Properties

Label 2-8001-1.1-c1-0-240
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  + 0.775·2-s − 1.39·4-s + 0.833·5-s + 7-s − 2.63·8-s + 0.646·10-s + 0.821·11-s − 3.40·13-s + 0.775·14-s + 0.752·16-s − 2.50·17-s + 1.40·19-s − 1.16·20-s + 0.636·22-s + 0.947·23-s − 4.30·25-s − 2.64·26-s − 1.39·28-s + 1.78·29-s + 8.51·31-s + 5.85·32-s − 1.94·34-s + 0.833·35-s − 2.89·37-s + 1.08·38-s − 2.19·40-s + 2.30·41-s + ⋯
L(s)  = 1  + 0.548·2-s − 0.699·4-s + 0.372·5-s + 0.377·7-s − 0.931·8-s + 0.204·10-s + 0.247·11-s − 0.945·13-s + 0.207·14-s + 0.188·16-s − 0.607·17-s + 0.321·19-s − 0.260·20-s + 0.135·22-s + 0.197·23-s − 0.861·25-s − 0.518·26-s − 0.264·28-s + 0.331·29-s + 1.53·31-s + 1.03·32-s − 0.332·34-s + 0.140·35-s − 0.475·37-s + 0.176·38-s − 0.347·40-s + 0.359·41-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 \( 1 - 0.775T + 2T^{2} \)
5 \( 1 - 0.833T + 5T^{2} \)
11 \( 1 - 0.821T + 11T^{2} \)
13 \( 1 + 3.40T + 13T^{2} \)
17 \( 1 + 2.50T + 17T^{2} \)
19 \( 1 - 1.40T + 19T^{2} \)
23 \( 1 - 0.947T + 23T^{2} \)
29 \( 1 - 1.78T + 29T^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 + 2.89T + 37T^{2} \)
41 \( 1 - 2.30T + 41T^{2} \)
43 \( 1 - 5.58T + 43T^{2} \)
47 \( 1 - 2.14T + 47T^{2} \)
53 \( 1 + 0.992T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 7.19T + 61T^{2} \)
67 \( 1 + 5.77T + 67T^{2} \)
71 \( 1 + 0.675T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 1.57T + 79T^{2} \)
83 \( 1 - 3.31T + 83T^{2} \)
89 \( 1 + 1.39T + 89T^{2} \)
97 \( 1 + 4.55T + 97T^{2} \)
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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

−7.57623897500493107979985451167, −6.61533375681355708334265429357, −5.99254634153604882424884093242, −5.29355216261997936746014283960, −4.59007160712681598180962648915, −4.18817671620810698914021787922, −3.10857898972745479761479085123, −2.42351803265814026152408113383, −1.27278531426221914198838988334, 0, 1.27278531426221914198838988334, 2.42351803265814026152408113383, 3.10857898972745479761479085123, 4.18817671620810698914021787922, 4.59007160712681598180962648915, 5.29355216261997936746014283960, 5.99254634153604882424884093242, 6.61533375681355708334265429357, 7.57623897500493107979985451167

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