Properties

Label 4-60e4-1.1-c1e2-0-1
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
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  − 8·19-s − 12·29-s − 16·31-s + 12·41-s − 2·49-s − 20·61-s + 16·79-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 + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 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 + 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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3578326089\)
\(L(\frac12)\) \(\approx\) \(0.3578326089\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−8.851360859386031912399177039006, −8.377956340485385274708632349968, −7.82418348307318354353349829140, −7.61771289620363914235698612092, −7.46178076909973119732509138908, −6.84408241163408889096336058762, −6.54567292886018637274414191838, −5.98272706848108211284266755523, −5.86604763262579823906092225746, −5.44009592867965687463818999012, −4.85538987404167509287345743113, −4.62332084524881264141357429385, −3.95914062930162466804642858021, −3.69961591616037270434502724535, −3.47182278170122975687984566840, −2.62470554339778977087531446619, −2.18276099877530284891619454517, −1.87179097192214152748665309522, −1.25912666591265772788830563511, −0.17443951772062310226240414068, 0.17443951772062310226240414068, 1.25912666591265772788830563511, 1.87179097192214152748665309522, 2.18276099877530284891619454517, 2.62470554339778977087531446619, 3.47182278170122975687984566840, 3.69961591616037270434502724535, 3.95914062930162466804642858021, 4.62332084524881264141357429385, 4.85538987404167509287345743113, 5.44009592867965687463818999012, 5.86604763262579823906092225746, 5.98272706848108211284266755523, 6.54567292886018637274414191838, 6.84408241163408889096336058762, 7.46178076909973119732509138908, 7.61771289620363914235698612092, 7.82418348307318354353349829140, 8.377956340485385274708632349968, 8.851360859386031912399177039006

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