Properties

Degree $4$
Conductor $27648$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s + 9-s − 4·15-s + 8·19-s + 16·23-s + 2·25-s + 27-s + 12·29-s − 8·43-s − 4·45-s − 14·49-s − 4·53-s + 8·57-s + 8·67-s + 16·69-s − 16·71-s + 20·73-s + 2·75-s + 81-s + 12·87-s − 32·95-s + 4·97-s − 36·101-s − 64·115-s − 6·121-s + 28·125-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.03·15-s + 1.83·19-s + 3.33·23-s + 2/5·25-s + 0.192·27-s + 2.22·29-s − 1.21·43-s − 0.596·45-s − 2·49-s − 0.549·53-s + 1.05·57-s + 0.977·67-s + 1.92·69-s − 1.89·71-s + 2.34·73-s + 0.230·75-s + 1/9·81-s + 1.28·87-s − 3.28·95-s + 0.406·97-s − 3.58·101-s − 5.96·115-s − 0.545·121-s + 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27648\)    =    \(2^{10} \cdot 3^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{27648} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27648,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160141318\)
\(L(\frac12)\) \(\approx\) \(1.160141318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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
   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92388151188979644343896213523, −9.780091138821383013974264992430, −9.656797755147918818405236990829, −8.884556993307409273860919130264, −8.173282349834764667604075089133, −8.157389656224392853374347080198, −7.26958585488936859387169650645, −7.06857822018944724974086155331, −6.38250152569089720234352942221, −5.01357758335939619070346679818, −4.99175853408267534988469991920, −4.02227786472289347756607333391, −3.14826068230388029805668446658, −3.04433904291477058087427796742, −1.12835632773744922111480113749, 1.12835632773744922111480113749, 3.04433904291477058087427796742, 3.14826068230388029805668446658, 4.02227786472289347756607333391, 4.99175853408267534988469991920, 5.01357758335939619070346679818, 6.38250152569089720234352942221, 7.06857822018944724974086155331, 7.26958585488936859387169650645, 8.157389656224392853374347080198, 8.173282349834764667604075089133, 8.884556993307409273860919130264, 9.656797755147918818405236990829, 9.780091138821383013974264992430, 10.92388151188979644343896213523

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