Properties

Degree 2
Conductor $ 2 \cdot 2003 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.94·3-s + 4-s − 0.275·5-s + 1.94·6-s − 2.80·7-s − 8-s + 0.763·9-s + 0.275·10-s − 2.43·11-s − 1.94·12-s + 1.63·13-s + 2.80·14-s + 0.533·15-s + 16-s − 1.12·17-s − 0.763·18-s + 6.24·19-s − 0.275·20-s + 5.43·21-s + 2.43·22-s − 5.13·23-s + 1.94·24-s − 4.92·25-s − 1.63·26-s + 4.33·27-s − 2.80·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.123·5-s + 0.792·6-s − 1.05·7-s − 0.353·8-s + 0.254·9-s + 0.0869·10-s − 0.733·11-s − 0.560·12-s + 0.453·13-s + 0.749·14-s + 0.137·15-s + 0.250·16-s − 0.272·17-s − 0.180·18-s + 1.43·19-s − 0.0615·20-s + 1.18·21-s + 0.518·22-s − 1.07·23-s + 0.396·24-s − 0.984·25-s − 0.320·26-s + 0.834·27-s − 0.529·28-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 \( 1 + 1.94T + 3T^{2} \)
5 \( 1 + 0.275T + 5T^{2} \)
7 \( 1 + 2.80T + 7T^{2} \)
11 \( 1 + 2.43T + 11T^{2} \)
13 \( 1 - 1.63T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 + 5.13T + 23T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 - 7.35T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 8.26T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 1.86T + 47T^{2} \)
53 \( 1 - 6.50T + 53T^{2} \)
59 \( 1 - 6.17T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 + 3.85T + 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + 0.249T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + 8.22T + 89T^{2} \)
97 \( 1 - 1.29T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.993084058726039462832767014614, −7.39349887033578075441666485748, −6.49530439921523924563517888529, −5.95122365561865397972526746336, −5.43962509862083208098922775734, −4.30749595057711961257981276952, −3.29171465524391555849229804715, −2.41223274885561934022078134374, −0.946096814006075786448225147874, 0, 0.946096814006075786448225147874, 2.41223274885561934022078134374, 3.29171465524391555849229804715, 4.30749595057711961257981276952, 5.43962509862083208098922775734, 5.95122365561865397972526746336, 6.49530439921523924563517888529, 7.39349887033578075441666485748, 7.993084058726039462832767014614

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