Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 2·7-s + 9-s − 11-s − 2·13-s − 2·15-s + 4·17-s + 6·19-s − 2·21-s − 25-s − 27-s − 8·29-s + 8·31-s + 33-s + 4·35-s + 10·37-s + 2·39-s + 8·41-s + 2·43-s + 2·45-s + 8·47-s − 3·49-s − 4·51-s − 2·53-s − 2·55-s − 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s + 0.970·17-s + 1.37·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.174·33-s + 0.676·35-s + 1.64·37-s + 0.320·39-s + 1.24·41-s + 0.304·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.794·57-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  =  0
Selberg data  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.520119812$
$L(\frac12)$  $\approx$  $1.520119812$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−19.38444454889214, −18.47605467544315, −17.94475977450932, −17.27967918086799, −16.65904414241619, −15.79139379163295, −14.81744642114547, −14.10751346067928, −13.38331384355751, −12.43669290422963, −11.65111924938002, −10.89684445757782, −9.857110987012249, −9.443535976607848, −7.956659964685635, −7.340873780111209, −5.929488144610813, −5.446205594115483, −4.374922056116919, −2.715529056536006, −1.299480841407063, 1.299480841407063, 2.715529056536006, 4.374922056116919, 5.446205594115483, 5.929488144610813, 7.340873780111209, 7.956659964685635, 9.443535976607848, 9.857110987012249, 10.89684445757782, 11.65111924938002, 12.43669290422963, 13.38331384355751, 14.10751346067928, 14.81744642114547, 15.79139379163295, 16.65904414241619, 17.27967918086799, 17.94475977450932, 18.47605467544315, 19.38444454889214

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