Properties

Label 4-40e4-1.1-c1e2-0-31
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

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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

\(L(1)\) \(\approx\) \(3.190837977\)
\(L(\frac12)\) \(\approx\) \(3.190837977\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.513871003874312272869143754425, −9.187612383945413102225316721841, −9.051407543055912242328520920807, −8.286011623142564029466650767447, −7.85560562529560915898058281441, −7.79477033434019170830710090174, −7.26643077585279397498803879420, −6.76219782009384726421444939190, −6.29381265691880785284170621135, −6.19029348020805220188607267134, −5.37008833478037499539663023357, −5.09860777925640542831425659063, −4.60232248809496972242441447434, −4.22043832916332029336124542381, −3.69931065778313444397527750891, −2.93304773998191735352605200088, −2.82914192240295278569376546481, −2.05140123874588340287551303767, −1.10178150921557309047208423578, −0.881076883402113655854665800668, 0.881076883402113655854665800668, 1.10178150921557309047208423578, 2.05140123874588340287551303767, 2.82914192240295278569376546481, 2.93304773998191735352605200088, 3.69931065778313444397527750891, 4.22043832916332029336124542381, 4.60232248809496972242441447434, 5.09860777925640542831425659063, 5.37008833478037499539663023357, 6.19029348020805220188607267134, 6.29381265691880785284170621135, 6.76219782009384726421444939190, 7.26643077585279397498803879420, 7.79477033434019170830710090174, 7.85560562529560915898058281441, 8.286011623142564029466650767447, 9.051407543055912242328520920807, 9.187612383945413102225316721841, 9.513871003874312272869143754425

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