Properties

Label 8-720e4-1.1-c0e4-0-1
Degree $8$
Conductor $268738560000$
Sign $1$
Analytic cond. $0.0166708$
Root an. cond. $0.599438$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16-s + 4·19-s + 4·49-s − 4·61-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 16-s + 4·19-s + 4·49-s − 4·61-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.0166708\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8045466795\)
\(L(\frac12)\) \(\approx\) \(0.8045466795\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3 \( 1 \)
5$C_2^2$ \( 1 + T^{4} \)
good7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.62033086590413236406189592231, −7.45237123227378816983632751686, −7.32067924072005349420657053653, −7.13931642202132727536844916255, −6.85744546489755928801315468096, −6.55672195386414276608856355551, −6.36693237660249894171426637406, −5.94177936700939655715355141630, −5.74710953651620593703347171559, −5.69949412272248130018087159535, −5.34522420386373569232417709751, −4.96641370577638982848358959965, −4.95770460266557193651478883978, −4.69447148435520395844182353373, −4.32768987079081907197032935784, −3.76971133479913460119449386306, −3.74322955803488103635970183538, −3.71103122058905116286250460304, −3.01702673398197661037316773425, −2.83457145538099995172097371699, −2.51040726453255872723452292787, −2.50138346045873189197283043621, −1.65089928933564993302833260922, −1.24946640468036301038987573849, −1.13445124268816426143868428178, 1.13445124268816426143868428178, 1.24946640468036301038987573849, 1.65089928933564993302833260922, 2.50138346045873189197283043621, 2.51040726453255872723452292787, 2.83457145538099995172097371699, 3.01702673398197661037316773425, 3.71103122058905116286250460304, 3.74322955803488103635970183538, 3.76971133479913460119449386306, 4.32768987079081907197032935784, 4.69447148435520395844182353373, 4.95770460266557193651478883978, 4.96641370577638982848358959965, 5.34522420386373569232417709751, 5.69949412272248130018087159535, 5.74710953651620593703347171559, 5.94177936700939655715355141630, 6.36693237660249894171426637406, 6.55672195386414276608856355551, 6.85744546489755928801315468096, 7.13931642202132727536844916255, 7.32067924072005349420657053653, 7.45237123227378816983632751686, 7.62033086590413236406189592231

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