Properties

Label 2-8001-1.1-c1-0-258
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  − 2.24·2-s + 3.03·4-s + 1.67·5-s + 7-s − 2.31·8-s − 3.74·10-s + 0.0139·11-s − 1.81·13-s − 2.24·14-s − 0.869·16-s + 6.81·17-s + 7.14·19-s + 5.06·20-s − 0.0312·22-s + 0.0905·23-s − 2.20·25-s + 4.06·26-s + 3.03·28-s − 7.70·29-s − 8.56·31-s + 6.58·32-s − 15.2·34-s + 1.67·35-s + 1.24·37-s − 16.0·38-s − 3.87·40-s + 6.73·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.51·4-s + 0.747·5-s + 0.377·7-s − 0.818·8-s − 1.18·10-s + 0.00419·11-s − 0.503·13-s − 0.599·14-s − 0.217·16-s + 1.65·17-s + 1.63·19-s + 1.13·20-s − 0.00665·22-s + 0.0188·23-s − 0.441·25-s + 0.798·26-s + 0.573·28-s − 1.42·29-s − 1.53·31-s + 1.16·32-s − 2.62·34-s + 0.282·35-s + 0.204·37-s − 2.60·38-s − 0.611·40-s + 1.05·41-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 \( 1 + 2.24T + 2T^{2} \)
5 \( 1 - 1.67T + 5T^{2} \)
11 \( 1 - 0.0139T + 11T^{2} \)
13 \( 1 + 1.81T + 13T^{2} \)
17 \( 1 - 6.81T + 17T^{2} \)
19 \( 1 - 7.14T + 19T^{2} \)
23 \( 1 - 0.0905T + 23T^{2} \)
29 \( 1 + 7.70T + 29T^{2} \)
31 \( 1 + 8.56T + 31T^{2} \)
37 \( 1 - 1.24T + 37T^{2} \)
41 \( 1 - 6.73T + 41T^{2} \)
43 \( 1 + 6.87T + 43T^{2} \)
47 \( 1 + 7.32T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 + 9.03T + 59T^{2} \)
61 \( 1 + 1.91T + 61T^{2} \)
67 \( 1 + 2.93T + 67T^{2} \)
71 \( 1 + 5.54T + 71T^{2} \)
73 \( 1 + 3.15T + 73T^{2} \)
79 \( 1 + 0.921T + 79T^{2} \)
83 \( 1 - 4.65T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + 8.25T + 97T^{2} \)
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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

−7.68329016166057972783003005301, −7.23811404038264431523504497148, −6.20960780597394742329570875479, −5.53048641872543318840466806535, −4.94742022252762225946177755032, −3.63688214735881721784280486472, −2.81949613533479793744580554461, −1.71362146251955903212444086090, −1.34801238512757822375336792981, 0, 1.34801238512757822375336792981, 1.71362146251955903212444086090, 2.81949613533479793744580554461, 3.63688214735881721784280486472, 4.94742022252762225946177755032, 5.53048641872543318840466806535, 6.20960780597394742329570875479, 7.23811404038264431523504497148, 7.68329016166057972783003005301

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