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 + 2.60·3-s + 4-s + 1.01·5-s − 2.60·6-s − 1.30·7-s − 8-s + 3.76·9-s − 1.01·10-s − 4.91·11-s + 2.60·12-s + 1.88·13-s + 1.30·14-s + 2.64·15-s + 16-s − 1.77·17-s − 3.76·18-s − 6.72·19-s + 1.01·20-s − 3.39·21-s + 4.91·22-s + 3.89·23-s − 2.60·24-s − 3.96·25-s − 1.88·26-s + 1.98·27-s − 1.30·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.454·5-s − 1.06·6-s − 0.493·7-s − 0.353·8-s + 1.25·9-s − 0.321·10-s − 1.48·11-s + 0.750·12-s + 0.522·13-s + 0.348·14-s + 0.682·15-s + 0.250·16-s − 0.430·17-s − 0.887·18-s − 1.54·19-s + 0.227·20-s − 0.740·21-s + 1.04·22-s + 0.812·23-s − 0.530·24-s − 0.793·25-s − 0.369·26-s + 0.382·27-s − 0.246·28-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 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 - 1.01T + 5T^{2} \)
7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
13 \( 1 - 1.88T + 13T^{2} \)
17 \( 1 + 1.77T + 17T^{2} \)
19 \( 1 + 6.72T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + 1.47T + 29T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 + 8.51T + 37T^{2} \)
41 \( 1 - 5.85T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + 1.16T + 47T^{2} \)
53 \( 1 + 3.42T + 53T^{2} \)
59 \( 1 + 3.61T + 59T^{2} \)
61 \( 1 - 5.74T + 61T^{2} \)
67 \( 1 + 4.91T + 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 - 4.06T + 73T^{2} \)
79 \( 1 - 8.00T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 - 5.76T + 89T^{2} \)
97 \( 1 + 0.273T + 97T^{2} \)
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

−8.188076661975975415402840504622, −7.66432264489819433690707183116, −6.83147042943419810074204278992, −6.07272353488892275255586418001, −5.10263284107291702994283025595, −3.94436106517796089421325444123, −3.12687210657045987665392961153, −2.38590891131306228364900738243, −1.77189370799285264516078228087, 0, 1.77189370799285264516078228087, 2.38590891131306228364900738243, 3.12687210657045987665392961153, 3.94436106517796089421325444123, 5.10263284107291702994283025595, 6.07272353488892275255586418001, 6.83147042943419810074204278992, 7.66432264489819433690707183116, 8.188076661975975415402840504622

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