Properties

Degree $4$
Conductor $86400$
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 − 2·5-s + 9-s − 8·11-s − 2·15-s + 4·17-s − 25-s + 27-s − 8·33-s − 8·43-s − 2·45-s − 14·49-s + 4·51-s − 4·53-s + 16·55-s − 8·59-s − 4·61-s + 8·67-s − 16·71-s − 75-s + 81-s − 8·85-s − 8·99-s − 32·103-s − 4·109-s + 36·113-s + 26·121-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 2.41·11-s − 0.516·15-s + 0.970·17-s − 1/5·25-s + 0.192·27-s − 1.39·33-s − 1.21·43-s − 0.298·45-s − 2·49-s + 0.560·51-s − 0.549·53-s + 2.15·55-s − 1.04·59-s − 0.512·61-s + 0.977·67-s − 1.89·71-s − 0.115·75-s + 1/9·81-s − 0.867·85-s − 0.804·99-s − 3.15·103-s − 0.383·109-s + 3.38·113-s + 2.36·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86400\)    =    \(2^{7} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{86400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 86400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.656797755147918818405236990829, −8.730633394111998328725718228031, −8.173282349834764667604075089133, −8.095549656698690551636863836910, −7.47308398754126662115910632702, −7.26958585488936859387169650645, −6.33548194788761977078999997603, −5.74245729062321480161908713186, −5.01357758335939619070346679818, −4.76184067616314722575220108003, −3.84771560491769769437106298900, −3.14826068230388029805668446658, −2.79864903411763483594224581992, −1.72979115909686153918413689995, 0, 1.72979115909686153918413689995, 2.79864903411763483594224581992, 3.14826068230388029805668446658, 3.84771560491769769437106298900, 4.76184067616314722575220108003, 5.01357758335939619070346679818, 5.74245729062321480161908713186, 6.33548194788761977078999997603, 7.26958585488936859387169650645, 7.47308398754126662115910632702, 8.095549656698690551636863836910, 8.173282349834764667604075089133, 8.730633394111998328725718228031, 9.656797755147918818405236990829

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