Properties

Label 2-8001-1.1-c1-0-179
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  − 0.840·2-s − 1.29·4-s − 2.74·5-s + 7-s + 2.76·8-s + 2.30·10-s + 5.62·11-s − 3.65·13-s − 0.840·14-s + 0.257·16-s + 5.86·17-s − 2.05·19-s + 3.55·20-s − 4.72·22-s − 2.78·23-s + 2.54·25-s + 3.07·26-s − 1.29·28-s + 4.83·29-s − 6.87·31-s − 5.75·32-s − 4.93·34-s − 2.74·35-s − 8.71·37-s + 1.73·38-s − 7.60·40-s + 4.09·41-s + ⋯
L(s)  = 1  − 0.594·2-s − 0.646·4-s − 1.22·5-s + 0.377·7-s + 0.978·8-s + 0.730·10-s + 1.69·11-s − 1.01·13-s − 0.224·14-s + 0.0644·16-s + 1.42·17-s − 0.472·19-s + 0.794·20-s − 1.00·22-s − 0.580·23-s + 0.509·25-s + 0.602·26-s − 0.244·28-s + 0.897·29-s − 1.23·31-s − 1.01·32-s − 0.845·34-s − 0.464·35-s − 1.43·37-s + 0.280·38-s − 1.20·40-s + 0.640·41-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 \( 1 + 0.840T + 2T^{2} \)
5 \( 1 + 2.74T + 5T^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + 3.65T + 13T^{2} \)
17 \( 1 - 5.86T + 17T^{2} \)
19 \( 1 + 2.05T + 19T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 - 4.83T + 29T^{2} \)
31 \( 1 + 6.87T + 31T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 - 4.09T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + 4.92T + 53T^{2} \)
59 \( 1 - 9.41T + 59T^{2} \)
61 \( 1 - 4.96T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 + 2.13T + 71T^{2} \)
73 \( 1 - 9.59T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 5.06T + 83T^{2} \)
89 \( 1 + 2.73T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.52006084767446143608963827652, −7.16730507560840956847881734027, −6.23494777040785515352059560829, −5.15844508203612161793623743119, −4.64165892858926490958881529416, −3.76954604134175659866502237157, −3.50662786607944416646579748805, −1.92792248334769426393719457636, −1.03589665077963974341060406936, 0, 1.03589665077963974341060406936, 1.92792248334769426393719457636, 3.50662786607944416646579748805, 3.76954604134175659866502237157, 4.64165892858926490958881529416, 5.15844508203612161793623743119, 6.23494777040785515352059560829, 7.16730507560840956847881734027, 7.52006084767446143608963827652

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