Properties

Label 2-8001-1.1-c1-0-247
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  + 0.691·2-s − 1.52·4-s + 0.236·5-s + 7-s − 2.43·8-s + 0.163·10-s + 4.73·11-s − 1.51·13-s + 0.691·14-s + 1.35·16-s − 4.88·17-s + 2.13·19-s − 0.360·20-s + 3.27·22-s − 1.62·23-s − 4.94·25-s − 1.05·26-s − 1.52·28-s + 0.481·29-s − 9.10·31-s + 5.81·32-s − 3.37·34-s + 0.236·35-s + 10.9·37-s + 1.47·38-s − 0.576·40-s + 4.78·41-s + ⋯
L(s)  = 1  + 0.489·2-s − 0.760·4-s + 0.105·5-s + 0.377·7-s − 0.861·8-s + 0.0517·10-s + 1.42·11-s − 0.421·13-s + 0.184·14-s + 0.339·16-s − 1.18·17-s + 0.490·19-s − 0.0805·20-s + 0.697·22-s − 0.337·23-s − 0.988·25-s − 0.206·26-s − 0.287·28-s + 0.0894·29-s − 1.63·31-s + 1.02·32-s − 0.579·34-s + 0.0400·35-s + 1.80·37-s + 0.239·38-s − 0.0911·40-s + 0.746·41-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 \( 1 - 0.691T + 2T^{2} \)
5 \( 1 - 0.236T + 5T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + 1.62T + 23T^{2} \)
29 \( 1 - 0.481T + 29T^{2} \)
31 \( 1 + 9.10T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 - 4.78T + 41T^{2} \)
43 \( 1 - 1.26T + 43T^{2} \)
47 \( 1 + 6.43T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 7.68T + 59T^{2} \)
61 \( 1 - 5.26T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 2.66T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 9.66T + 97T^{2} \)
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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

−7.50307010205427591832263403108, −6.65061649402412208465009787998, −6.00305982061410108299613073708, −5.37950974133863637072081978063, −4.42643415634495477655977330659, −4.16311630772601057797788855158, −3.32167762417548066963593158872, −2.26393676121810750621959709920, −1.28736542700736629530522306373, 0, 1.28736542700736629530522306373, 2.26393676121810750621959709920, 3.32167762417548066963593158872, 4.16311630772601057797788855158, 4.42643415634495477655977330659, 5.37950974133863637072081978063, 6.00305982061410108299613073708, 6.65061649402412208465009787998, 7.50307010205427591832263403108

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