Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 13^{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 − 4·14-s − 16-s + 2·17-s + 2·20-s + 22-s − 8·23-s − 25-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 − 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 − 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 + 0.755·28-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 − 0.150·44-s − 1.17·46-s + 1.16·47-s + 9/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16731\)    =    \(3^{2} \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{16731} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 16731,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−15.98342347841261, −15.52966668373592, −15.29566763920036, −14.30624520228902, −13.85967313962213, −13.62507039211831, −12.67668367963981, −12.36624224493970, −12.00974951081771, −11.44915340764688, −10.38806136770981, −9.951265955753097, −9.518139551357039, −8.713369726738920, −8.228971752543685, −7.580535636301132, −6.657876375550096, −6.289695084816430, −5.679564601672022, −4.820053606306035, −4.160729708899151, −3.620059834618266, −3.201492364454646, −2.330973427594975, −0.8115044004384301, 0, 0.8115044004384301, 2.330973427594975, 3.201492364454646, 3.620059834618266, 4.160729708899151, 4.820053606306035, 5.679564601672022, 6.289695084816430, 6.657876375550096, 7.580535636301132, 8.228971752543685, 8.713369726738920, 9.518139551357039, 9.951265955753097, 10.38806136770981, 11.44915340764688, 12.00974951081771, 12.36624224493970, 12.67668367963981, 13.62507039211831, 13.85967313962213, 14.30624520228902, 15.29566763920036, 15.52966668373592, 15.98342347841261

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