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Properties

Label 8-1920e4-1.1-c0e4-0-2
Degree $8$
Conductor $135895.450\times 10^{8}$
Sign $1$
Analytic cond. $0.843011$
Root an. cond. $0.978879$
Motivic weight $0$
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  + 4·19-s − 4·49-s − 4·61-s + 8·79-s − 81-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 + ⋯
L(s)  = 1  + 4·19-s − 4·49-s − 4·61-s + 8·79-s − 81-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.843011\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540452346\)
\(L(\frac12)\) \(\approx\) \(1.540452346\)
\(L(1)\) 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 \( 1 \)
3$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
good7$C_2$ \( ( 1 + T^{2} )^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
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$C_1$ \( ( 1 - T )^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
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

−6.70403297876323933341092950439, −6.60406978827796684763351198882, −6.22870055801824354069459311506, −6.15482831036092311680169594509, −6.02832272998309290544903564838, −5.53044713783399615094760808943, −5.48446583179660042208604885098, −5.32846993092474744399879322591, −4.93037100735730426237105588860, −4.89964597872888503742949498674, −4.66616075109678715808851924896, −4.54147855639560133824732523319, −4.21035481134347670454651788897, −3.72322816787976880019680435729, −3.57781372830606672855267773889, −3.25647727923477409085927961514, −3.13451694512454482195420610889, −3.08555575098576558348423473765, −2.98967766695846537264953587272, −2.17186569299943386037840278972, −1.95913707563857676612491659535, −1.95778654842782115418464481243, −1.38653300217874830927539256674, −1.02629549577599849375829471900, −0.77884833184188450067356919233, 0.77884833184188450067356919233, 1.02629549577599849375829471900, 1.38653300217874830927539256674, 1.95778654842782115418464481243, 1.95913707563857676612491659535, 2.17186569299943386037840278972, 2.98967766695846537264953587272, 3.08555575098576558348423473765, 3.13451694512454482195420610889, 3.25647727923477409085927961514, 3.57781372830606672855267773889, 3.72322816787976880019680435729, 4.21035481134347670454651788897, 4.54147855639560133824732523319, 4.66616075109678715808851924896, 4.89964597872888503742949498674, 4.93037100735730426237105588860, 5.32846993092474744399879322591, 5.48446583179660042208604885098, 5.53044713783399615094760808943, 6.02832272998309290544903564838, 6.15482831036092311680169594509, 6.22870055801824354069459311506, 6.60406978827796684763351198882, 6.70403297876323933341092950439

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