Properties

Degree 4
Conductor $ 3^{3} \cdot 7^{4} $
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  − 3-s − 3·4-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 8·19-s − 6·25-s − 27-s − 3·36-s + 12·37-s − 4·39-s − 8·43-s − 5·48-s − 12·52-s + 8·57-s + 4·61-s − 3·64-s + 8·67-s + 12·73-s + 6·75-s + 24·76-s − 32·79-s + 81-s − 36·97-s + 18·100-s − 16·103-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 1.83·19-s − 6/5·25-s − 0.192·27-s − 1/2·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s − 0.721·48-s − 1.66·52-s + 1.05·57-s + 0.512·61-s − 3/8·64-s + 0.977·67-s + 1.40·73-s + 0.692·75-s + 2.75·76-s − 3.60·79-s + 1/9·81-s − 3.65·97-s + 9/5·100-s − 1.57·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64827\)    =    \(3^{3} \cdot 7^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{64827} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 64827,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 + T \)
7 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
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

−9.800766905340478341584155035885, −9.163045893798789171919999202805, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −7.977686193886306753143677629185, −6.81422856027217448872168795499, −6.53484921474790582970192984243, −5.73267711633132091527857634664, −5.48510765590071063887274763802, −4.61914309368534933978095799917, −4.01671863219499060051377207353, −3.94328917656394473634657471263, −2.64839746138688614832708523229, −1.40786771277750203137263322761, 0, 1.40786771277750203137263322761, 2.64839746138688614832708523229, 3.94328917656394473634657471263, 4.01671863219499060051377207353, 4.61914309368534933978095799917, 5.48510765590071063887274763802, 5.73267711633132091527857634664, 6.53484921474790582970192984243, 6.81422856027217448872168795499, 7.977686193886306753143677629185, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 9.163045893798789171919999202805, 9.800766905340478341584155035885

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