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 − 0.535·3-s + 4-s + 2.84·5-s + 0.535·6-s + 2.46·7-s − 8-s − 2.71·9-s − 2.84·10-s + 3.17·11-s − 0.535·12-s − 7.14·13-s − 2.46·14-s − 1.51·15-s + 16-s + 0.244·17-s + 2.71·18-s − 7.69·19-s + 2.84·20-s − 1.32·21-s − 3.17·22-s + 4.96·23-s + 0.535·24-s + 3.06·25-s + 7.14·26-s + 3.05·27-s + 2.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.308·3-s + 0.5·4-s + 1.27·5-s + 0.218·6-s + 0.932·7-s − 0.353·8-s − 0.904·9-s − 0.898·10-s + 0.958·11-s − 0.154·12-s − 1.98·13-s − 0.659·14-s − 0.392·15-s + 0.250·16-s + 0.0593·17-s + 0.639·18-s − 1.76·19-s + 0.635·20-s − 0.288·21-s − 0.677·22-s + 1.03·23-s + 0.109·24-s + 0.613·25-s + 1.40·26-s + 0.588·27-s + 0.466·28-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 \( 1 + 0.535T + 3T^{2} \)
5 \( 1 - 2.84T + 5T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 - 3.17T + 11T^{2} \)
13 \( 1 + 7.14T + 13T^{2} \)
17 \( 1 - 0.244T + 17T^{2} \)
19 \( 1 + 7.69T + 19T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 - 3.48T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 + 3.01T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 2.04T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 4.34T + 71T^{2} \)
73 \( 1 - 8.56T + 73T^{2} \)
79 \( 1 + 1.56T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 8.19T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−8.172583971396913734107067711592, −7.40231758295501917389098815603, −6.49730232678762255449909584408, −6.08779847137843223454447507355, −5.04915871662088916127020425169, −4.64974364593150345731112473389, −3.04058495849549129521982221578, −2.17785980610808115840767010019, −1.54108554403060096164386657849, 0, 1.54108554403060096164386657849, 2.17785980610808115840767010019, 3.04058495849549129521982221578, 4.64974364593150345731112473389, 5.04915871662088916127020425169, 6.08779847137843223454447507355, 6.49730232678762255449909584408, 7.40231758295501917389098815603, 8.172583971396913734107067711592

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