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 − 0.956·3-s + 4-s − 0.0953·5-s + 0.956·6-s − 4.20·7-s − 8-s − 2.08·9-s + 0.0953·10-s − 0.121·11-s − 0.956·12-s − 1.63·13-s + 4.20·14-s + 0.0911·15-s + 16-s + 3.22·17-s + 2.08·18-s − 1.33·19-s − 0.0953·20-s + 4.02·21-s + 0.121·22-s + 7.51·23-s + 0.956·24-s − 4.99·25-s + 1.63·26-s + 4.86·27-s − 4.20·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.552·3-s + 0.5·4-s − 0.0426·5-s + 0.390·6-s − 1.58·7-s − 0.353·8-s − 0.694·9-s + 0.0301·10-s − 0.0366·11-s − 0.276·12-s − 0.453·13-s + 1.12·14-s + 0.0235·15-s + 0.250·16-s + 0.781·17-s + 0.491·18-s − 0.307·19-s − 0.0213·20-s + 0.878·21-s + 0.0259·22-s + 1.56·23-s + 0.195·24-s − 0.998·25-s + 0.320·26-s + 0.936·27-s − 0.794·28-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 \( 1 + 0.956T + 3T^{2} \)
5 \( 1 + 0.0953T + 5T^{2} \)
7 \( 1 + 4.20T + 7T^{2} \)
11 \( 1 + 0.121T + 11T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 + 1.33T + 19T^{2} \)
23 \( 1 - 7.51T + 23T^{2} \)
29 \( 1 - 9.09T + 29T^{2} \)
31 \( 1 - 1.34T + 31T^{2} \)
37 \( 1 + 0.847T + 37T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 - 6.69T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 + 3.92T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 - 5.68T + 71T^{2} \)
73 \( 1 - 0.786T + 73T^{2} \)
79 \( 1 + 1.19T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 5.77T + 89T^{2} \)
97 \( 1 - 4.53T + 97T^{2} \)
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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

−8.135530260218022035547418575549, −7.24893646284680643260603239676, −6.66184853418188082744164916439, −5.97624789093461912837587342322, −5.39114047057030885342063642568, −4.22772183149786087778981095997, −3.04463585073104024336303867832, −2.69107197553148783513230364722, −1.00080182422731033960325648561, 0, 1.00080182422731033960325648561, 2.69107197553148783513230364722, 3.04463585073104024336303867832, 4.22772183149786087778981095997, 5.39114047057030885342063642568, 5.97624789093461912837587342322, 6.66184853418188082744164916439, 7.24893646284680643260603239676, 8.135530260218022035547418575549

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