Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 7-s − 11-s + 6·13-s − 2·17-s − 4·19-s − 25-s + 2·29-s − 8·31-s − 2·35-s + 6·37-s − 10·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s − 2·55-s + 4·59-s − 10·61-s + 12·65-s + 12·67-s + 2·73-s + 77-s − 16·79-s + 4·83-s − 4·85-s − 18·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 0.301·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 1.28·61-s + 1.48·65-s + 1.46·67-s + 0.234·73-s + 0.113·77-s − 1.80·79-s + 0.439·83-s − 0.433·85-s − 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11088\)    =    \(2^{4} \cdot 3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11088} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 11088,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−16.82759215414346, −16.10325413811254, −15.78652044263309, −15.06609596572819, −14.46941011824982, −13.77920424709360, −13.29254129026418, −12.99311946053245, −12.34022460263748, −11.37707982364244, −10.96780165350123, −10.43605819396073, −9.687623676400174, −9.251713428292094, −8.465154684976048, −8.125515422190505, −7.023191568642536, −6.480705177047420, −5.938502532778549, −5.403952381138842, −4.436996247081940, −3.741081155588825, −2.960655723164154, −2.033007707536944, −1.372569488902831, 0, 1.372569488902831, 2.033007707536944, 2.960655723164154, 3.741081155588825, 4.436996247081940, 5.403952381138842, 5.938502532778549, 6.480705177047420, 7.023191568642536, 8.125515422190505, 8.465154684976048, 9.251713428292094, 9.687623676400174, 10.43605819396073, 10.96780165350123, 11.37707982364244, 12.34022460263748, 12.99311946053245, 13.29254129026418, 13.77920424709360, 14.46941011824982, 15.06609596572819, 15.78652044263309, 16.10325413811254, 16.82759215414346

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