Properties

Label 2-4006-1.1-c1-0-125
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 + 3.41·3-s + 4-s + 3.53·5-s + 3.41·6-s − 2.95·7-s + 8-s + 8.65·9-s + 3.53·10-s − 1.60·11-s + 3.41·12-s − 2.27·13-s − 2.95·14-s + 12.0·15-s + 16-s − 0.568·17-s + 8.65·18-s − 1.94·19-s + 3.53·20-s − 10.0·21-s − 1.60·22-s + 5.52·23-s + 3.41·24-s + 7.52·25-s − 2.27·26-s + 19.3·27-s − 2.95·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.97·3-s + 0.5·4-s + 1.58·5-s + 1.39·6-s − 1.11·7-s + 0.353·8-s + 2.88·9-s + 1.11·10-s − 0.483·11-s + 0.985·12-s − 0.631·13-s − 0.789·14-s + 3.11·15-s + 0.250·16-s − 0.137·17-s + 2.03·18-s − 0.446·19-s + 0.791·20-s − 2.20·21-s − 0.341·22-s + 1.15·23-s + 0.696·24-s + 1.50·25-s − 0.446·26-s + 3.71·27-s − 0.558·28-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

\(L(1)\) \(\approx\) \(7.376042911\)
\(L(\frac12)\) \(\approx\) \(7.376042911\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 + 2.95T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + 2.27T + 13T^{2} \)
17 \( 1 + 0.568T + 17T^{2} \)
19 \( 1 + 1.94T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 + 8.68T + 29T^{2} \)
31 \( 1 + 3.38T + 31T^{2} \)
37 \( 1 + 2.20T + 37T^{2} \)
41 \( 1 - 7.24T + 41T^{2} \)
43 \( 1 - 8.72T + 43T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 + 5.94T + 61T^{2} \)
67 \( 1 - 3.53T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 2.38T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 6.46T + 83T^{2} \)
89 \( 1 - 2.75T + 89T^{2} \)
97 \( 1 + 0.390T + 97T^{2} \)
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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

−8.699404511724924642441691074100, −7.38212433841541502096170038971, −7.24747026284863255883289160791, −6.21567265056113881947988217793, −5.50353753718438885992972356155, −4.47487978753139006363193326405, −3.63533670255588419292886239790, −2.69773564863659707208538190060, −2.48951310146209370548028271615, −1.51806279573173971882050964151, 1.51806279573173971882050964151, 2.48951310146209370548028271615, 2.69773564863659707208538190060, 3.63533670255588419292886239790, 4.47487978753139006363193326405, 5.50353753718438885992972356155, 6.21567265056113881947988217793, 7.24747026284863255883289160791, 7.38212433841541502096170038971, 8.699404511724924642441691074100

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