Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \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  − 3-s − 2·7-s + 9-s − 11-s − 2·17-s − 8·19-s + 2·21-s + 2·23-s − 5·25-s − 27-s − 6·29-s + 33-s − 2·37-s + 2·41-s − 4·43-s + 6·47-s − 3·49-s + 2·51-s − 8·53-s + 8·57-s + 8·59-s − 4·61-s − 2·63-s − 12·67-s − 2·69-s + 10·71-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.485·17-s − 1.83·19-s + 0.436·21-s + 0.417·23-s − 25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.09·53-s + 1.05·57-s + 1.04·59-s − 0.512·61-s − 0.251·63-s − 1.46·67-s − 0.240·69-s + 1.18·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{528} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 528,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.63719119455017, −19.14403707552659, −18.39398213720454, −17.44805647221183, −16.94495462458409, −16.11273212842575, −15.40510686135108, −14.68783787656561, −13.43144649335491, −12.95914699594227, −12.16894308741470, −11.15810730451035, −10.52073741751256, −9.609430582163952, −8.723597096299050, −7.599791130571006, −6.578215104841494, −5.912517184745220, −4.711476240698665, −3.637449016766687, −2.123571327074728, 0, 2.123571327074728, 3.637449016766687, 4.711476240698665, 5.912517184745220, 6.578215104841494, 7.599791130571006, 8.723597096299050, 9.609430582163952, 10.52073741751256, 11.15810730451035, 12.16894308741470, 12.95914699594227, 13.43144649335491, 14.68783787656561, 15.40510686135108, 16.11273212842575, 16.94495462458409, 17.44805647221183, 18.39398213720454, 19.14403707552659, 19.63719119455017

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