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 − 2.96·3-s + 4-s + 0.539·5-s + 2.96·6-s − 0.220·7-s − 8-s + 5.79·9-s − 0.539·10-s + 5.11·11-s − 2.96·12-s − 1.83·13-s + 0.220·14-s − 1.59·15-s + 16-s + 4.75·17-s − 5.79·18-s − 4.79·19-s + 0.539·20-s + 0.652·21-s − 5.11·22-s + 2.67·23-s + 2.96·24-s − 4.70·25-s + 1.83·26-s − 8.28·27-s − 0.220·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.71·3-s + 0.5·4-s + 0.241·5-s + 1.21·6-s − 0.0831·7-s − 0.353·8-s + 1.93·9-s − 0.170·10-s + 1.54·11-s − 0.856·12-s − 0.510·13-s + 0.0588·14-s − 0.412·15-s + 0.250·16-s + 1.15·17-s − 1.36·18-s − 1.10·19-s + 0.120·20-s + 0.142·21-s − 1.08·22-s + 0.557·23-s + 0.605·24-s − 0.941·25-s + 0.360·26-s − 1.59·27-s − 0.0415·28-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 \( 1 + 2.96T + 3T^{2} \)
5 \( 1 - 0.539T + 5T^{2} \)
7 \( 1 + 0.220T + 7T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 + 1.83T + 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 + 4.79T + 19T^{2} \)
23 \( 1 - 2.67T + 23T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 + 6.62T + 31T^{2} \)
37 \( 1 + 1.59T + 37T^{2} \)
41 \( 1 + 4.76T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 2.31T + 47T^{2} \)
53 \( 1 - 2.96T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 2.30T + 61T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 - 5.87T + 71T^{2} \)
73 \( 1 + 4.62T + 73T^{2} \)
79 \( 1 + 9.45T + 79T^{2} \)
83 \( 1 - 7.62T + 83T^{2} \)
89 \( 1 - 9.82T + 89T^{2} \)
97 \( 1 + 3.10T + 97T^{2} \)
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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

−7.937010514660902968891820748615, −7.14858258414877665024332789739, −6.62304793472048623505990051530, −5.89290388154380578054752822767, −5.42607997923189079613914394840, −4.36663505790314702427280639088, −3.57597125348805448605710371146, −1.96949717104001575526532156619, −1.14995734287139036213914705274, 0, 1.14995734287139036213914705274, 1.96949717104001575526532156619, 3.57597125348805448605710371146, 4.36663505790314702427280639088, 5.42607997923189079613914394840, 5.89290388154380578054752822767, 6.62304793472048623505990051530, 7.14858258414877665024332789739, 7.937010514660902968891820748615

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