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 + 1.32·3-s + 4-s + 2.16·5-s − 1.32·6-s − 2.98·7-s − 8-s − 1.25·9-s − 2.16·10-s + 2.12·11-s + 1.32·12-s + 2.25·13-s + 2.98·14-s + 2.85·15-s + 16-s + 1.94·17-s + 1.25·18-s − 4.50·19-s + 2.16·20-s − 3.94·21-s − 2.12·22-s − 2.25·23-s − 1.32·24-s − 0.312·25-s − 2.25·26-s − 5.62·27-s − 2.98·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.762·3-s + 0.5·4-s + 0.968·5-s − 0.538·6-s − 1.13·7-s − 0.353·8-s − 0.419·9-s − 0.684·10-s + 0.641·11-s + 0.381·12-s + 0.626·13-s + 0.799·14-s + 0.737·15-s + 0.250·16-s + 0.472·17-s + 0.296·18-s − 1.03·19-s + 0.484·20-s − 0.861·21-s − 0.453·22-s − 0.469·23-s − 0.269·24-s − 0.0625·25-s − 0.442·26-s − 1.08·27-s − 0.565·28-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 \( 1 - 1.32T + 3T^{2} \)
5 \( 1 - 2.16T + 5T^{2} \)
7 \( 1 + 2.98T + 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 - 2.25T + 13T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
23 \( 1 + 2.25T + 23T^{2} \)
29 \( 1 + 3.38T + 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + 2.51T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 5.35T + 43T^{2} \)
47 \( 1 - 2.67T + 47T^{2} \)
53 \( 1 + 7.74T + 53T^{2} \)
59 \( 1 - 8.45T + 59T^{2} \)
61 \( 1 + 7.65T + 61T^{2} \)
67 \( 1 - 0.113T + 67T^{2} \)
71 \( 1 - 5.80T + 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 8.08T + 83T^{2} \)
89 \( 1 + 9.85T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.332941854280332126832166159773, −7.45977077372082843160813503027, −6.51912227742671325536221966451, −6.13242863064396602505494373265, −5.38375551722953541186562764052, −3.83365974726459686771584511414, −3.32664459622169101835432620091, −2.30849907494635608727497267921, −1.61172384905982804628450271105, 0, 1.61172384905982804628450271105, 2.30849907494635608727497267921, 3.32664459622169101835432620091, 3.83365974726459686771584511414, 5.38375551722953541186562764052, 6.13242863064396602505494373265, 6.51912227742671325536221966451, 7.45977077372082843160813503027, 8.332941854280332126832166159773

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