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 − 2.30·3-s + 4-s + 3.88·5-s + 2.30·6-s + 0.245·7-s − 8-s + 2.31·9-s − 3.88·10-s − 2.29·11-s − 2.30·12-s − 5.34·13-s − 0.245·14-s − 8.95·15-s + 16-s − 2.60·17-s − 2.31·18-s + 8.55·19-s + 3.88·20-s − 0.565·21-s + 2.29·22-s + 2.94·23-s + 2.30·24-s + 10.0·25-s + 5.34·26-s + 1.57·27-s + 0.245·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.33·3-s + 0.5·4-s + 1.73·5-s + 0.941·6-s + 0.0927·7-s − 0.353·8-s + 0.772·9-s − 1.22·10-s − 0.693·11-s − 0.665·12-s − 1.48·13-s − 0.0655·14-s − 2.31·15-s + 0.250·16-s − 0.632·17-s − 0.546·18-s + 1.96·19-s + 0.867·20-s − 0.123·21-s + 0.490·22-s + 0.614·23-s + 0.470·24-s + 2.01·25-s + 1.04·26-s + 0.302·27-s + 0.0463·28-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 \( 1 + 2.30T + 3T^{2} \)
5 \( 1 - 3.88T + 5T^{2} \)
7 \( 1 - 0.245T + 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 + 5.34T + 13T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 - 8.55T + 19T^{2} \)
23 \( 1 - 2.94T + 23T^{2} \)
29 \( 1 + 7.74T + 29T^{2} \)
31 \( 1 + 5.79T + 31T^{2} \)
37 \( 1 - 9.58T + 37T^{2} \)
41 \( 1 + 6.77T + 41T^{2} \)
43 \( 1 - 3.09T + 43T^{2} \)
47 \( 1 + 7.31T + 47T^{2} \)
53 \( 1 - 5.03T + 53T^{2} \)
59 \( 1 + 8.17T + 59T^{2} \)
61 \( 1 + 8.95T + 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 7.56T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 7.26T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 7.83T + 97T^{2} \)
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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

−7.943891175568409750762787670943, −7.21014254914755489039050013927, −6.60776311005909655905090151438, −5.75451040488116062525342182136, −5.29861286767664033773068828677, −4.84960944413960310428818144839, −3.03968238586360065992573419159, −2.20569969317452194078790432126, −1.27561758885413168962428294600, 0, 1.27561758885413168962428294600, 2.20569969317452194078790432126, 3.03968238586360065992573419159, 4.84960944413960310428818144839, 5.29861286767664033773068828677, 5.75451040488116062525342182136, 6.60776311005909655905090151438, 7.21014254914755489039050013927, 7.943891175568409750762787670943

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