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 − 1.69·3-s + 4-s + 3.10·5-s + 1.69·6-s + 1.10·7-s − 8-s − 0.120·9-s − 3.10·10-s − 4.72·11-s − 1.69·12-s − 3.35·13-s − 1.10·14-s − 5.26·15-s + 16-s − 0.0655·17-s + 0.120·18-s − 0.989·19-s + 3.10·20-s − 1.86·21-s + 4.72·22-s + 5.99·23-s + 1.69·24-s + 4.63·25-s + 3.35·26-s + 5.29·27-s + 1.10·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.979·3-s + 0.5·4-s + 1.38·5-s + 0.692·6-s + 0.416·7-s − 0.353·8-s − 0.0402·9-s − 0.981·10-s − 1.42·11-s − 0.489·12-s − 0.929·13-s − 0.294·14-s − 1.35·15-s + 0.250·16-s − 0.0158·17-s + 0.0284·18-s − 0.227·19-s + 0.693·20-s − 0.407·21-s + 1.00·22-s + 1.25·23-s + 0.346·24-s + 0.926·25-s + 0.657·26-s + 1.01·27-s + 0.208·28-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 \( 1 + 1.69T + 3T^{2} \)
5 \( 1 - 3.10T + 5T^{2} \)
7 \( 1 - 1.10T + 7T^{2} \)
11 \( 1 + 4.72T + 11T^{2} \)
13 \( 1 + 3.35T + 13T^{2} \)
17 \( 1 + 0.0655T + 17T^{2} \)
19 \( 1 + 0.989T + 19T^{2} \)
23 \( 1 - 5.99T + 23T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 - 2.81T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 8.08T + 41T^{2} \)
43 \( 1 - 2.83T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + 9.29T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 0.610T + 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + 4.59T + 89T^{2} \)
97 \( 1 + 2.77T + 97T^{2} \)
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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

−8.299296508384605319287515386341, −7.16049746887851984077350123722, −6.68135590855884330240328703652, −5.80181047214876085323472739984, −5.17513464402969419679023901855, −4.83934943908584039813931690125, −2.93359123211684113551942962491, −2.36678667000687870302222753014, −1.26536314080310528775825936220, 0, 1.26536314080310528775825936220, 2.36678667000687870302222753014, 2.93359123211684113551942962491, 4.83934943908584039813931690125, 5.17513464402969419679023901855, 5.80181047214876085323472739984, 6.68135590855884330240328703652, 7.16049746887851984077350123722, 8.299296508384605319287515386341

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