Properties

Label 2-8001-1.1-c1-0-0
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  + 1.42·2-s + 0.0348·4-s − 4.07·5-s + 7-s − 2.80·8-s − 5.81·10-s − 3.50·11-s − 5.98·13-s + 1.42·14-s − 4.06·16-s − 7.04·17-s + 3.99·19-s − 0.142·20-s − 4.99·22-s − 7.32·23-s + 11.6·25-s − 8.53·26-s + 0.0348·28-s − 7.77·29-s + 2.84·31-s − 0.197·32-s − 10.0·34-s − 4.07·35-s − 1.23·37-s + 5.70·38-s + 11.4·40-s − 1.75·41-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.0174·4-s − 1.82·5-s + 0.377·7-s − 0.991·8-s − 1.83·10-s − 1.05·11-s − 1.65·13-s + 0.381·14-s − 1.01·16-s − 1.70·17-s + 0.917·19-s − 0.0318·20-s − 1.06·22-s − 1.52·23-s + 2.32·25-s − 1.67·26-s + 0.00659·28-s − 1.44·29-s + 0.510·31-s − 0.0348·32-s − 1.72·34-s − 0.689·35-s − 0.202·37-s + 0.925·38-s + 1.80·40-s − 0.274·41-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

\(L(1)\) \(\approx\) \(0.008726446836\)
\(L(\frac12)\) \(\approx\) \(0.008726446836\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.42T + 2T^{2} \)
5 \( 1 + 4.07T + 5T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 + 5.98T + 13T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 - 3.99T + 19T^{2} \)
23 \( 1 + 7.32T + 23T^{2} \)
29 \( 1 + 7.77T + 29T^{2} \)
31 \( 1 - 2.84T + 31T^{2} \)
37 \( 1 + 1.23T + 37T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 4.77T + 47T^{2} \)
53 \( 1 - 4.77T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 - 8.07T + 79T^{2} \)
83 \( 1 - 5.09T + 83T^{2} \)
89 \( 1 + 0.679T + 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−7.87612806636030688845934082918, −7.17511603130160938690311574540, −6.49116763865047515895178445275, −5.32009214632513175255703988226, −4.91083029559909931337297146697, −4.35987755804841071413730803410, −3.71198579186851536089002988750, −2.94333632376135452135371649599, −2.14071888337167092366438786528, −0.03645257865752169123783842547, 0.03645257865752169123783842547, 2.14071888337167092366438786528, 2.94333632376135452135371649599, 3.71198579186851536089002988750, 4.35987755804841071413730803410, 4.91083029559909931337297146697, 5.32009214632513175255703988226, 6.49116763865047515895178445275, 7.17511603130160938690311574540, 7.87612806636030688845934082918

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