Properties

Label 2-8001-1.1-c1-0-290
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  + 1.69·2-s + 0.875·4-s + 2.42·5-s + 7-s − 1.90·8-s + 4.10·10-s − 1.73·11-s − 5.13·13-s + 1.69·14-s − 4.98·16-s − 0.851·17-s + 2.28·19-s + 2.12·20-s − 2.94·22-s + 0.763·23-s + 0.869·25-s − 8.70·26-s + 0.875·28-s − 1.78·29-s − 0.617·31-s − 4.64·32-s − 1.44·34-s + 2.42·35-s − 0.416·37-s + 3.87·38-s − 4.61·40-s − 10.0·41-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.437·4-s + 1.08·5-s + 0.377·7-s − 0.673·8-s + 1.29·10-s − 0.522·11-s − 1.42·13-s + 0.453·14-s − 1.24·16-s − 0.206·17-s + 0.524·19-s + 0.474·20-s − 0.626·22-s + 0.159·23-s + 0.173·25-s − 1.70·26-s + 0.165·28-s − 0.330·29-s − 0.110·31-s − 0.820·32-s − 0.247·34-s + 0.409·35-s − 0.0684·37-s + 0.628·38-s − 0.730·40-s − 1.57·41-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 \( 1 - 1.69T + 2T^{2} \)
5 \( 1 - 2.42T + 5T^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + 5.13T + 13T^{2} \)
17 \( 1 + 0.851T + 17T^{2} \)
19 \( 1 - 2.28T + 19T^{2} \)
23 \( 1 - 0.763T + 23T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + 0.617T + 31T^{2} \)
37 \( 1 + 0.416T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + 7.96T + 53T^{2} \)
59 \( 1 - 4.85T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 2.64T + 67T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 - 5.81T + 73T^{2} \)
79 \( 1 + 9.99T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−7.25565032674831305397859011069, −6.64801361382451263871665623710, −5.76024353529775377640935321522, −5.34627760672089172902270298530, −4.83100170369299588961263999427, −4.08945599679165475595532894072, −3.03711807329924805141349893821, −2.48138184763936443956976468822, −1.64506902712139071497672957882, 0, 1.64506902712139071497672957882, 2.48138184763936443956976468822, 3.03711807329924805141349893821, 4.08945599679165475595532894072, 4.83100170369299588961263999427, 5.34627760672089172902270298530, 5.76024353529775377640935321522, 6.64801361382451263871665623710, 7.25565032674831305397859011069

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