Properties

Label 4-1200e2-1.1-c1e2-0-38
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
Motivic weight $1$
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  + 3·3-s + 6·9-s + 6·11-s + 4·13-s + 12·23-s + 9·27-s + 18·33-s − 16·37-s + 12·39-s − 12·47-s + 14·49-s + 24·59-s + 16·61-s + 36·69-s − 12·71-s − 2·73-s + 9·81-s − 18·83-s − 20·97-s + 36·99-s + 6·107-s − 16·109-s − 48·111-s + 24·117-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s + 1.80·11-s + 1.10·13-s + 2.50·23-s + 1.73·27-s + 3.13·33-s − 2.63·37-s + 1.92·39-s − 1.75·47-s + 2·49-s + 3.12·59-s + 2.04·61-s + 4.33·69-s − 1.42·71-s − 0.234·73-s + 81-s − 1.97·83-s − 2.03·97-s + 3.61·99-s + 0.580·107-s − 1.53·109-s − 4.55·111-s + 2.21·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.997420365\)
\(L(\frac12)\) \(\approx\) \(5.997420365\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 151 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−9.636056766947829873974712111778, −9.567972938619317561013746762792, −8.946556696855410891653361674147, −8.589486773596952376594479886760, −8.561096986885643761215032663331, −8.302801054496911758025187629576, −7.22448923444748345220654041407, −7.13625477715377596711215652509, −6.90407213384807497456515070758, −6.46396752624243489975258546626, −5.66419977881165032911271124808, −5.30339302420644617768063060101, −4.64677931818851887307434951068, −4.05808383313783800148740209705, −3.62616091572041836974630010664, −3.50249802131170676754177676213, −2.79897978593093859497624378701, −2.23714710739069456453926632505, −1.33592265847091080086535612718, −1.21580348837780398920690467909, 1.21580348837780398920690467909, 1.33592265847091080086535612718, 2.23714710739069456453926632505, 2.79897978593093859497624378701, 3.50249802131170676754177676213, 3.62616091572041836974630010664, 4.05808383313783800148740209705, 4.64677931818851887307434951068, 5.30339302420644617768063060101, 5.66419977881165032911271124808, 6.46396752624243489975258546626, 6.90407213384807497456515070758, 7.13625477715377596711215652509, 7.22448923444748345220654041407, 8.302801054496911758025187629576, 8.561096986885643761215032663331, 8.589486773596952376594479886760, 8.946556696855410891653361674147, 9.567972938619317561013746762792, 9.636056766947829873974712111778

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