Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{6} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 58·7-s + 658·13-s + 3.59e3·19-s + 1.52e3·25-s + 1.04e4·31-s + 1.75e4·37-s + 3.99e4·43-s − 3.10e4·49-s − 2.13e3·61-s − 1.24e5·67-s − 9.61e4·73-s + 9.99e4·79-s + 3.81e4·91-s + 2.58e4·97-s − 1.55e5·103-s − 3.54e4·109-s − 3.14e5·121-s + 127-s + 131-s + 2.08e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.447·7-s + 1.07·13-s + 2.28·19-s + 0.488·25-s + 1.95·31-s + 2.10·37-s + 3.29·43-s − 1.84·49-s − 0.0735·61-s − 3.37·67-s − 2.11·73-s + 1.80·79-s + 0.483·91-s + 0.278·97-s − 1.43·103-s − 0.285·109-s − 1.95·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 1.02·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11664\)    =    \(2^{4} \cdot 3^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 11664,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(3.61929\)
\(L(\frac12)\)  \(\approx\)  \(3.61929\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 1526 T^{2} + p^{10} T^{4} \)
7$C_2$ \( ( 1 - 29 T + p^{5} T^{2} )^{2} \)
11$C_2^2$ \( 1 + 314326 T^{2} + p^{10} T^{4} \)
13$C_2$ \( ( 1 - 329 T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2020286 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1799 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 198770 T^{2} + p^{10} T^{4} \)
29$C_2^2$ \( 1 + 39031642 T^{2} + p^{10} T^{4} \)
31$C_2$ \( ( 1 - 5228 T + p^{5} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8783 T + p^{5} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9156974 T^{2} + p^{10} T^{4} \)
43$C_2$ \( ( 1 - 19976 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 341046910 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 31068470 T^{2} + p^{10} T^{4} \)
59$C_2^2$ \( 1 + 1396994998 T^{2} + p^{10} T^{4} \)
61$C_2$ \( ( 1 + 1069 T + p^{5} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 62077 T + p^{5} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 1457026126 T^{2} + p^{10} T^{4} \)
73$C_2$ \( ( 1 + 48079 T + p^{5} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 49979 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 4552161670 T^{2} + p^{10} T^{4} \)
89$C_2^2$ \( 1 + 3438704914 T^{2} + p^{10} T^{4} \)
97$C_2$ \( ( 1 - 12917 T + p^{5} T^{2} )^{2} \)
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

−13.19511601443184025649020698767, −12.42439022651627505476971439061, −11.81780717363408966068934463178, −11.59436403364964920292591836039, −10.84350465638465981989280208824, −10.58233259051719470918443246567, −9.539194038455000706818034120299, −9.483382057989998486003113401563, −8.689545064228936692991465487764, −7.929420413152470749238584238208, −7.69214051862478737708636673214, −6.93499565280773669993965702308, −5.94610751945138490172786722389, −5.85762870602224772323332024007, −4.71498481783240816581428689664, −4.32387649703875644622264018298, −3.21335658878699340620447031106, −2.69793736432408052759390939110, −1.30593061073762158092596882428, −0.863284644929729163948244231784, 0.863284644929729163948244231784, 1.30593061073762158092596882428, 2.69793736432408052759390939110, 3.21335658878699340620447031106, 4.32387649703875644622264018298, 4.71498481783240816581428689664, 5.85762870602224772323332024007, 5.94610751945138490172786722389, 6.93499565280773669993965702308, 7.69214051862478737708636673214, 7.929420413152470749238584238208, 8.689545064228936692991465487764, 9.483382057989998486003113401563, 9.539194038455000706818034120299, 10.58233259051719470918443246567, 10.84350465638465981989280208824, 11.59436403364964920292591836039, 11.81780717363408966068934463178, 12.42439022651627505476971439061, 13.19511601443184025649020698767

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