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 − 0.712·3-s + 4-s − 4.09·5-s + 0.712·6-s + 2.33·7-s − 8-s − 2.49·9-s + 4.09·10-s + 4.23·11-s − 0.712·12-s − 0.870·13-s − 2.33·14-s + 2.91·15-s + 16-s − 6.84·17-s + 2.49·18-s + 1.01·19-s − 4.09·20-s − 1.65·21-s − 4.23·22-s − 2.53·23-s + 0.712·24-s + 11.7·25-s + 0.870·26-s + 3.91·27-s + 2.33·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.411·3-s + 0.5·4-s − 1.83·5-s + 0.290·6-s + 0.880·7-s − 0.353·8-s − 0.830·9-s + 1.29·10-s + 1.27·11-s − 0.205·12-s − 0.241·13-s − 0.622·14-s + 0.752·15-s + 0.250·16-s − 1.65·17-s + 0.587·18-s + 0.231·19-s − 0.915·20-s − 0.362·21-s − 0.902·22-s − 0.528·23-s + 0.145·24-s + 2.34·25-s + 0.170·26-s + 0.752·27-s + 0.440·28-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 \( 1 + 0.712T + 3T^{2} \)
5 \( 1 + 4.09T + 5T^{2} \)
7 \( 1 - 2.33T + 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 + 0.870T + 13T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 + 2.53T + 23T^{2} \)
29 \( 1 - 0.501T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 - 7.79T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 0.979T + 43T^{2} \)
47 \( 1 - 0.948T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 3.12T + 61T^{2} \)
67 \( 1 + 2.12T + 67T^{2} \)
71 \( 1 + 0.135T + 71T^{2} \)
73 \( 1 + 5.71T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 6.20T + 83T^{2} \)
89 \( 1 - 3.24T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−8.165129605130704072430246929732, −7.51356562055658025226880999852, −6.74649447137763979521384628845, −6.13786332663720384569813044788, −4.83397352458445962100729596166, −4.34773186206034058182311553065, −3.47102982902720528644598716745, −2.38134757862113028391792265274, −1.03678139129449249808626775661, 0, 1.03678139129449249808626775661, 2.38134757862113028391792265274, 3.47102982902720528644598716745, 4.34773186206034058182311553065, 4.83397352458445962100729596166, 6.13786332663720384569813044788, 6.74649447137763979521384628845, 7.51356562055658025226880999852, 8.165129605130704072430246929732

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