Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 29^{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 + 11-s − 2·13-s + 4·14-s − 16-s − 2·17-s − 2·20-s + 22-s − 8·23-s − 25-s − 2·26-s − 4·28-s + 8·31-s + 5·32-s − 2·34-s + 8·35-s − 6·37-s − 6·40-s − 2·41-s − 44-s − 8·46-s + 8·47-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 − 0.485·17-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 + 1.43·31-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 − 0.150·44-s − 1.17·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 83259 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 83259 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(83259\)    =    \(3^{2} \cdot 11 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{83259} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 83259,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
29 \( 1 \)
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 \( 1 + 8 T + p T^{2} \)
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

−14.11974920973879, −13.79405374365135, −13.51093333855134, −12.68957731875564, −12.25909636207690, −11.81739847934702, −11.43918473265077, −10.70481698016150, −10.11749980386064, −9.807244688351867, −9.123301011341708, −8.697616478661693, −8.088007863798719, −7.785899972473105, −6.896800480128440, −6.319780971067083, −5.785405123586994, −5.347435116249319, −4.775927479037715, −4.362485108816903, −3.877803878807875, −3.015465067093289, −2.229681474822411, −1.875705896743316, −1.037277606598723, 0, 1.037277606598723, 1.875705896743316, 2.229681474822411, 3.015465067093289, 3.877803878807875, 4.362485108816903, 4.775927479037715, 5.347435116249319, 5.785405123586994, 6.319780971067083, 6.896800480128440, 7.785899972473105, 8.088007863798719, 8.697616478661693, 9.123301011341708, 9.807244688351867, 10.11749980386064, 10.70481698016150, 11.43918473265077, 11.81739847934702, 12.25909636207690, 12.68957731875564, 13.51093333855134, 13.79405374365135, 14.11974920973879

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