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 + 6·13-s + 2·15-s − 4·17-s + 2·19-s − 2·21-s + 8·23-s − 25-s + 27-s + 33-s − 4·35-s − 6·37-s + 6·39-s − 10·43-s + 2·45-s − 3·49-s − 4·51-s + 14·53-s + 2·55-s + 2·57-s + 12·59-s − 14·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.970·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.174·33-s − 0.676·35-s − 0.986·37-s + 0.960·39-s − 1.52·43-s + 0.298·45-s − 3/7·49-s − 0.560·51-s + 1.92·53-s + 0.269·55-s + 0.264·57-s + 1.56·59-s − 1.79·61-s − 0.251·63-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  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.990658435$
$L(\frac12)$  $\approx$  $1.990658435$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
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

−19.57361988851758, −18.66229534214065, −18.12264877617630, −17.25218984767081, −16.42658180638099, −15.64495726520162, −14.98385814848480, −13.83300220025145, −13.42357788863875, −12.90147931790474, −11.61757648713906, −10.71879143206400, −9.853168983015715, −9.019094517389160, −8.520481971363138, −7.023395604302586, −6.383838754734028, −5.344747114348273, −3.900416577160038, −2.918545645514896, −1.499853133157270, 1.499853133157270, 2.918545645514896, 3.900416577160038, 5.344747114348273, 6.383838754734028, 7.023395604302586, 8.520481971363138, 9.019094517389160, 9.853168983015715, 10.71879143206400, 11.61757648713906, 12.90147931790474, 13.42357788863875, 13.83300220025145, 14.98385814848480, 15.64495726520162, 16.42658180638099, 17.25218984767081, 18.12264877617630, 18.66229534214065, 19.57361988851758

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