Properties

Label 2-4006-1.1-c1-0-14
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 − 0.267·3-s + 4-s − 0.925·5-s + 0.267·6-s − 3.16·7-s − 8-s − 2.92·9-s + 0.925·10-s + 6.35·11-s − 0.267·12-s + 3.11·13-s + 3.16·14-s + 0.247·15-s + 16-s − 5.68·17-s + 2.92·18-s − 7.58·19-s − 0.925·20-s + 0.845·21-s − 6.35·22-s − 7.47·23-s + 0.267·24-s − 4.14·25-s − 3.11·26-s + 1.58·27-s − 3.16·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.154·3-s + 0.5·4-s − 0.414·5-s + 0.109·6-s − 1.19·7-s − 0.353·8-s − 0.976·9-s + 0.292·10-s + 1.91·11-s − 0.0771·12-s + 0.863·13-s + 0.846·14-s + 0.0638·15-s + 0.250·16-s − 1.37·17-s + 0.690·18-s − 1.73·19-s − 0.207·20-s + 0.184·21-s − 1.35·22-s − 1.55·23-s + 0.0545·24-s − 0.828·25-s − 0.610·26-s + 0.304·27-s − 0.598·28-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

\(L(1)\) \(\approx\) \(0.6060638473\)
\(L(\frac12)\) \(\approx\) \(0.6060638473\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.267T + 3T^{2} \)
5 \( 1 + 0.925T + 5T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 - 6.35T + 11T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 + 7.58T + 19T^{2} \)
23 \( 1 + 7.47T + 23T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 - 7.50T + 31T^{2} \)
37 \( 1 + 1.89T + 37T^{2} \)
41 \( 1 - 5.59T + 41T^{2} \)
43 \( 1 - 9.76T + 43T^{2} \)
47 \( 1 + 9.16T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 - 6.13T + 59T^{2} \)
61 \( 1 - 4.28T + 61T^{2} \)
67 \( 1 + 2.73T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 7.30T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 1.92T + 97T^{2} \)
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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

−8.626647134669080372619232325550, −7.963501215166330105424359253942, −6.66192363796835584025730492500, −6.30688812695836640577695729724, −6.11723096625311299720471653453, −4.32937060379581851652280569357, −3.91021178666378793934576166588, −2.88340303545953804208951041165, −1.87401960161153660957918439187, −0.48215519749704538373498486231, 0.48215519749704538373498486231, 1.87401960161153660957918439187, 2.88340303545953804208951041165, 3.91021178666378793934576166588, 4.32937060379581851652280569357, 6.11723096625311299720471653453, 6.30688812695836640577695729724, 6.66192363796835584025730492500, 7.963501215166330105424359253942, 8.626647134669080372619232325550

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