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 + 3.09·3-s + 4-s − 1.23·5-s − 3.09·6-s − 4.11·7-s − 8-s + 6.60·9-s + 1.23·10-s − 2.95·11-s + 3.09·12-s − 4.28·13-s + 4.11·14-s − 3.82·15-s + 16-s + 7.58·17-s − 6.60·18-s + 6.52·19-s − 1.23·20-s − 12.7·21-s + 2.95·22-s + 3.37·23-s − 3.09·24-s − 3.47·25-s + 4.28·26-s + 11.1·27-s − 4.11·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.78·3-s + 0.5·4-s − 0.552·5-s − 1.26·6-s − 1.55·7-s − 0.353·8-s + 2.20·9-s + 0.390·10-s − 0.890·11-s + 0.894·12-s − 1.18·13-s + 1.09·14-s − 0.987·15-s + 0.250·16-s + 1.84·17-s − 1.55·18-s + 1.49·19-s − 0.276·20-s − 2.78·21-s + 0.629·22-s + 0.702·23-s − 0.632·24-s − 0.695·25-s + 0.840·26-s + 2.14·27-s − 0.777·28-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 \( 1 - 3.09T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + 4.11T + 7T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + 4.28T + 13T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
19 \( 1 - 6.52T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + 6.69T + 29T^{2} \)
31 \( 1 + 3.82T + 31T^{2} \)
37 \( 1 - 1.89T + 37T^{2} \)
41 \( 1 - 0.398T + 41T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + 5.86T + 47T^{2} \)
53 \( 1 + 2.92T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 8.90T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 4.74T + 71T^{2} \)
73 \( 1 + 5.73T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 - 1.59T + 97T^{2} \)
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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

−7.82813032212501279741029257115, −7.54444157955268183910642665145, −7.30775041515244213675343171847, −6.02209631558834966992073410622, −5.03902942039716922994159496943, −3.71125260485163949739548796950, −3.10833889444662082849171909861, −2.82201827502530372225194348870, −1.53643246181142282152936166983, 0, 1.53643246181142282152936166983, 2.82201827502530372225194348870, 3.10833889444662082849171909861, 3.71125260485163949739548796950, 5.03902942039716922994159496943, 6.02209631558834966992073410622, 7.30775041515244213675343171847, 7.54444157955268183910642665145, 7.82813032212501279741029257115

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