Properties

Label 4-882e2-1.1-c3e2-0-10
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 6·5-s + 8·8-s + 12·10-s − 30·11-s + 4·13-s − 16·16-s − 66·17-s + 52·19-s + 60·22-s − 114·23-s + 125·25-s − 8·26-s + 144·29-s + 196·31-s + 132·34-s + 286·37-s − 104·38-s − 48·40-s − 756·41-s + 328·43-s + 228·46-s + 228·47-s − 250·50-s + 348·53-s + 180·55-s − 288·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.536·5-s + 0.353·8-s + 0.379·10-s − 0.822·11-s + 0.0853·13-s − 1/4·16-s − 0.941·17-s + 0.627·19-s + 0.581·22-s − 1.03·23-s + 25-s − 0.0603·26-s + 0.922·29-s + 1.13·31-s + 0.665·34-s + 1.27·37-s − 0.443·38-s − 0.189·40-s − 2.87·41-s + 1.16·43-s + 0.730·46-s + 0.707·47-s − 0.707·50-s + 0.901·53-s + 0.441·55-s − 0.652·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.109569399\)
\(L(\frac12)\) \(\approx\) \(1.109569399\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 66 T - 557 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 52 T - 4155 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 114 T + 829 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 72 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 196 T + 8625 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 286 T + 31143 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 378 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 228 T - 51839 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 348 T - 27773 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 348 T - 84275 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 106 T - 215745 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 630 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 1042 T + 696747 T^{2} - 1042 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 88 T - 485295 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 1440 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 1374 T + 1182907 T^{2} + 1374 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
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

−9.902790238805984039776109117079, −9.740992415260192216760967640354, −9.057431468015248125979292916824, −8.421329719653034666935414173357, −8.303940316831252033972606853681, −8.240383328163199529541607929321, −7.36776001539196694753564458443, −7.04495507028091005635289317297, −6.79882935808722504087402662393, −6.01771529573056678937269654869, −5.71739176850383292411010324277, −4.92811501049603110385809074857, −4.72738928702710004218330272263, −4.10960330366790843041070634525, −3.64499683281847913923765425896, −2.78522023454477561243176007189, −2.56448878178511533615433797122, −1.71707677918271716372354227296, −0.908403316575332959191683866485, −0.39633018111258639945708327238, 0.39633018111258639945708327238, 0.908403316575332959191683866485, 1.71707677918271716372354227296, 2.56448878178511533615433797122, 2.78522023454477561243176007189, 3.64499683281847913923765425896, 4.10960330366790843041070634525, 4.72738928702710004218330272263, 4.92811501049603110385809074857, 5.71739176850383292411010324277, 6.01771529573056678937269654869, 6.79882935808722504087402662393, 7.04495507028091005635289317297, 7.36776001539196694753564458443, 8.240383328163199529541607929321, 8.303940316831252033972606853681, 8.421329719653034666935414173357, 9.057431468015248125979292916824, 9.740992415260192216760967640354, 9.902790238805984039776109117079

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