Properties

Degree 2
Conductor 28571
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 2·5-s + 2·6-s − 4·7-s − 2·9-s + 4·10-s − 6·11-s − 2·12-s − 2·13-s + 8·14-s + 2·15-s − 4·16-s − 4·17-s + 4·18-s + 4·19-s − 4·20-s + 4·21-s + 12·22-s − 7·23-s − 25-s + 4·26-s + 5·27-s − 8·28-s − 9·29-s − 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s − 2/3·9-s + 1.26·10-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 2.13·14-s + 0.516·15-s − 16-s − 0.970·17-s + 0.942·18-s + 0.917·19-s − 0.894·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s − 1/5·25-s + 0.784·26-s + 0.962·27-s − 1.51·28-s − 1.67·29-s − 0.730·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28571 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28571 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28571\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{28571} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 28571,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 28571$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 28571$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad28571 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.15874548540822, −15.81632142322497, −15.30395664786347, −14.61152690489028, −13.55706119307350, −13.36847144633163, −12.72919098234650, −11.98173678475215, −11.68999961654631, −10.94068804477375, −10.54628139706805, −10.07589246838031, −9.517345731880381, −8.965142527738077, −8.389762340093655, −7.782321339147279, −7.236138194410104, −7.006677443627949, −5.925077980371910, −5.590260945338720, −4.819499468377150, −3.894146840960006, −3.230997136962011, −2.519280440445147, −1.749297790829379, 0, 0, 0, 1.749297790829379, 2.519280440445147, 3.230997136962011, 3.894146840960006, 4.819499468377150, 5.590260945338720, 5.925077980371910, 7.006677443627949, 7.236138194410104, 7.782321339147279, 8.389762340093655, 8.965142527738077, 9.517345731880381, 10.07589246838031, 10.54628139706805, 10.94068804477375, 11.68999961654631, 11.98173678475215, 12.72919098234650, 13.36847144633163, 13.55706119307350, 14.61152690489028, 15.30395664786347, 15.81632142322497, 16.15874548540822

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