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 + 0.327·3-s + 4-s − 1.35·5-s − 0.327·6-s + 3.43·7-s − 8-s − 2.89·9-s + 1.35·10-s + 5.09·11-s + 0.327·12-s − 3.01·13-s − 3.43·14-s − 0.443·15-s + 16-s + 7.56·17-s + 2.89·18-s − 6.52·19-s − 1.35·20-s + 1.12·21-s − 5.09·22-s − 5.33·23-s − 0.327·24-s − 3.15·25-s + 3.01·26-s − 1.92·27-s + 3.43·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.188·3-s + 0.5·4-s − 0.606·5-s − 0.133·6-s + 1.29·7-s − 0.353·8-s − 0.964·9-s + 0.429·10-s + 1.53·11-s + 0.0944·12-s − 0.835·13-s − 0.918·14-s − 0.114·15-s + 0.250·16-s + 1.83·17-s + 0.681·18-s − 1.49·19-s − 0.303·20-s + 0.245·21-s − 1.08·22-s − 1.11·23-s − 0.0667·24-s − 0.631·25-s + 0.591·26-s − 0.370·27-s + 0.649·28-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 \( 1 - 0.327T + 3T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 - 5.09T + 11T^{2} \)
13 \( 1 + 3.01T + 13T^{2} \)
17 \( 1 - 7.56T + 17T^{2} \)
19 \( 1 + 6.52T + 19T^{2} \)
23 \( 1 + 5.33T + 23T^{2} \)
29 \( 1 + 4.11T + 29T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 + 9.67T + 37T^{2} \)
41 \( 1 + 8.83T + 41T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 + 6.20T + 53T^{2} \)
59 \( 1 + 0.844T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 3.77T + 67T^{2} \)
71 \( 1 + 0.624T + 71T^{2} \)
73 \( 1 - 7.05T + 73T^{2} \)
79 \( 1 - 5.34T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−8.297244567323708931958900150213, −7.60624318221155310076304435670, −6.81365985203159686651824018091, −5.90671317398183638552003360770, −5.15917227416355103435409934789, −4.09686827954779649646750735521, −3.46624489982157746910540270969, −2.18578890383263228109598431935, −1.44532468184315241351087286240, 0, 1.44532468184315241351087286240, 2.18578890383263228109598431935, 3.46624489982157746910540270969, 4.09686827954779649646750735521, 5.15917227416355103435409934789, 5.90671317398183638552003360770, 6.81365985203159686651824018091, 7.60624318221155310076304435670, 8.297244567323708931958900150213

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