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 − 4·5-s + 2·7-s + 9-s − 11-s + 4·13-s + 4·15-s − 2·17-s − 2·21-s + 6·23-s + 11·25-s − 27-s + 10·29-s + 8·31-s + 33-s − 8·35-s − 2·37-s − 4·39-s + 2·41-s − 4·43-s − 4·45-s + 2·47-s − 3·49-s + 2·51-s + 4·53-s + 4·55-s − 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.03·15-s − 0.485·17-s − 0.436·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.174·33-s − 1.35·35-s − 0.328·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s − 0.596·45-s + 0.291·47-s − 3/7·49-s + 0.280·51-s + 0.549·53-s + 0.539·55-s − 1.02·61-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  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9402796052$
$L(\frac12)$  $\approx$  $0.9402796052$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
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

−19.37639365582777, −18.79382860654346, −18.01799186270572, −17.26081762825424, −16.29327132865886, −15.61863403744739, −15.29498376151711, −14.20620004554899, −13.19274347613154, −12.26536367333248, −11.56701648477122, −11.08902156269882, −10.33368578606213, −8.682484661204744, −8.250767983667014, −7.278092389375405, −6.398125726845332, −4.925927170927614, −4.324824828293880, −3.116348144306563, −0.9338305816952735, 0.9338305816952735, 3.116348144306563, 4.324824828293880, 4.925927170927614, 6.398125726845332, 7.278092389375405, 8.250767983667014, 8.682484661204744, 10.33368578606213, 11.08902156269882, 11.56701648477122, 12.26536367333248, 13.19274347613154, 14.20620004554899, 15.29498376151711, 15.61863403744739, 16.29327132865886, 17.26081762825424, 18.01799186270572, 18.79382860654346, 19.37639365582777

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