Properties

Degree $4$
Conductor $419904$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 24·7-s − 44·11-s − 22·13-s − 100·17-s + 88·19-s − 56·23-s + 125·25-s + 198·29-s + 160·31-s + 48·35-s − 324·37-s − 198·41-s − 52·43-s + 528·47-s + 343·49-s + 484·53-s + 88·55-s − 668·59-s − 550·61-s + 44·65-s − 188·67-s − 1.45e3·71-s + 308·73-s + 1.05e3·77-s + 656·79-s + 236·83-s + ⋯
L(s)  = 1  − 0.178·5-s − 1.29·7-s − 1.20·11-s − 0.469·13-s − 1.42·17-s + 1.06·19-s − 0.507·23-s + 25-s + 1.26·29-s + 0.926·31-s + 0.231·35-s − 1.43·37-s − 0.754·41-s − 0.184·43-s + 1.63·47-s + 49-s + 1.25·53-s + 0.215·55-s − 1.47·59-s − 1.15·61-s + 0.0839·65-s − 0.342·67-s − 2.43·71-s + 0.493·73-s + 1.56·77-s + 0.934·79-s + 0.312·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(419904\)    =    \(2^{6} \cdot 3^{8}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{648} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 419904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4355583427\)
\(L(\frac12)\) \(\approx\) \(0.4355583427\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - 121 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 24 T + 233 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 4 p T + 5 p^{2} T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 22 T - 1713 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 50 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 44 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 56 T - 9031 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 198 T + 14815 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 162 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 198 T - 29717 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 52 T - 76803 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 528 T + 174961 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 242 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 668 T + 240845 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 550 T + 75519 T^{2} + 550 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 188 T - 265419 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 728 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 154 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 656 T - 62703 T^{2} - 656 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 236 T - 516091 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 714 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 478 T - 684189 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \)
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

−10.52292302953065660462191370122, −9.980151991246154127191365621933, −9.578734759383615817765128385501, −9.005143251180905756070549648234, −8.704723714868392880586206525073, −8.309988181002088436631245163516, −7.62214447456477314984682258437, −7.21529938937790780939825899586, −6.93510175111827270636373129756, −6.31705470112171227026398270020, −5.99655875446824650775123878046, −5.31629077333186188592788258123, −4.82984354306585173770823619764, −4.44725423681082230657047285755, −3.71161943398741022857133480950, −2.96005027876523609254972131000, −2.84189157796045839808486460297, −2.12923742344217128470869358102, −1.08974842838240832900997237513, −0.20753574627567907397600727409, 0.20753574627567907397600727409, 1.08974842838240832900997237513, 2.12923742344217128470869358102, 2.84189157796045839808486460297, 2.96005027876523609254972131000, 3.71161943398741022857133480950, 4.44725423681082230657047285755, 4.82984354306585173770823619764, 5.31629077333186188592788258123, 5.99655875446824650775123878046, 6.31705470112171227026398270020, 6.93510175111827270636373129756, 7.21529938937790780939825899586, 7.62214447456477314984682258437, 8.309988181002088436631245163516, 8.704723714868392880586206525073, 9.005143251180905756070549648234, 9.578734759383615817765128385501, 9.980151991246154127191365621933, 10.52292302953065660462191370122

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