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 + 2.02·3-s + 4-s − 3.63·5-s − 2.02·6-s + 2.67·7-s − 8-s + 1.09·9-s + 3.63·10-s + 1.35·11-s + 2.02·12-s + 0.698·13-s − 2.67·14-s − 7.35·15-s + 16-s − 2.67·17-s − 1.09·18-s + 5.59·19-s − 3.63·20-s + 5.41·21-s − 1.35·22-s − 8.27·23-s − 2.02·24-s + 8.21·25-s − 0.698·26-s − 3.86·27-s + 2.67·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.16·3-s + 0.5·4-s − 1.62·5-s − 0.825·6-s + 1.01·7-s − 0.353·8-s + 0.363·9-s + 1.14·10-s + 0.407·11-s + 0.583·12-s + 0.193·13-s − 0.715·14-s − 1.89·15-s + 0.250·16-s − 0.648·17-s − 0.257·18-s + 1.28·19-s − 0.812·20-s + 1.18·21-s − 0.287·22-s − 1.72·23-s − 0.412·24-s + 1.64·25-s − 0.136·26-s − 0.743·27-s + 0.505·28-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 \( 1 - 2.02T + 3T^{2} \)
5 \( 1 + 3.63T + 5T^{2} \)
7 \( 1 - 2.67T + 7T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 - 0.698T + 13T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 - 5.59T + 19T^{2} \)
23 \( 1 + 8.27T + 23T^{2} \)
29 \( 1 + 4.22T + 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 0.202T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 5.05T + 47T^{2} \)
53 \( 1 + 9.65T + 53T^{2} \)
59 \( 1 - 3.66T + 59T^{2} \)
61 \( 1 - 0.421T + 61T^{2} \)
67 \( 1 - 7.87T + 67T^{2} \)
71 \( 1 + 5.64T + 71T^{2} \)
73 \( 1 + 8.92T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 9.01T + 89T^{2} \)
97 \( 1 + 5.97T + 97T^{2} \)
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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

−8.078259142341430260810730613620, −7.55367286783786034878875339749, −7.26373196107643254782904630484, −5.97065875613523946621980815479, −4.89488094497907759815953064887, −3.83677876875278966650060777432, −3.57862309334043812214895923577, −2.38378663548664788009525061038, −1.47644253471436725255526731398, 0, 1.47644253471436725255526731398, 2.38378663548664788009525061038, 3.57862309334043812214895923577, 3.83677876875278966650060777432, 4.89488094497907759815953064887, 5.97065875613523946621980815479, 7.26373196107643254782904630484, 7.55367286783786034878875339749, 8.078259142341430260810730613620

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