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 + 1.84·3-s + 4-s + 1.47·5-s − 1.84·6-s + 1.09·7-s − 8-s + 0.390·9-s − 1.47·10-s − 0.101·11-s + 1.84·12-s − 3.54·13-s − 1.09·14-s + 2.71·15-s + 16-s − 3.92·17-s − 0.390·18-s − 4.99·19-s + 1.47·20-s + 2.01·21-s + 0.101·22-s − 6.77·23-s − 1.84·24-s − 2.83·25-s + 3.54·26-s − 4.80·27-s + 1.09·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.658·5-s − 0.751·6-s + 0.413·7-s − 0.353·8-s + 0.130·9-s − 0.465·10-s − 0.0305·11-s + 0.531·12-s − 0.981·13-s − 0.292·14-s + 0.700·15-s + 0.250·16-s − 0.951·17-s − 0.0919·18-s − 1.14·19-s + 0.329·20-s + 0.440·21-s + 0.0215·22-s − 1.41·23-s − 0.375·24-s − 0.566·25-s + 0.694·26-s − 0.924·27-s + 0.206·28-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 \( 1 - 1.84T + 3T^{2} \)
5 \( 1 - 1.47T + 5T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 + 0.101T + 11T^{2} \)
13 \( 1 + 3.54T + 13T^{2} \)
17 \( 1 + 3.92T + 17T^{2} \)
19 \( 1 + 4.99T + 19T^{2} \)
23 \( 1 + 6.77T + 23T^{2} \)
29 \( 1 - 7.27T + 29T^{2} \)
31 \( 1 - 7.41T + 31T^{2} \)
37 \( 1 - 2.39T + 37T^{2} \)
41 \( 1 + 9.66T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 1.56T + 47T^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + 3.27T + 59T^{2} \)
61 \( 1 + 2.11T + 61T^{2} \)
67 \( 1 - 4.55T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + 1.14T + 73T^{2} \)
79 \( 1 - 4.89T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 4.38T + 89T^{2} \)
97 \( 1 + 4.01T + 97T^{2} \)
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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

−8.256738380996560508460972018078, −7.69565535942828271087430688261, −6.62141508819304850794127950529, −6.18495017721810002474698376357, −5.00123579707009791831052267993, −4.22771397617330078568819993630, −3.05082355926008325027804780994, −2.25656850252401365854856491925, −1.79222727132922896208324961463, 0, 1.79222727132922896208324961463, 2.25656850252401365854856491925, 3.05082355926008325027804780994, 4.22771397617330078568819993630, 5.00123579707009791831052267993, 6.18495017721810002474698376357, 6.62141508819304850794127950529, 7.69565535942828271087430688261, 8.256738380996560508460972018078

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