Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 2·11-s − 12-s − 4·13-s − 16-s − 18-s + 2·22-s − 16·23-s + 3·24-s − 6·25-s + 4·26-s + 27-s − 5·32-s − 2·33-s − 36-s + 12·37-s − 4·39-s + 2·44-s + 16·46-s − 16·47-s − 48-s + 2·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s − 0.235·18-s + 0.426·22-s − 3.33·23-s + 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s − 0.883·32-s − 0.348·33-s − 1/6·36-s + 1.97·37-s − 0.640·39-s + 0.301·44-s + 2.35·46-s − 2.33·47-s − 0.144·48-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(52272\)    =    \(2^{4} \cdot 3^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{52272} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 52272,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 - T \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.723139098205724899141727122835, −9.637990407765945223554197990480, −8.548353092044889979587894449296, −8.389268452519734683124359977272, −7.75720782023225106841673661541, −7.60731726518247812897599595849, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −5.64913594526162848677189952595, −4.52395871965258729630923756023, −4.35461089374668409702872860477, −3.53074178558790394542885743604, −2.45250023112523203194083612695, −1.82001999803537272093437549398, 0, 1.82001999803537272093437549398, 2.45250023112523203194083612695, 3.53074178558790394542885743604, 4.35461089374668409702872860477, 4.52395871965258729630923756023, 5.64913594526162848677189952595, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 7.60731726518247812897599595849, 7.75720782023225106841673661541, 8.389268452519734683124359977272, 8.548353092044889979587894449296, 9.637990407765945223554197990480, 9.723139098205724899141727122835

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