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 + 1.81·3-s + 4-s + 1.21·5-s − 1.81·6-s − 2.11·7-s − 8-s + 0.288·9-s − 1.21·10-s + 1.44·11-s + 1.81·12-s − 1.15·13-s + 2.11·14-s + 2.19·15-s + 16-s − 2.16·17-s − 0.288·18-s + 4.58·19-s + 1.21·20-s − 3.82·21-s − 1.44·22-s − 3.56·23-s − 1.81·24-s − 3.53·25-s + 1.15·26-s − 4.91·27-s − 2.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.541·5-s − 0.740·6-s − 0.798·7-s − 0.353·8-s + 0.0962·9-s − 0.383·10-s + 0.437·11-s + 0.523·12-s − 0.319·13-s + 0.564·14-s + 0.567·15-s + 0.250·16-s − 0.524·17-s − 0.0680·18-s + 1.05·19-s + 0.270·20-s − 0.835·21-s − 0.309·22-s − 0.742·23-s − 0.370·24-s − 0.706·25-s + 0.226·26-s − 0.946·27-s − 0.399·28-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 \( 1 - 1.81T + 3T^{2} \)
5 \( 1 - 1.21T + 5T^{2} \)
7 \( 1 + 2.11T + 7T^{2} \)
11 \( 1 - 1.44T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + 2.16T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
23 \( 1 + 3.56T + 23T^{2} \)
29 \( 1 + 2.53T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 - 1.48T + 37T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
43 \( 1 - 0.511T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 + 6.75T + 59T^{2} \)
61 \( 1 + 0.358T + 61T^{2} \)
67 \( 1 + 1.39T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 - 2.26T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 - 9.24T + 97T^{2} \)
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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

−8.083949925666564622452432523871, −7.60691929773381507769559239548, −6.72453310074470754336650408625, −6.06528790770275680395669553983, −5.23900234466838921705715983379, −3.88673129526390236234258102628, −3.25029164417035926068921482416, −2.37924492825677895459739898670, −1.62594757864617393046661277163, 0, 1.62594757864617393046661277163, 2.37924492825677895459739898670, 3.25029164417035926068921482416, 3.88673129526390236234258102628, 5.23900234466838921705715983379, 6.06528790770275680395669553983, 6.72453310074470754336650408625, 7.60691929773381507769559239548, 8.083949925666564622452432523871

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