Properties

Label 2-8001-1.1-c1-0-180
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  − 2.70·2-s + 5.33·4-s − 2.87·5-s + 7-s − 9.04·8-s + 7.79·10-s + 1.50·11-s + 5.43·13-s − 2.70·14-s + 13.8·16-s − 1.68·17-s − 3.30·19-s − 15.3·20-s − 4.06·22-s + 5.07·23-s + 3.28·25-s − 14.7·26-s + 5.33·28-s + 8.28·29-s − 9.29·31-s − 19.3·32-s + 4.56·34-s − 2.87·35-s − 7.22·37-s + 8.94·38-s + 26.0·40-s + 2.03·41-s + ⋯
L(s)  = 1  − 1.91·2-s + 2.66·4-s − 1.28·5-s + 0.377·7-s − 3.19·8-s + 2.46·10-s + 0.452·11-s + 1.50·13-s − 0.723·14-s + 3.45·16-s − 0.408·17-s − 0.757·19-s − 3.43·20-s − 0.866·22-s + 1.05·23-s + 0.657·25-s − 2.88·26-s + 1.00·28-s + 1.53·29-s − 1.66·31-s − 3.41·32-s + 0.782·34-s − 0.486·35-s − 1.18·37-s + 1.45·38-s + 4.11·40-s + 0.317·41-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 \( 1 + 2.70T + 2T^{2} \)
5 \( 1 + 2.87T + 5T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
13 \( 1 - 5.43T + 13T^{2} \)
17 \( 1 + 1.68T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + 9.29T + 31T^{2} \)
37 \( 1 + 7.22T + 37T^{2} \)
41 \( 1 - 2.03T + 41T^{2} \)
43 \( 1 - 0.859T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 8.32T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 9.99T + 61T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + 5.53T + 71T^{2} \)
73 \( 1 - 6.56T + 73T^{2} \)
79 \( 1 + 9.27T + 79T^{2} \)
83 \( 1 + 7.17T + 83T^{2} \)
89 \( 1 + 9.76T + 89T^{2} \)
97 \( 1 + 9.08T + 97T^{2} \)
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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

−7.63227367376549006477614539478, −7.11589163697469304633968158694, −6.50333343299231071566135160526, −5.77964970785956133227289661155, −4.51150545462456089130948811328, −3.66292437634884631888547891532, −2.93663163621314274017159023406, −1.76761288060562782071768442327, −1.03816421937151758153099926746, 0, 1.03816421937151758153099926746, 1.76761288060562782071768442327, 2.93663163621314274017159023406, 3.66292437634884631888547891532, 4.51150545462456089130948811328, 5.77964970785956133227289661155, 6.50333343299231071566135160526, 7.11589163697469304633968158694, 7.63227367376549006477614539478

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