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 + 4·7-s + 9-s + 11-s − 6·13-s + 2·15-s + 2·17-s − 4·19-s + 4·21-s − 4·23-s − 25-s + 27-s + 6·29-s + 33-s + 8·35-s + 6·37-s − 6·39-s − 6·41-s − 4·43-s + 2·45-s + 12·47-s + 9·49-s + 2·51-s + 2·53-s + 2·55-s − 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s + 1.35·35-s + 0.986·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s + 0.269·55-s − 0.529·57-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  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.194059710$
$L(\frac12)$  $\approx$  $2.194059710$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.66393208827131, −18.63405902190552, −17.94718916268101, −17.20134691519721, −16.82879283736247, −15.37381509187497, −14.77510999836406, −14.14212602518802, −13.66231959022759, −12.39733912466367, −11.86282404516060, −10.63457399256276, −9.964740834749841, −9.101258160534375, −8.126100348919151, −7.474994724522290, −6.227568693314560, −5.086600208096707, −4.309775382142797, −2.561944642271234, −1.695732545137944, 1.695732545137944, 2.561944642271234, 4.309775382142797, 5.086600208096707, 6.227568693314560, 7.474994724522290, 8.126100348919151, 9.101258160534375, 9.964740834749841, 10.63457399256276, 11.86282404516060, 12.39733912466367, 13.66231959022759, 14.14212602518802, 14.77510999836406, 15.37381509187497, 16.82879283736247, 17.20134691519721, 17.94718916268101, 18.63405902190552, 19.66393208827131

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