - 3.57T + 23T^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 - 4.14T + 31T^{2} \)
37 \( 1 + 9.31T + 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 - 3.87T + 47T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + 0.382T + 61T^{2} \)
67 \( 1 - 4.40T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 3.37T + 73T^{2} \)
79 \( 1 - 9.73T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 2.53T + 89T^{2} \)
97 \( 1 + 5.85T + 97T^{2} \)
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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

−8.052532186233334682214576798659, −7.14989968241308973980985829639, −6.63605139148213800208905131191, −5.72733297796064517780725767319, −4.85476021062614653025031102192, −3.90954083641305768615854610831, −3.81802129466203600632576080932, −3.15680801202117280234749076885, −1.23056849851885052941351951071, −0.72601382777373585214400352891, 0.72601382777373585214400352891, 1.23056849851885052941351951071, 3.15680801202117280234749076885, 3.81802129466203600632576080932, 3.90954083641305768615854610831, 4.85476021062614653025031102192, 5.72733297796064517780725767319, 6.63605139148213800208905131191, 7.14989968241308973980985829639, 8.052532186233334682214576798659

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