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 + 1.65·3-s + 4-s + 2.27·5-s − 1.65·6-s − 2.18·7-s − 8-s − 0.250·9-s − 2.27·10-s + 0.389·11-s + 1.65·12-s − 6.44·13-s + 2.18·14-s + 3.77·15-s + 16-s + 4.88·17-s + 0.250·18-s + 1.82·19-s + 2.27·20-s − 3.62·21-s − 0.389·22-s − 3.56·23-s − 1.65·24-s + 0.190·25-s + 6.44·26-s − 5.38·27-s − 2.18·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.957·3-s + 0.5·4-s + 1.01·5-s − 0.676·6-s − 0.825·7-s − 0.353·8-s − 0.0834·9-s − 0.720·10-s + 0.117·11-s + 0.478·12-s − 1.78·13-s + 0.583·14-s + 0.975·15-s + 0.250·16-s + 1.18·17-s + 0.0590·18-s + 0.418·19-s + 0.509·20-s − 0.790·21-s − 0.0831·22-s − 0.742·23-s − 0.338·24-s + 0.0381·25-s + 1.26·26-s − 1.03·27-s − 0.412·28-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 \( 1 - 1.65T + 3T^{2} \)
5 \( 1 - 2.27T + 5T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 - 0.389T + 11T^{2} \)
13 \( 1 + 6.44T + 13T^{2} \)
17 \( 1 - 4.88T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 + 3.56T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 + 6.39T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 - 3.34T + 43T^{2} \)
47 \( 1 - 4.52T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 7.73T + 59T^{2} \)
61 \( 1 - 4.63T + 61T^{2} \)
67 \( 1 - 0.0366T + 67T^{2} \)
71 \( 1 + 7.83T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.225043559593015114960904963521, −7.29437663011100933299564843053, −7.01871319737111254511145730895, −5.63756928889891243694868085769, −5.56959258356456931197641766775, −3.99442581235636113209199141481, −3.02344593515506906458017434527, −2.48428096510741536613668941381, −1.62574330170056375554805070870, 0, 1.62574330170056375554805070870, 2.48428096510741536613668941381, 3.02344593515506906458017434527, 3.99442581235636113209199141481, 5.56959258356456931197641766775, 5.63756928889891243694868085769, 7.01871319737111254511145730895, 7.29437663011100933299564843053, 8.225043559593015114960904963521

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