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 − 1.01·3-s + 4-s − 1.91·5-s + 1.01·6-s + 4.92·7-s − 8-s − 1.97·9-s + 1.91·10-s − 0.589·11-s − 1.01·12-s + 2.74·13-s − 4.92·14-s + 1.93·15-s + 16-s − 1.63·17-s + 1.97·18-s − 1.23·19-s − 1.91·20-s − 4.97·21-s + 0.589·22-s + 1.68·23-s + 1.01·24-s − 1.32·25-s − 2.74·26-s + 5.03·27-s + 4.92·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.583·3-s + 0.5·4-s − 0.857·5-s + 0.412·6-s + 1.86·7-s − 0.353·8-s − 0.659·9-s + 0.606·10-s − 0.177·11-s − 0.291·12-s + 0.760·13-s − 1.31·14-s + 0.500·15-s + 0.250·16-s − 0.395·17-s + 0.466·18-s − 0.284·19-s − 0.428·20-s − 1.08·21-s + 0.125·22-s + 0.350·23-s + 0.206·24-s − 0.264·25-s − 0.537·26-s + 0.968·27-s + 0.930·28-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 \( 1 + 1.01T + 3T^{2} \)
5 \( 1 + 1.91T + 5T^{2} \)
7 \( 1 - 4.92T + 7T^{2} \)
11 \( 1 + 0.589T + 11T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 1.68T + 23T^{2} \)
29 \( 1 + 7.46T + 29T^{2} \)
31 \( 1 - 1.54T + 31T^{2} \)
37 \( 1 - 3.02T + 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 + 9.29T + 43T^{2} \)
47 \( 1 + 3.13T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 7.96T + 61T^{2} \)
67 \( 1 - 2.77T + 67T^{2} \)
71 \( 1 - 2.80T + 71T^{2} \)
73 \( 1 - 0.952T + 73T^{2} \)
79 \( 1 + 9.00T + 79T^{2} \)
83 \( 1 - 3.06T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 + 8.87T + 97T^{2} \)
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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

−7.958683203006362786361998515558, −7.77183689664662842375784687063, −6.69570770732408390795265798053, −5.87709958248005721392046086456, −5.09769595011872196424989861655, −4.40452855877381410550394123205, −3.41802720457098998303406235015, −2.17897338324948972598599307894, −1.24289943189213891202065801965, 0, 1.24289943189213891202065801965, 2.17897338324948972598599307894, 3.41802720457098998303406235015, 4.40452855877381410550394123205, 5.09769595011872196424989861655, 5.87709958248005721392046086456, 6.69570770732408390795265798053, 7.77183689664662842375784687063, 7.958683203006362786361998515558

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