Properties

Label 2-8001-1.1-c1-0-200
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  − 2.49·2-s + 4.22·4-s − 0.373·5-s − 7-s − 5.54·8-s + 0.932·10-s + 6.40·11-s + 2.54·13-s + 2.49·14-s + 5.38·16-s − 4.88·17-s − 5.83·19-s − 1.57·20-s − 15.9·22-s + 8.35·23-s − 4.86·25-s − 6.34·26-s − 4.22·28-s + 1.63·29-s − 6.96·31-s − 2.35·32-s + 12.1·34-s + 0.373·35-s + 7.46·37-s + 14.5·38-s + 2.07·40-s + 9.02·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 2.11·4-s − 0.167·5-s − 0.377·7-s − 1.96·8-s + 0.294·10-s + 1.93·11-s + 0.705·13-s + 0.666·14-s + 1.34·16-s − 1.18·17-s − 1.33·19-s − 0.352·20-s − 3.40·22-s + 1.74·23-s − 0.972·25-s − 1.24·26-s − 0.798·28-s + 0.303·29-s − 1.25·31-s − 0.415·32-s + 2.09·34-s + 0.0631·35-s + 1.22·37-s + 2.36·38-s + 0.327·40-s + 1.40·41-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: $\chi_{8001} (1, \cdot )$
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 \( 1 + 2.49T + 2T^{2} \)
5 \( 1 + 0.373T + 5T^{2} \)
11 \( 1 - 6.40T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 - 8.35T + 23T^{2} \)
29 \( 1 - 1.63T + 29T^{2} \)
31 \( 1 + 6.96T + 31T^{2} \)
37 \( 1 - 7.46T + 37T^{2} \)
41 \( 1 - 9.02T + 41T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 + 6.14T + 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 - 5.49T + 73T^{2} \)
79 \( 1 + 7.21T + 79T^{2} \)
83 \( 1 + 4.98T + 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.59812970908898895485377680528, −6.71541692473629138659909450092, −6.63203292893840354455706428314, −5.87208673809112509434680885907, −4.44393883238393125821278735787, −3.83539322885809506495770430486, −2.79186717337905710957243148017, −1.81790832717354732237654157519, −1.13223073178406970027683502675, 0, 1.13223073178406970027683502675, 1.81790832717354732237654157519, 2.79186717337905710957243148017, 3.83539322885809506495770430486, 4.44393883238393125821278735787, 5.87208673809112509434680885907, 6.63203292893840354455706428314, 6.71541692473629138659909450092, 7.59812970908898895485377680528

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