Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 4·7-s + 9-s + 11-s + 6·13-s + 2·15-s + 6·17-s + 8·19-s + 4·21-s − 25-s − 27-s − 6·29-s − 33-s + 8·35-s + 6·37-s − 6·39-s − 10·41-s + 8·43-s − 2·45-s + 9·49-s − 6·51-s + 6·53-s − 2·55-s − 8·57-s − 4·59-s − 2·61-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 + 1.45·17-s + 1.83·19-s + 0.872·21-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 − 1.56·41-s + 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.269·55-s − 1.05·57-s − 0.520·59-s − 0.256·61-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  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9267338296$
$L(\frac12)$  $\approx$  $0.9267338296$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.41220926309793, −18.59337920071639, −18.34108420768483, −16.96201332129084, −16.30458290716829, −15.93037470510244, −15.23086648154461, −13.95790162970664, −13.26056002284423, −12.37136695349348, −11.73600155015172, −10.98884473874776, −9.903740287069165, −9.298012922137153, −8.024470565850336, −7.198190715068514, −6.190154852495861, −5.464051902583235, −3.763054263379257, −3.381087598173365, −0.9189920406593867, 0.9189920406593867, 3.381087598173365, 3.763054263379257, 5.464051902583235, 6.190154852495861, 7.198190715068514, 8.024470565850336, 9.298012922137153, 9.903740287069165, 10.98884473874776, 11.73600155015172, 12.37136695349348, 13.26056002284423, 13.95790162970664, 15.23086648154461, 15.93037470510244, 16.30458290716829, 16.96201332129084, 18.34108420768483, 18.59337920071639, 19.41220926309793

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