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 + 2.91·3-s + 4-s − 4.29·5-s − 2.91·6-s − 2.38·7-s − 8-s + 5.47·9-s + 4.29·10-s + 1.43·11-s + 2.91·12-s + 1.78·13-s + 2.38·14-s − 12.4·15-s + 16-s + 4.38·17-s − 5.47·18-s − 4.76·19-s − 4.29·20-s − 6.93·21-s − 1.43·22-s − 6.83·23-s − 2.91·24-s + 13.4·25-s − 1.78·26-s + 7.19·27-s − 2.38·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.68·3-s + 0.5·4-s − 1.92·5-s − 1.18·6-s − 0.900·7-s − 0.353·8-s + 1.82·9-s + 1.35·10-s + 0.433·11-s + 0.840·12-s + 0.494·13-s + 0.636·14-s − 3.22·15-s + 0.250·16-s + 1.06·17-s − 1.28·18-s − 1.09·19-s − 0.960·20-s − 1.51·21-s − 0.306·22-s − 1.42·23-s − 0.594·24-s + 2.68·25-s − 0.349·26-s + 1.38·27-s − 0.450·28-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 \( 1 - 2.91T + 3T^{2} \)
5 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
13 \( 1 - 1.78T + 13T^{2} \)
17 \( 1 - 4.38T + 17T^{2} \)
19 \( 1 + 4.76T + 19T^{2} \)
23 \( 1 + 6.83T + 23T^{2} \)
29 \( 1 - 7.89T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 3.64T + 37T^{2} \)
41 \( 1 + 0.826T + 41T^{2} \)
43 \( 1 + 7.98T + 43T^{2} \)
47 \( 1 + 8.14T + 47T^{2} \)
53 \( 1 + 5.76T + 53T^{2} \)
59 \( 1 - 0.874T + 59T^{2} \)
61 \( 1 + 9.10T + 61T^{2} \)
67 \( 1 - 0.114T + 67T^{2} \)
71 \( 1 - 0.382T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 + 5.90T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.208987668265899568201757751252, −7.80984507456013698255338598407, −6.83266965088097703554337536735, −6.40973483092789814091627440197, −4.66639410568035095070915662689, −3.77419042173062359100600117635, −3.44278031564409304305359112578, −2.70294199291119991156076250626, −1.39693982209664861377455756516, 0, 1.39693982209664861377455756516, 2.70294199291119991156076250626, 3.44278031564409304305359112578, 3.77419042173062359100600117635, 4.66639410568035095070915662689, 6.40973483092789814091627440197, 6.83266965088097703554337536735, 7.80984507456013698255338598407, 8.208987668265899568201757751252

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