Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 17^{2} $
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 − 4-s − 2·5-s − 4·7-s + 3·8-s + 2·10-s + 11-s − 2·13-s + 4·14-s − 16-s + 2·20-s − 22-s + 8·23-s − 25-s + 2·26-s + 4·28-s − 6·29-s + 8·31-s − 5·32-s + 8·35-s − 6·37-s − 6·40-s − 2·41-s − 44-s − 8·46-s − 8·47-s + 9·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s − 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.447·20-s − 0.213·22-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s + 1.43·31-s − 0.883·32-s + 1.35·35-s − 0.986·37-s − 0.948·40-s − 0.312·41-s − 0.150·44-s − 1.17·46-s − 1.16·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28611 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28611 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28611\)    =    \(3^{2} \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{28611} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 28611,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.62917211349366, −15.05657401731747, −14.46499374751222, −13.77262059647261, −13.19384962433433, −12.91764395066596, −12.23266261388096, −11.77980157751257, −11.01832014026391, −10.56056076318227, −9.790357226572451, −9.556465006718971, −9.032517766001466, −8.389414205191639, −7.876440097657171, −7.163493511293461, −6.842918035535572, −6.124592621883279, −5.207649521473470, −4.695234759100363, −3.917624738767493, −3.404809517973098, −2.804504399307671, −1.652814410627100, −0.6601644000349602, 0, 0.6601644000349602, 1.652814410627100, 2.804504399307671, 3.404809517973098, 3.917624738767493, 4.695234759100363, 5.207649521473470, 6.124592621883279, 6.842918035535572, 7.163493511293461, 7.876440097657171, 8.389414205191639, 9.032517766001466, 9.556465006718971, 9.790357226572451, 10.56056076318227, 11.01832014026391, 11.77980157751257, 12.23266261388096, 12.91764395066596, 13.19384962433433, 13.77262059647261, 14.46499374751222, 15.05657401731747, 15.62917211349366

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