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 + 0.388·3-s + 4-s − 3.74·5-s − 0.388·6-s − 4.20·7-s − 8-s − 2.84·9-s + 3.74·10-s + 0.0915·11-s + 0.388·12-s + 3.00·13-s + 4.20·14-s − 1.45·15-s + 16-s + 5.06·17-s + 2.84·18-s + 4.20·19-s − 3.74·20-s − 1.63·21-s − 0.0915·22-s + 3.33·23-s − 0.388·24-s + 9.01·25-s − 3.00·26-s − 2.27·27-s − 4.20·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.224·3-s + 0.5·4-s − 1.67·5-s − 0.158·6-s − 1.58·7-s − 0.353·8-s − 0.949·9-s + 1.18·10-s + 0.0276·11-s + 0.112·12-s + 0.834·13-s + 1.12·14-s − 0.375·15-s + 0.250·16-s + 1.22·17-s + 0.671·18-s + 0.963·19-s − 0.837·20-s − 0.356·21-s − 0.0195·22-s + 0.695·23-s − 0.0793·24-s + 1.80·25-s − 0.589·26-s − 0.437·27-s − 0.794·28-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 \( 1 - 0.388T + 3T^{2} \)
5 \( 1 + 3.74T + 5T^{2} \)
7 \( 1 + 4.20T + 7T^{2} \)
11 \( 1 - 0.0915T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 - 4.20T + 19T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 + 3.95T + 29T^{2} \)
31 \( 1 + 0.812T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 + 5.81T + 41T^{2} \)
43 \( 1 - 7.46T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 - 7.02T + 53T^{2} \)
59 \( 1 - 4.12T + 59T^{2} \)
61 \( 1 - 7.36T + 61T^{2} \)
67 \( 1 - 5.07T + 67T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 + 2.39T + 73T^{2} \)
79 \( 1 - 0.844T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.185892810803055103073676286677, −7.42622404370939667927103429825, −6.89960350797601492023522853277, −6.00921597000620937094179976354, −5.22894612705924746093776152160, −3.66500913775029850236588967927, −3.50606434853629879384413520555, −2.76937617364416000497042940170, −0.971902858685020803984390040302, 0, 0.971902858685020803984390040302, 2.76937617364416000497042940170, 3.50606434853629879384413520555, 3.66500913775029850236588967927, 5.22894612705924746093776152160, 6.00921597000620937094179976354, 6.89960350797601492023522853277, 7.42622404370939667927103429825, 8.185892810803055103073676286677

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