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 − 2.86·3-s + 4-s − 2.90·5-s + 2.86·6-s + 0.924·7-s − 8-s + 5.18·9-s + 2.90·10-s − 1.96·11-s − 2.86·12-s + 4.75·13-s − 0.924·14-s + 8.31·15-s + 16-s − 3.12·17-s − 5.18·18-s − 2.40·19-s − 2.90·20-s − 2.64·21-s + 1.96·22-s − 5.24·23-s + 2.86·24-s + 3.45·25-s − 4.75·26-s − 6.24·27-s + 0.924·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.65·3-s + 0.5·4-s − 1.30·5-s + 1.16·6-s + 0.349·7-s − 0.353·8-s + 1.72·9-s + 0.919·10-s − 0.591·11-s − 0.825·12-s + 1.31·13-s − 0.247·14-s + 2.14·15-s + 0.250·16-s − 0.757·17-s − 1.22·18-s − 0.552·19-s − 0.650·20-s − 0.577·21-s + 0.418·22-s − 1.09·23-s + 0.583·24-s + 0.690·25-s − 0.931·26-s − 1.20·27-s + 0.174·28-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 \( 1 + 2.86T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 - 0.924T + 7T^{2} \)
11 \( 1 + 1.96T + 11T^{2} \)
13 \( 1 - 4.75T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + 2.40T + 19T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 - 5.39T + 31T^{2} \)
37 \( 1 + 1.56T + 37T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 - 0.246T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 0.976T + 53T^{2} \)
59 \( 1 - 7.22T + 59T^{2} \)
61 \( 1 - 1.80T + 61T^{2} \)
67 \( 1 - 6.89T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 - 8.90T + 73T^{2} \)
79 \( 1 + 2.07T + 79T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 - 8.42T + 89T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−8.100915227577844783584669330137, −7.36322619171868082264715821438, −6.59833585670464573011840359224, −5.99776456081372839538148579562, −5.20977442775081019692080563260, −4.26671566594552100015252340260, −3.71255445433271262223717353174, −2.13694789134998071271297598733, −0.881122066402227543314111782456, 0, 0.881122066402227543314111782456, 2.13694789134998071271297598733, 3.71255445433271262223717353174, 4.26671566594552100015252340260, 5.20977442775081019692080563260, 5.99776456081372839538148579562, 6.59833585670464573011840359224, 7.36322619171868082264715821438, 8.100915227577844783584669330137

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