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 + 1.92·3-s + 4-s − 0.783·5-s − 1.92·6-s + 1.63·7-s − 8-s + 0.706·9-s + 0.783·10-s − 5.84·11-s + 1.92·12-s + 4.41·13-s − 1.63·14-s − 1.50·15-s + 16-s + 3.42·17-s − 0.706·18-s + 3.30·19-s − 0.783·20-s + 3.15·21-s + 5.84·22-s − 8.31·23-s − 1.92·24-s − 4.38·25-s − 4.41·26-s − 4.41·27-s + 1.63·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.11·3-s + 0.5·4-s − 0.350·5-s − 0.785·6-s + 0.619·7-s − 0.353·8-s + 0.235·9-s + 0.247·10-s − 1.76·11-s + 0.555·12-s + 1.22·13-s − 0.438·14-s − 0.389·15-s + 0.250·16-s + 0.829·17-s − 0.166·18-s + 0.757·19-s − 0.175·20-s + 0.688·21-s + 1.24·22-s − 1.73·23-s − 0.392·24-s − 0.877·25-s − 0.865·26-s − 0.849·27-s + 0.309·28-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 \( 1 - 1.92T + 3T^{2} \)
5 \( 1 + 0.783T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 5.84T + 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 + 8.31T + 23T^{2} \)
29 \( 1 + 5.42T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 7.48T + 43T^{2} \)
47 \( 1 + 2.15T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 4.96T + 59T^{2} \)
61 \( 1 - 3.45T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 1.81T + 73T^{2} \)
79 \( 1 - 0.540T + 79T^{2} \)
83 \( 1 + 5.02T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.106063317220931331609191123105, −7.900115851494475541137440833434, −6.98022616727778777702058773261, −5.72553769967932499612746927945, −5.31310428209167261615542884234, −3.85568188403231959537060559597, −3.36279515885668037683809014798, −2.34392002123914382194509100590, −1.60471075964663672251564019062, 0, 1.60471075964663672251564019062, 2.34392002123914382194509100590, 3.36279515885668037683809014798, 3.85568188403231959537060559597, 5.31310428209167261615542884234, 5.72553769967932499612746927945, 6.98022616727778777702058773261, 7.900115851494475541137440833434, 8.106063317220931331609191123105

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