Properties

Label 4-1200e2-1.1-c1e2-0-6
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 about

Normalization:  

Dirichlet series

L(s)  = 1  − 9-s − 8·19-s + 12·29-s − 16·31-s − 12·41-s − 2·49-s − 20·61-s + 16·79-s + 81-s − 36·89-s + 36·101-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 8·171-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.83·19-s + 2.22·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s − 2.56·61-s + 1.80·79-s + 1/9·81-s − 3.81·89-s + 3.58·101-s + 1.91·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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

\(L(1)\) \(\approx\) \(1.123555713\)
\(L(\frac12)\) \(\approx\) \(1.123555713\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−10.19174573345121234503406816936, −9.412415368895010110577584049983, −9.102583885333548575319203067640, −8.634492796115289623641541591247, −8.513959693519003072099003833275, −7.923960150013263745472702459032, −7.53290580843691848524365423607, −7.03106501189624728289311285097, −6.52816372776241258618775144025, −6.31622233499071781882115027900, −5.82500079550649568890455986869, −5.09997914259797008869091954507, −5.00958422924308679007974735678, −4.19621709129075032138817413281, −4.00079934849080517680346014977, −3.11151715435156831813944606012, −2.95446901053280278700842005809, −1.90070444860112036439956286397, −1.75916814902390267633954093228, −0.43816537375513400236523009121, 0.43816537375513400236523009121, 1.75916814902390267633954093228, 1.90070444860112036439956286397, 2.95446901053280278700842005809, 3.11151715435156831813944606012, 4.00079934849080517680346014977, 4.19621709129075032138817413281, 5.00958422924308679007974735678, 5.09997914259797008869091954507, 5.82500079550649568890455986869, 6.31622233499071781882115027900, 6.52816372776241258618775144025, 7.03106501189624728289311285097, 7.53290580843691848524365423607, 7.923960150013263745472702459032, 8.513959693519003072099003833275, 8.634492796115289623641541591247, 9.102583885333548575319203067640, 9.412415368895010110577584049983, 10.19174573345121234503406816936

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