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 + 1.55·3-s + 4-s − 1.55·5-s − 1.55·6-s + 1.46·7-s − 8-s − 0.586·9-s + 1.55·10-s + 1.61·11-s + 1.55·12-s − 1.39·13-s − 1.46·14-s − 2.41·15-s + 16-s − 0.623·17-s + 0.586·18-s + 1.68·19-s − 1.55·20-s + 2.27·21-s − 1.61·22-s + 1.66·23-s − 1.55·24-s − 2.58·25-s + 1.39·26-s − 5.57·27-s + 1.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.896·3-s + 0.5·4-s − 0.695·5-s − 0.634·6-s + 0.553·7-s − 0.353·8-s − 0.195·9-s + 0.491·10-s + 0.488·11-s + 0.448·12-s − 0.388·13-s − 0.391·14-s − 0.623·15-s + 0.250·16-s − 0.151·17-s + 0.138·18-s + 0.385·19-s − 0.347·20-s + 0.496·21-s − 0.345·22-s + 0.347·23-s − 0.317·24-s − 0.516·25-s + 0.274·26-s − 1.07·27-s + 0.276·28-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 \( 1 - 1.55T + 3T^{2} \)
5 \( 1 + 1.55T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + 1.39T + 13T^{2} \)
17 \( 1 + 0.623T + 17T^{2} \)
19 \( 1 - 1.68T + 19T^{2} \)
23 \( 1 - 1.66T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 2.33T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 - 5.35T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 + 8.95T + 47T^{2} \)
53 \( 1 - 1.11T + 53T^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 5.97T + 67T^{2} \)
71 \( 1 + 8.75T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 8.36T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 3.11T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−7.937998125078021724355511662800, −7.77919315761363982422102879745, −6.93875444082635784858646455477, −5.97782914536266725465812626877, −5.04777489551949568944961138408, −4.01141568637446817103256055027, −3.33077281647394750874332692785, −2.39198856640236127526460743997, −1.48858235303623212569325405459, 0, 1.48858235303623212569325405459, 2.39198856640236127526460743997, 3.33077281647394750874332692785, 4.01141568637446817103256055027, 5.04777489551949568944961138408, 5.97782914536266725465812626877, 6.93875444082635784858646455477, 7.77919315761363982422102879745, 7.937998125078021724355511662800

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