Properties

Label 4-3120e2-1.1-c1e2-0-0
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
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  − 4·5-s − 6·7-s − 9-s − 6·13-s + 11·25-s + 24·35-s + 6·37-s + 4·45-s − 16·47-s + 13·49-s − 6·61-s + 6·63-s + 24·65-s + 24·67-s − 12·73-s − 30·79-s + 81-s − 8·83-s + 36·91-s + 6·97-s + 24·101-s + 6·117-s − 3·121-s − 24·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.26·7-s − 1/3·9-s − 1.66·13-s + 11/5·25-s + 4.05·35-s + 0.986·37-s + 0.596·45-s − 2.33·47-s + 13/7·49-s − 0.768·61-s + 0.755·63-s + 2.97·65-s + 2.93·67-s − 1.40·73-s − 3.37·79-s + 1/9·81-s − 0.878·83-s + 3.77·91-s + 0.609·97-s + 2.38·101-s + 0.554·117-s − 0.272·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.05682899733\)
\(L(\frac12)\) \(\approx\) \(0.05682899733\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 177 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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

−9.610875655153587179872753532007, −8.377346363819862429331601921879, −8.207960417147461859007466228689, −7.67627774964196119136531151262, −7.28575463076288826635226459460, −7.11063598978960415044995989270, −6.73577070795522839884670585312, −6.26827814234845853291634767943, −6.05627846078621860102002908410, −5.37544663091936709678045041423, −4.98864235403725474970793800424, −4.46161971148202931605443683765, −4.24372691868397224377443368508, −3.64526661755535967956231440113, −3.13768625253344074599359553333, −3.09936762043146093806057794328, −2.63173403053387437006948673871, −1.88210786682119234839347269830, −0.801781087417461336246036292767, −0.10123670417556176823877703294, 0.10123670417556176823877703294, 0.801781087417461336246036292767, 1.88210786682119234839347269830, 2.63173403053387437006948673871, 3.09936762043146093806057794328, 3.13768625253344074599359553333, 3.64526661755535967956231440113, 4.24372691868397224377443368508, 4.46161971148202931605443683765, 4.98864235403725474970793800424, 5.37544663091936709678045041423, 6.05627846078621860102002908410, 6.26827814234845853291634767943, 6.73577070795522839884670585312, 7.11063598978960415044995989270, 7.28575463076288826635226459460, 7.67627774964196119136531151262, 8.207960417147461859007466228689, 8.377346363819862429331601921879, 9.610875655153587179872753532007

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