Properties

Label 2-8001-1.1-c1-0-139
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  − 0.946·2-s − 1.10·4-s + 2.90·5-s + 7-s + 2.93·8-s − 2.74·10-s + 4.94·11-s + 2.17·13-s − 0.946·14-s − 0.572·16-s + 3.30·17-s − 3.52·19-s − 3.20·20-s − 4.68·22-s − 0.742·23-s + 3.44·25-s − 2.05·26-s − 1.10·28-s − 0.319·29-s − 7.09·31-s − 5.33·32-s − 3.12·34-s + 2.90·35-s + 5.44·37-s + 3.34·38-s + 8.53·40-s − 4.63·41-s + ⋯
L(s)  = 1  − 0.669·2-s − 0.552·4-s + 1.29·5-s + 0.377·7-s + 1.03·8-s − 0.869·10-s + 1.49·11-s + 0.603·13-s − 0.252·14-s − 0.143·16-s + 0.800·17-s − 0.809·19-s − 0.717·20-s − 0.998·22-s − 0.154·23-s + 0.688·25-s − 0.403·26-s − 0.208·28-s − 0.0592·29-s − 1.27·31-s − 0.942·32-s − 0.535·34-s + 0.491·35-s + 0.895·37-s + 0.541·38-s + 1.34·40-s − 0.723·41-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

\(L(1)\) \(\approx\) \(2.079504329\)
\(L(\frac12)\) \(\approx\) \(2.079504329\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.946T + 2T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 - 4.94T + 11T^{2} \)
13 \( 1 - 2.17T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 + 0.742T + 23T^{2} \)
29 \( 1 + 0.319T + 29T^{2} \)
31 \( 1 + 7.09T + 31T^{2} \)
37 \( 1 - 5.44T + 37T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 - 2.59T + 43T^{2} \)
47 \( 1 - 2.05T + 47T^{2} \)
53 \( 1 - 8.76T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 3.00T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 - 6.60T + 73T^{2} \)
79 \( 1 - 2.76T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.102521746377315572677355293983, −7.10069277218890230135116441720, −6.53830972181897371787953375577, −5.66206319067471416109758501887, −5.27963796652946154208203707016, −4.10855367386774790414105994877, −3.76638932305876288798700660115, −2.30549484801610836968924695963, −1.56708833045715430323760860606, −0.885821341012196396600156838799, 0.885821341012196396600156838799, 1.56708833045715430323760860606, 2.30549484801610836968924695963, 3.76638932305876288798700660115, 4.10855367386774790414105994877, 5.27963796652946154208203707016, 5.66206319067471416109758501887, 6.53830972181897371787953375577, 7.10069277218890230135116441720, 8.102521746377315572677355293983

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