Properties

Label 2-4001-1.1-c1-0-123
Degree $2$
Conductor $4001$
Sign $1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
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  − 1.56·2-s − 2.13·3-s + 0.440·4-s + 3.07·5-s + 3.34·6-s + 3.18·7-s + 2.43·8-s + 1.57·9-s − 4.80·10-s + 0.942·11-s − 0.942·12-s + 0.539·13-s − 4.97·14-s − 6.57·15-s − 4.68·16-s + 7.09·17-s − 2.45·18-s − 6.61·19-s + 1.35·20-s − 6.80·21-s − 1.47·22-s + 6.33·23-s − 5.20·24-s + 4.46·25-s − 0.842·26-s + 3.05·27-s + 1.40·28-s + ⋯
L(s)  = 1  − 1.10·2-s − 1.23·3-s + 0.220·4-s + 1.37·5-s + 1.36·6-s + 1.20·7-s + 0.861·8-s + 0.523·9-s − 1.51·10-s + 0.284·11-s − 0.272·12-s + 0.149·13-s − 1.32·14-s − 1.69·15-s − 1.17·16-s + 1.72·17-s − 0.578·18-s − 1.51·19-s + 0.303·20-s − 1.48·21-s − 0.313·22-s + 1.32·23-s − 1.06·24-s + 0.893·25-s − 0.165·26-s + 0.587·27-s + 0.265·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098641635\)
\(L(\frac12)\) \(\approx\) \(1.098641635\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 + 1.56T + 2T^{2} \)
3 \( 1 + 2.13T + 3T^{2} \)
5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
11 \( 1 - 0.942T + 11T^{2} \)
13 \( 1 - 0.539T + 13T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 6.33T + 23T^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 + 3.51T + 31T^{2} \)
37 \( 1 + 5.49T + 37T^{2} \)
41 \( 1 - 1.95T + 41T^{2} \)
43 \( 1 - 6.66T + 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 1.46T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 0.331T + 83T^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.673452447800569796173962897730, −7.81007103130547083692461878090, −6.98872525289580980079080502234, −6.24289845453962706452349007478, −5.40657371628882315496146396826, −5.09036478651301986080304517653, −4.11639579011165614074390438146, −2.45350206028468091272970499548, −1.45826472027421396400312074012, −0.870853432547502287515960925873, 0.870853432547502287515960925873, 1.45826472027421396400312074012, 2.45350206028468091272970499548, 4.11639579011165614074390438146, 5.09036478651301986080304517653, 5.40657371628882315496146396826, 6.24289845453962706452349007478, 6.98872525289580980079080502234, 7.81007103130547083692461878090, 8.673452447800569796173962897730

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