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 + 3.27·3-s + 4-s − 1.91·5-s − 3.27·6-s + 1.09·7-s − 8-s + 7.75·9-s + 1.91·10-s − 3.44·11-s + 3.27·12-s − 2.45·13-s − 1.09·14-s − 6.26·15-s + 16-s − 5.97·17-s − 7.75·18-s − 1.07·19-s − 1.91·20-s + 3.60·21-s + 3.44·22-s − 5.86·23-s − 3.27·24-s − 1.34·25-s + 2.45·26-s + 15.6·27-s + 1.09·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.89·3-s + 0.5·4-s − 0.854·5-s − 1.33·6-s + 0.415·7-s − 0.353·8-s + 2.58·9-s + 0.604·10-s − 1.04·11-s + 0.946·12-s − 0.681·13-s − 0.293·14-s − 1.61·15-s + 0.250·16-s − 1.44·17-s − 1.82·18-s − 0.246·19-s − 0.427·20-s + 0.785·21-s + 0.735·22-s − 1.22·23-s − 0.669·24-s − 0.269·25-s + 0.481·26-s + 3.00·27-s + 0.207·28-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 \( 1 - 3.27T + 3T^{2} \)
5 \( 1 + 1.91T + 5T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 + 2.45T + 13T^{2} \)
17 \( 1 + 5.97T + 17T^{2} \)
19 \( 1 + 1.07T + 19T^{2} \)
23 \( 1 + 5.86T + 23T^{2} \)
29 \( 1 + 1.05T + 29T^{2} \)
31 \( 1 + 0.801T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 + 4.64T + 59T^{2} \)
61 \( 1 - 9.17T + 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 + 7.43T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 2.91T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 - 2.04T + 97T^{2} \)
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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

−8.222572052195519842697016332062, −7.59556349338778661761448758012, −7.22851162395292851583105186227, −6.11790400843967352108269794672, −4.61365562899224692133134846205, −4.20786643805764914797661819530, −3.12690351900689759270757262621, −2.43245231821408606360650129780, −1.75804806154337297338365350004, 0, 1.75804806154337297338365350004, 2.43245231821408606360650129780, 3.12690351900689759270757262621, 4.20786643805764914797661819530, 4.61365562899224692133134846205, 6.11790400843967352108269794672, 7.22851162395292851583105186227, 7.59556349338778661761448758012, 8.222572052195519842697016332062

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