Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.84·3-s + 4-s + 2.20·5-s + 1.84·6-s − 0.412·7-s − 8-s + 0.405·9-s − 2.20·10-s + 0.613·11-s − 1.84·12-s + 3.23·13-s + 0.412·14-s − 4.06·15-s + 16-s − 2.81·17-s − 0.405·18-s − 1.83·19-s + 2.20·20-s + 0.761·21-s − 0.613·22-s + 2.84·23-s + 1.84·24-s − 0.141·25-s − 3.23·26-s + 4.78·27-s − 0.412·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.985·5-s + 0.753·6-s − 0.156·7-s − 0.353·8-s + 0.135·9-s − 0.696·10-s + 0.185·11-s − 0.532·12-s + 0.898·13-s + 0.110·14-s − 1.05·15-s + 0.250·16-s − 0.681·17-s − 0.0956·18-s − 0.421·19-s + 0.492·20-s + 0.166·21-s − 0.130·22-s + 0.593·23-s + 0.376·24-s − 0.0283·25-s − 0.634·26-s + 0.921·27-s − 0.0780·28-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 \( 1 + 1.84T + 3T^{2} \)
5 \( 1 - 2.20T + 5T^{2} \)
7 \( 1 + 0.412T + 7T^{2} \)
11 \( 1 - 0.613T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
19 \( 1 + 1.83T + 19T^{2} \)
23 \( 1 - 2.84T + 23T^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 - 2.50T + 43T^{2} \)
47 \( 1 + 3.96T + 47T^{2} \)
53 \( 1 + 5.16T + 53T^{2} \)
59 \( 1 + 2.37T + 59T^{2} \)
61 \( 1 + 0.510T + 61T^{2} \)
67 \( 1 - 2.29T + 67T^{2} \)
71 \( 1 + 4.88T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 7.76T + 79T^{2} \)
83 \( 1 + 0.852T + 83T^{2} \)
89 \( 1 + 0.595T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.190659586447072854379774019789, −7.20986736697921930516837427720, −6.41927680947208481183212261407, −6.03407414744055646047202609406, −5.40702400580684156557739157243, −4.43353979465655324588913288288, −3.27129726475285822376056557604, −2.15427456616827587913513530102, −1.26482455512136247382591424744, 0, 1.26482455512136247382591424744, 2.15427456616827587913513530102, 3.27129726475285822376056557604, 4.43353979465655324588913288288, 5.40702400580684156557739157243, 6.03407414744055646047202609406, 6.41927680947208481183212261407, 7.20986736697921930516837427720, 8.190659586447072854379774019789

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