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 − 2.23·3-s + 4-s − 2.75·5-s + 2.23·6-s − 2.11·7-s − 8-s + 1.98·9-s + 2.75·10-s − 1.85·11-s − 2.23·12-s − 5.20·13-s + 2.11·14-s + 6.14·15-s + 16-s + 3.15·17-s − 1.98·18-s − 7.18·19-s − 2.75·20-s + 4.71·21-s + 1.85·22-s + 5.98·23-s + 2.23·24-s + 2.57·25-s + 5.20·26-s + 2.27·27-s − 2.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.28·3-s + 0.5·4-s − 1.23·5-s + 0.911·6-s − 0.797·7-s − 0.353·8-s + 0.660·9-s + 0.870·10-s − 0.558·11-s − 0.644·12-s − 1.44·13-s + 0.563·14-s + 1.58·15-s + 0.250·16-s + 0.765·17-s − 0.467·18-s − 1.64·19-s − 0.615·20-s + 1.02·21-s + 0.395·22-s + 1.24·23-s + 0.455·24-s + 0.515·25-s + 1.02·26-s + 0.437·27-s − 0.398·28-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 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 + 2.75T + 5T^{2} \)
7 \( 1 + 2.11T + 7T^{2} \)
11 \( 1 + 1.85T + 11T^{2} \)
13 \( 1 + 5.20T + 13T^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + 7.18T + 19T^{2} \)
23 \( 1 - 5.98T + 23T^{2} \)
29 \( 1 - 0.982T + 29T^{2} \)
31 \( 1 - 8.19T + 31T^{2} \)
37 \( 1 - 4.83T + 37T^{2} \)
41 \( 1 - 1.04T + 41T^{2} \)
43 \( 1 + 0.753T + 43T^{2} \)
47 \( 1 + 3.72T + 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
59 \( 1 + 8.16T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 0.0465T + 71T^{2} \)
73 \( 1 + 3.43T + 73T^{2} \)
79 \( 1 - 0.984T + 79T^{2} \)
83 \( 1 + 1.05T + 83T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−8.016470787748025994429318637497, −7.33362085797342885154895875996, −6.65991349387488779716414593041, −6.09246885083562573621587063929, −5.02389404818090466657905700902, −4.51517092359100714467764855955, −3.31128680363656395966674892820, −2.47460413215811177740536119018, −0.74435366245040824810840571940, 0, 0.74435366245040824810840571940, 2.47460413215811177740536119018, 3.31128680363656395966674892820, 4.51517092359100714467764855955, 5.02389404818090466657905700902, 6.09246885083562573621587063929, 6.65991349387488779716414593041, 7.33362085797342885154895875996, 8.016470787748025994429318637497

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