Properties

Label 2-8001-1.1-c1-0-289
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  + 2.44·2-s + 3.98·4-s − 1.74·5-s − 7-s + 4.85·8-s − 4.26·10-s + 0.135·11-s − 2.02·13-s − 2.44·14-s + 3.90·16-s − 0.659·17-s − 3.26·19-s − 6.94·20-s + 0.332·22-s + 2.00·23-s − 1.96·25-s − 4.95·26-s − 3.98·28-s + 6.31·29-s + 6.78·31-s − 0.149·32-s − 1.61·34-s + 1.74·35-s + 2.08·37-s − 7.98·38-s − 8.46·40-s − 2.96·41-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.99·4-s − 0.779·5-s − 0.377·7-s + 1.71·8-s − 1.34·10-s + 0.0409·11-s − 0.561·13-s − 0.653·14-s + 0.977·16-s − 0.159·17-s − 0.748·19-s − 1.55·20-s + 0.0708·22-s + 0.418·23-s − 0.392·25-s − 0.971·26-s − 0.753·28-s + 1.17·29-s + 1.21·31-s − 0.0263·32-s − 0.276·34-s + 0.294·35-s + 0.342·37-s − 1.29·38-s − 1.33·40-s − 0.462·41-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: $\chi_{8001} (1, \cdot )$
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 \( 1 - 2.44T + 2T^{2} \)
5 \( 1 + 1.74T + 5T^{2} \)
11 \( 1 - 0.135T + 11T^{2} \)
13 \( 1 + 2.02T + 13T^{2} \)
17 \( 1 + 0.659T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 - 6.31T + 29T^{2} \)
31 \( 1 - 6.78T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 1.28T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 0.817T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 3.94T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 - 4.98T + 97T^{2} \)
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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

−7.26110303154009766723762621421, −6.43195732466562984243053170622, −6.23273669465655003864475038295, −5.11693317759136199382864592903, −4.58902367215264171955057174975, −4.09686951993344969793710811777, −3.14862030530262339541243071550, −2.77322296546735944012084109949, −1.63160999653097351357865243570, 0, 1.63160999653097351357865243570, 2.77322296546735944012084109949, 3.14862030530262339541243071550, 4.09686951993344969793710811777, 4.58902367215264171955057174975, 5.11693317759136199382864592903, 6.23273669465655003864475038295, 6.43195732466562984243053170622, 7.26110303154009766723762621421

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