Properties

Label 2-4006-1.1-c1-0-115
Degree $2$
Conductor $4006$
Sign $1$
Analytic cond. $31.9880$
Root an. cond. $5.65579$
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  + 2-s + 0.949·3-s + 4-s + 3.98·5-s + 0.949·6-s − 0.436·7-s + 8-s − 2.09·9-s + 3.98·10-s + 5.12·11-s + 0.949·12-s + 7.08·13-s − 0.436·14-s + 3.78·15-s + 16-s − 3.09·17-s − 2.09·18-s + 3.41·19-s + 3.98·20-s − 0.414·21-s + 5.12·22-s − 1.25·23-s + 0.949·24-s + 10.8·25-s + 7.08·26-s − 4.84·27-s − 0.436·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.548·3-s + 0.5·4-s + 1.78·5-s + 0.387·6-s − 0.164·7-s + 0.353·8-s − 0.699·9-s + 1.26·10-s + 1.54·11-s + 0.274·12-s + 1.96·13-s − 0.116·14-s + 0.977·15-s + 0.250·16-s − 0.749·17-s − 0.494·18-s + 0.784·19-s + 0.891·20-s − 0.0904·21-s + 1.09·22-s − 0.262·23-s + 0.193·24-s + 2.17·25-s + 1.39·26-s − 0.931·27-s − 0.0824·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4006\)    =    \(2 \cdot 2003\)
Sign: $1$
Analytic conductor: \(31.9880\)
Root analytic conductor: \(5.65579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.644697060\)
\(L(\frac12)\) \(\approx\) \(5.644697060\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2003 \( 1 + T \)
good3 \( 1 - 0.949T + 3T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 + 0.436T + 7T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 - 7.08T + 13T^{2} \)
17 \( 1 + 3.09T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 + 1.25T + 23T^{2} \)
29 \( 1 + 5.71T + 29T^{2} \)
31 \( 1 + 5.45T + 31T^{2} \)
37 \( 1 + 7.29T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 3.47T + 43T^{2} \)
47 \( 1 + 8.37T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 0.298T + 71T^{2} \)
73 \( 1 + 0.271T + 73T^{2} \)
79 \( 1 + 4.92T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + 6.47T + 97T^{2} \)
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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

−8.726047288724074675592259620918, −7.65803486924739174166006228932, −6.55655403037399128811001176151, −6.10181650198914633868220264142, −5.78025981237789760526991977765, −4.70066374598079395333339068493, −3.60080780746036653012187285381, −3.15165780390321476958038715385, −1.89420310401037806332873808148, −1.46456483337051585370539415343, 1.46456483337051585370539415343, 1.89420310401037806332873808148, 3.15165780390321476958038715385, 3.60080780746036653012187285381, 4.70066374598079395333339068493, 5.78025981237789760526991977765, 6.10181650198914633868220264142, 6.55655403037399128811001176151, 7.65803486924739174166006228932, 8.726047288724074675592259620918

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