Properties

Label 4-40e4-1.1-c1e2-0-11
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  + 2·9-s − 8·19-s + 12·29-s − 8·31-s + 12·41-s + 10·49-s + 24·59-s − 4·61-s − 24·71-s − 16·79-s − 5·81-s + 12·89-s − 12·101-s + 4·109-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 − 16·171-s + 173-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.83·19-s + 2.22·29-s − 1.43·31-s + 1.87·41-s + 10/7·49-s + 3.12·59-s − 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s + 1.27·89-s − 1.19·101-s + 0.383·109-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 − 1.22·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068730941\)
\(L(\frac12)\) \(\approx\) \(2.068730941\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
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

−9.796311374032066566307355733499, −9.003077847639080326930611714784, −8.864181538070128002851988081823, −8.465104758803893787805325964614, −8.148814130656262106238896035439, −7.37457652619397995200233679116, −7.34619343535064306042996878642, −6.85224633743913090682177552781, −6.30625831673245170586613956228, −6.09434938433328747182718098873, −5.49539059958553844134401934777, −5.12032888678831571270740769770, −4.39855579451985909396387400672, −4.12878098925751391315630894068, −3.98231194533867730950445797965, −2.98265302954566036668057108245, −2.63561954949853613425876041889, −2.05174102390501109448796544699, −1.38482072697619602370463073392, −0.58128801619554062481232200609, 0.58128801619554062481232200609, 1.38482072697619602370463073392, 2.05174102390501109448796544699, 2.63561954949853613425876041889, 2.98265302954566036668057108245, 3.98231194533867730950445797965, 4.12878098925751391315630894068, 4.39855579451985909396387400672, 5.12032888678831571270740769770, 5.49539059958553844134401934777, 6.09434938433328747182718098873, 6.30625831673245170586613956228, 6.85224633743913090682177552781, 7.34619343535064306042996878642, 7.37457652619397995200233679116, 8.148814130656262106238896035439, 8.465104758803893787805325964614, 8.864181538070128002851988081823, 9.003077847639080326930611714784, 9.796311374032066566307355733499

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