Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 2003 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 4·5-s + 3·6-s − 7-s − 8-s + 6·9-s + 4·10-s − 6·11-s − 3·12-s − 3·13-s + 14-s + 12·15-s + 16-s − 6·17-s − 6·18-s − 4·19-s − 4·20-s + 3·21-s + 6·22-s − 6·23-s + 3·24-s + 11·25-s + 3·26-s − 9·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 1.26·10-s − 1.80·11-s − 0.866·12-s − 0.832·13-s + 0.267·14-s + 3.09·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 0.917·19-s − 0.894·20-s + 0.654·21-s + 1.27·22-s − 1.25·23-s + 0.612·24-s + 11/5·25-s + 0.588·26-s − 1.73·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28042\)    =    \(2 \cdot 7 \cdot 2003\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{28042} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 28042,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;2003\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;2003\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
2003 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.09015436361428, −15.61541206141731, −15.21733564513386, −15.02973727779648, −13.70577522228544, −12.89472872702216, −12.67759199005924, −12.18152178038496, −11.55155102696743, −11.11374663458392, −10.86073636806947, −10.27185014790960, −9.754437499476991, −8.897089375084428, −8.145962631992349, −7.692066514116583, −7.299585900145627, −6.642979938512620, −6.108449371183158, −5.363965835333898, −4.640543122763443, −4.369860815073653, −3.437039737945569, −2.544931157552177, −1.655058685167693, 0, 0, 0, 1.655058685167693, 2.544931157552177, 3.437039737945569, 4.369860815073653, 4.640543122763443, 5.363965835333898, 6.108449371183158, 6.642979938512620, 7.299585900145627, 7.692066514116583, 8.145962631992349, 8.897089375084428, 9.754437499476991, 10.27185014790960, 10.86073636806947, 11.11374663458392, 11.55155102696743, 12.18152178038496, 12.67759199005924, 12.89472872702216, 13.70577522228544, 15.02973727779648, 15.21733564513386, 15.61541206141731, 16.09015436361428

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