Properties

Label 4-1200e2-1.1-c1e2-0-30
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  − 9-s + 8·11-s + 8·19-s + 4·29-s + 20·41-s + 14·49-s − 8·59-s − 4·61-s + 16·71-s + 81-s + 12·89-s − 8·99-s + 12·101-s − 28·109-s + 26·121-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 + ⋯
L(s)  = 1  − 1/3·9-s + 2.41·11-s + 1.83·19-s + 0.742·29-s + 3.12·41-s + 2·49-s − 1.04·59-s − 0.512·61-s + 1.89·71-s + 1/9·81-s + 1.27·89-s − 0.804·99-s + 1.19·101-s − 2.68·109-s + 2.36·121-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 + ⋯

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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.138702211\)
\(L(\frac12)\) \(\approx\) \(3.138702211\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−9.725148343900171711124221399126, −9.379947411315422652347026087288, −9.152717564513935782288900288903, −9.057609742322537072649819836430, −8.283633133016687228917985778814, −7.939610145420223523523086760885, −7.34537144146800530912049430699, −7.20684727315671799935516232823, −6.42869984759876819019552063864, −6.39798139076487484662415891584, −5.65923955247976653081254887585, −5.52868566302576137975914323986, −4.60964085589104656837769710331, −4.41462643387234700486070338188, −3.68706878407768117812414703809, −3.52142704407444770719251388374, −2.73115294775246656089604331924, −2.20075590483635444950086539102, −1.07959321079286452203840082072, −1.04722992885745284208468418137, 1.04722992885745284208468418137, 1.07959321079286452203840082072, 2.20075590483635444950086539102, 2.73115294775246656089604331924, 3.52142704407444770719251388374, 3.68706878407768117812414703809, 4.41462643387234700486070338188, 4.60964085589104656837769710331, 5.52868566302576137975914323986, 5.65923955247976653081254887585, 6.39798139076487484662415891584, 6.42869984759876819019552063864, 7.20684727315671799935516232823, 7.34537144146800530912049430699, 7.939610145420223523523086760885, 8.283633133016687228917985778814, 9.057609742322537072649819836430, 9.152717564513935782288900288903, 9.379947411315422652347026087288, 9.725148343900171711124221399126

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