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 + 0.961·3-s + 4-s + 2.47·5-s − 0.961·6-s − 3.64·7-s − 8-s − 2.07·9-s − 2.47·10-s − 0.534·11-s + 0.961·12-s + 6.07·13-s + 3.64·14-s + 2.38·15-s + 16-s − 5.92·17-s + 2.07·18-s + 0.706·19-s + 2.47·20-s − 3.50·21-s + 0.534·22-s + 2.43·23-s − 0.961·24-s + 1.14·25-s − 6.07·26-s − 4.87·27-s − 3.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.554·3-s + 0.5·4-s + 1.10·5-s − 0.392·6-s − 1.37·7-s − 0.353·8-s − 0.692·9-s − 0.784·10-s − 0.161·11-s + 0.277·12-s + 1.68·13-s + 0.974·14-s + 0.615·15-s + 0.250·16-s − 1.43·17-s + 0.489·18-s + 0.162·19-s + 0.554·20-s − 0.764·21-s + 0.114·22-s + 0.508·23-s − 0.196·24-s + 0.229·25-s − 1.19·26-s − 0.938·27-s − 0.689·28-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 \( 1 - 0.961T + 3T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 + 0.534T + 11T^{2} \)
13 \( 1 - 6.07T + 13T^{2} \)
17 \( 1 + 5.92T + 17T^{2} \)
19 \( 1 - 0.706T + 19T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + 5.02T + 29T^{2} \)
31 \( 1 - 4.49T + 31T^{2} \)
37 \( 1 + 5.93T + 37T^{2} \)
41 \( 1 + 7.28T + 41T^{2} \)
43 \( 1 - 7.34T + 43T^{2} \)
47 \( 1 - 1.28T + 47T^{2} \)
53 \( 1 - 7.69T + 53T^{2} \)
59 \( 1 + 8.59T + 59T^{2} \)
61 \( 1 + 3.03T + 61T^{2} \)
67 \( 1 + 6.51T + 67T^{2} \)
71 \( 1 + 8.18T + 71T^{2} \)
73 \( 1 + 1.72T + 73T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 + 3.12T + 97T^{2} \)
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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

−8.433813009441560889690934767827, −7.35594701742413643663991268193, −6.47820390460229721539707698554, −6.13673195271272800295263900674, −5.42952110971139560296502401910, −3.96175230335543746654986849276, −3.13630285815638072503638301180, −2.46815846242975550754444404778, −1.49368603366334255675491277744, 0, 1.49368603366334255675491277744, 2.46815846242975550754444404778, 3.13630285815638072503638301180, 3.96175230335543746654986849276, 5.42952110971139560296502401910, 6.13673195271272800295263900674, 6.47820390460229721539707698554, 7.35594701742413643663991268193, 8.433813009441560889690934767827

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