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 + 4·5-s + 2·7-s + 9-s + 11-s − 4·15-s − 6·17-s − 4·19-s − 2·21-s + 6·23-s + 11·25-s − 27-s + 6·29-s − 33-s + 8·35-s + 6·37-s − 10·41-s + 8·43-s + 4·45-s − 6·47-s − 3·49-s + 6·51-s − 12·53-s + 4·55-s + 4·57-s + 8·59-s + 4·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.03·15-s − 1.45·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s − 0.174·33-s + 1.35·35-s + 0.986·37-s − 1.56·41-s + 1.21·43-s + 0.596·45-s − 0.875·47-s − 3/7·49-s + 0.840·51-s − 1.64·53-s + 0.539·55-s + 0.529·57-s + 1.04·59-s + 0.512·61-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  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.731441910$
$L(\frac12)$  $\approx$  $1.731441910$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−19.39744227283985, −18.46876530410808, −17.69670202966667, −17.41140072172330, −16.82696846894041, −15.73594432294163, −14.74730317688478, −14.13883831900817, −13.18821398029828, −12.80290091031987, −11.45326542898956, −10.85767698459354, −10.03932570061289, −9.171820100888388, −8.391236878349637, −6.788340375983685, −6.339657717581418, −5.231839581119871, −4.527065508415994, −2.531192177429876, −1.461040030087125, 1.461040030087125, 2.531192177429876, 4.527065508415994, 5.231839581119871, 6.339657717581418, 6.788340375983685, 8.391236878349637, 9.171820100888388, 10.03932570061289, 10.85767698459354, 11.45326542898956, 12.80290091031987, 13.18821398029828, 14.13883831900817, 14.74730317688478, 15.73594432294163, 16.82696846894041, 17.41140072172330, 17.69670202966667, 18.46876530410808, 19.39744227283985

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