Properties

Degree 2
Conductor $ 3^{2} \cdot 11^{2} $
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 − 4-s + 2·5-s − 4·7-s − 3·8-s + 2·10-s + 2·13-s − 4·14-s − 16-s − 2·17-s − 2·20-s − 8·23-s − 25-s + 2·26-s + 4·28-s − 6·29-s − 8·31-s + 5·32-s − 2·34-s − 8·35-s + 6·37-s − 6·40-s − 2·41-s − 8·46-s − 8·47-s + 9·49-s − 50-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 + 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.447·20-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 − 0.342·34-s − 1.35·35-s + 0.986·37-s − 0.948·40-s − 0.312·41-s − 1.17·46-s − 1.16·47-s + 9/7·49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1089\)    =    \(3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1089} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1089,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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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

−19.77045007766621, −18.86832583409695, −18.19264738570565, −17.79172115612019, −16.74430051544550, −16.19797392173991, −15.40582876703533, −14.58303150132660, −13.81906674327057, −13.27095550470699, −12.88481833610392, −12.11526429954149, −11.07074874943342, −9.979784146917935, −9.531112831864385, −8.938244517857001, −7.777447076567768, −6.388660215311124, −6.119825565394706, −5.259093025507723, −3.989945349267733, −3.366622419451778, −2.096663337801894, 0, 2.096663337801894, 3.366622419451778, 3.989945349267733, 5.259093025507723, 6.119825565394706, 6.388660215311124, 7.777447076567768, 8.938244517857001, 9.531112831864385, 9.979784146917935, 11.07074874943342, 12.11526429954149, 12.88481833610392, 13.27095550470699, 13.81906674327057, 14.58303150132660, 15.40582876703533, 16.19797392173991, 16.74430051544550, 17.79172115612019, 18.19264738570565, 18.86832583409695, 19.77045007766621

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