Properties

Degree 2
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 2·13-s − 16-s + 2·17-s + 2·20-s − 8·23-s − 25-s − 2·26-s − 6·29-s + 8·31-s + 5·32-s + 2·34-s + 6·37-s + 6·40-s + 2·41-s − 8·46-s + 8·47-s − 50-s + 2·52-s − 6·53-s − 6·58-s − 4·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 0.554·13-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 − 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 0.986·37-s + 0.948·40-s + 0.312·41-s − 1.17·46-s + 1.16·47-s − 0.141·50-s + 0.277·52-s − 0.824·53-s − 0.787·58-s − 0.520·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(53361\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{53361} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 53361,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7223344678$
$L(\frac12)$  $\approx$  $0.7223344678$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.42857663342633, −13.91538684077652, −13.52156375397412, −12.90435102303871, −12.27154831047292, −12.13711372554740, −11.53908250645419, −11.08792775390205, −10.18774199202446, −9.817909748889228, −9.357047312138147, −8.575125485858346, −8.157303165849052, −7.625897740248053, −7.185163660926686, −6.137446966621650, −5.973497764684955, −5.234419549490064, −4.553235335170780, −4.106085229356769, −3.740959696628894, −2.932276251029419, −2.380673740129956, −1.293068284211579, −0.2758002527941792, 0.2758002527941792, 1.293068284211579, 2.380673740129956, 2.932276251029419, 3.740959696628894, 4.106085229356769, 4.553235335170780, 5.234419549490064, 5.973497764684955, 6.137446966621650, 7.185163660926686, 7.625897740248053, 8.157303165849052, 8.575125485858346, 9.357047312138147, 9.817909748889228, 10.18774199202446, 11.08792775390205, 11.53908250645419, 12.13711372554740, 12.27154831047292, 12.90435102303871, 13.52156375397412, 13.91538684077652, 14.42857663342633

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