Properties

Label 2-8001-1.1-c1-0-280
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  + 1.82·2-s + 1.32·4-s − 0.630·5-s + 7-s − 1.23·8-s − 1.14·10-s + 2.60·11-s − 4.34·13-s + 1.82·14-s − 4.89·16-s + 1.67·17-s + 7.62·19-s − 0.834·20-s + 4.75·22-s − 4.50·23-s − 4.60·25-s − 7.92·26-s + 1.32·28-s + 1.00·29-s − 6.18·31-s − 6.45·32-s + 3.05·34-s − 0.630·35-s − 0.667·37-s + 13.9·38-s + 0.778·40-s − 8.44·41-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.661·4-s − 0.282·5-s + 0.377·7-s − 0.436·8-s − 0.363·10-s + 0.786·11-s − 1.20·13-s + 0.487·14-s − 1.22·16-s + 0.406·17-s + 1.74·19-s − 0.186·20-s + 1.01·22-s − 0.939·23-s − 0.920·25-s − 1.55·26-s + 0.250·28-s + 0.186·29-s − 1.11·31-s − 1.14·32-s + 0.523·34-s − 0.106·35-s − 0.109·37-s + 2.25·38-s + 0.123·40-s − 1.31·41-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 \( 1 - 1.82T + 2T^{2} \)
5 \( 1 + 0.630T + 5T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
17 \( 1 - 1.67T + 17T^{2} \)
19 \( 1 - 7.62T + 19T^{2} \)
23 \( 1 + 4.50T + 23T^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 + 6.18T + 31T^{2} \)
37 \( 1 + 0.667T + 37T^{2} \)
41 \( 1 + 8.44T + 41T^{2} \)
43 \( 1 - 0.767T + 43T^{2} \)
47 \( 1 + 1.27T + 47T^{2} \)
53 \( 1 + 9.23T + 53T^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 + 6.45T + 61T^{2} \)
67 \( 1 - 5.31T + 67T^{2} \)
71 \( 1 - 9.52T + 71T^{2} \)
73 \( 1 - 4.29T + 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 - 4.96T + 83T^{2} \)
89 \( 1 - 0.366T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.45592824963754769267812834542, −6.63690831134507308911297379105, −5.91583626562855160802795882664, −5.16289310999395099612536837109, −4.81875917247986681868483345526, −3.81042581503503199371985553815, −3.46698042487706113217306921652, −2.47929709767864894918569567434, −1.51399440962788534261189351418, 0, 1.51399440962788534261189351418, 2.47929709767864894918569567434, 3.46698042487706113217306921652, 3.81042581503503199371985553815, 4.81875917247986681868483345526, 5.16289310999395099612536837109, 5.91583626562855160802795882664, 6.63690831134507308911297379105, 7.45592824963754769267812834542

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