Properties

Degree 2
Conductor $ 2 \cdot 2003 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.91·3-s + 4-s − 1.39·5-s + 1.91·6-s + 3.46·7-s − 8-s + 0.652·9-s + 1.39·10-s − 5.38·11-s − 1.91·12-s + 0.707·13-s − 3.46·14-s + 2.67·15-s + 16-s + 1.30·17-s − 0.652·18-s + 1.25·19-s − 1.39·20-s − 6.62·21-s + 5.38·22-s − 1.63·23-s + 1.91·24-s − 3.04·25-s − 0.707·26-s + 4.48·27-s + 3.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.10·3-s + 0.5·4-s − 0.624·5-s + 0.780·6-s + 1.31·7-s − 0.353·8-s + 0.217·9-s + 0.441·10-s − 1.62·11-s − 0.551·12-s + 0.196·13-s − 0.927·14-s + 0.689·15-s + 0.250·16-s + 0.315·17-s − 0.153·18-s + 0.287·19-s − 0.312·20-s − 1.44·21-s + 1.14·22-s − 0.340·23-s + 0.390·24-s − 0.609·25-s − 0.138·26-s + 0.863·27-s + 0.655·28-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

\( d \)  =  \(2\)
\( N \)  =  \(4006\)    =    \(2 \cdot 2003\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4006} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4006,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;2003\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;2003\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
2003 \( 1 + T \)
good3 \( 1 + 1.91T + 3T^{2} \)
5 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
13 \( 1 - 0.707T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 - 1.25T + 19T^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 - 9.27T + 29T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 + 5.07T + 41T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + 9.82T + 47T^{2} \)
53 \( 1 - 2.68T + 53T^{2} \)
59 \( 1 + 4.53T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 + 7.12T + 67T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 1.68T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 4.62T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.129146925387390203492825972053, −7.60461143813055641389754872174, −6.64444742424468432147445062151, −5.84936444694027856373721378685, −5.04122383286582937255022309137, −4.67740386295752647696242363002, −3.29923494252027306647894563727, −2.26045931126908092584692228526, −1.06811128757607116262477863287, 0, 1.06811128757607116262477863287, 2.26045931126908092584692228526, 3.29923494252027306647894563727, 4.67740386295752647696242363002, 5.04122383286582937255022309137, 5.84936444694027856373721378685, 6.64444742424468432147445062151, 7.60461143813055641389754872174, 8.129146925387390203492825972053

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