Properties

Degree 2
Conductor $ 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s − 2·5-s + 2·7-s + 9-s − 5·11-s − 4·12-s + 2·13-s − 4·15-s + 4·16-s − 17-s − 5·19-s + 4·20-s + 4·21-s − 5·23-s − 25-s − 4·27-s − 4·28-s − 6·29-s + 8·31-s − 10·33-s − 4·35-s − 2·36-s + 2·37-s + 4·39-s − 6·41-s − 2·43-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 1.15·12-s + 0.554·13-s − 1.03·15-s + 16-s − 0.242·17-s − 1.14·19-s + 0.894·20-s + 0.872·21-s − 1.04·23-s − 1/5·25-s − 0.769·27-s − 0.755·28-s − 1.11·29-s + 1.43·31-s − 1.74·33-s − 0.676·35-s − 1/3·36-s + 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1003\)    =    \(17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1003} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad17 \( 1 + T \)
59 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−19.79838085291528, −19.01295714644338, −18.68734516096000, −17.88484197497099, −17.22839013245343, −16.14768481266310, −15.26992950378594, −15.03169825390258, −14.07196381336270, −13.54437722871713, −12.95432379532034, −12.01800797324161, −11.05924645565760, −10.28861140667567, −9.350338958223821, −8.392005801640334, −8.136466910523484, −7.638528373068945, −6.014283750274253, −4.881545601885033, −4.132152935509112, −3.276295837094660, −2.054394782215165, 0, 2.054394782215165, 3.276295837094660, 4.132152935509112, 4.881545601885033, 6.014283750274253, 7.638528373068945, 8.136466910523484, 8.392005801640334, 9.350338958223821, 10.28861140667567, 11.05924645565760, 12.01800797324161, 12.95432379532034, 13.54437722871713, 14.07196381336270, 15.03169825390258, 15.26992950378594, 16.14768481266310, 17.22839013245343, 17.88484197497099, 18.68734516096000, 19.01295714644338, 19.79838085291528

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