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 − 0.941·3-s + 4-s − 0.129·5-s + 0.941·6-s + 0.350·7-s − 8-s − 2.11·9-s + 0.129·10-s + 3.66·11-s − 0.941·12-s − 0.342·13-s − 0.350·14-s + 0.122·15-s + 16-s − 0.324·17-s + 2.11·18-s − 0.916·19-s − 0.129·20-s − 0.330·21-s − 3.66·22-s + 1.25·23-s + 0.941·24-s − 4.98·25-s + 0.342·26-s + 4.81·27-s + 0.350·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.543·3-s + 0.5·4-s − 0.0580·5-s + 0.384·6-s + 0.132·7-s − 0.353·8-s − 0.704·9-s + 0.0410·10-s + 1.10·11-s − 0.271·12-s − 0.0950·13-s − 0.0936·14-s + 0.0315·15-s + 0.250·16-s − 0.0787·17-s + 0.498·18-s − 0.210·19-s − 0.0290·20-s − 0.0720·21-s − 0.782·22-s + 0.260·23-s + 0.192·24-s − 0.996·25-s + 0.0671·26-s + 0.926·27-s + 0.0662·28-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 \( 1 + 0.941T + 3T^{2} \)
5 \( 1 + 0.129T + 5T^{2} \)
7 \( 1 - 0.350T + 7T^{2} \)
11 \( 1 - 3.66T + 11T^{2} \)
13 \( 1 + 0.342T + 13T^{2} \)
17 \( 1 + 0.324T + 17T^{2} \)
19 \( 1 + 0.916T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 - 2.88T + 29T^{2} \)
31 \( 1 + 8.36T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 8.00T + 43T^{2} \)
47 \( 1 + 2.84T + 47T^{2} \)
53 \( 1 - 7.31T + 53T^{2} \)
59 \( 1 - 3.01T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 2.69T + 67T^{2} \)
71 \( 1 - 5.69T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 1.47T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−8.293509190203989313225220955184, −7.31624233206908236194968170568, −6.64549006855110791232670068864, −5.98861634172547277951322761119, −5.28010985734822988991017366369, −4.24866340631370994936786425267, −3.35581691948260380144503998827, −2.27442615285556702615792180700, −1.22214239555870270671850095324, 0, 1.22214239555870270671850095324, 2.27442615285556702615792180700, 3.35581691948260380144503998827, 4.24866340631370994936786425267, 5.28010985734822988991017366369, 5.98861634172547277951322761119, 6.64549006855110791232670068864, 7.31624233206908236194968170568, 8.293509190203989313225220955184

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