Properties

Label 8-1014e4-1.1-c1e4-0-5
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $4297.93$
Root an. cond. $2.84549$
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  + 2·3-s + 4-s + 9-s + 2·12-s + 4·17-s + 12·25-s − 2·27-s − 12·29-s + 36-s + 8·43-s + 2·49-s + 8·51-s − 40·53-s + 4·61-s − 64-s + 4·68-s + 24·75-s + 32·79-s − 4·81-s − 24·87-s + 12·100-s − 4·101-s − 64·103-s − 24·107-s − 2·108-s + 12·113-s − 12·116-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 0.970·17-s + 12/5·25-s − 0.384·27-s − 2.22·29-s + 1/6·36-s + 1.21·43-s + 2/7·49-s + 1.12·51-s − 5.49·53-s + 0.512·61-s − 1/8·64-s + 0.485·68-s + 2.77·75-s + 3.60·79-s − 4/9·81-s − 2.57·87-s + 6/5·100-s − 0.398·101-s − 6.30·103-s − 2.32·107-s − 0.192·108-s + 1.12·113-s − 1.11·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(4297.93\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.607131743\)
\(L(\frac12)\) \(\approx\) \(3.607131743\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \)
41$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
43$C_2^2$ \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 + 102 T^{2} + 6923 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
71$C_2^3$ \( 1 + 78 T^{2} + 1043 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 18 T^{2} - 7597 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.98505464745093035386424058786, −6.95557565334194861427303394131, −6.72617146178320099045695754933, −6.67558110156609001274842377349, −6.19735001497661621941806821516, −6.03395454614493201863073695107, −5.87340093876526780398250693671, −5.46263376969867313293770309364, −5.19794544272011510995124710335, −5.18839788149181929584059259586, −4.90062169100557344684646140803, −4.45597645166152038457802064950, −4.35507349827546269520540980801, −3.86582321560939600417178721843, −3.85871708507883369556338997599, −3.30553108629550334747084292505, −3.27624232899731830590955182173, −3.01959727394717631804046158324, −2.76477429205027418481081924013, −2.31512185939463015109385080253, −2.28953336633807563442895864782, −1.60290371862982725135754429192, −1.49784687045497768096058282702, −1.14810231307952175467939165403, −0.36798407069961602701292391149, 0.36798407069961602701292391149, 1.14810231307952175467939165403, 1.49784687045497768096058282702, 1.60290371862982725135754429192, 2.28953336633807563442895864782, 2.31512185939463015109385080253, 2.76477429205027418481081924013, 3.01959727394717631804046158324, 3.27624232899731830590955182173, 3.30553108629550334747084292505, 3.85871708507883369556338997599, 3.86582321560939600417178721843, 4.35507349827546269520540980801, 4.45597645166152038457802064950, 4.90062169100557344684646140803, 5.18839788149181929584059259586, 5.19794544272011510995124710335, 5.46263376969867313293770309364, 5.87340093876526780398250693671, 6.03395454614493201863073695107, 6.19735001497661621941806821516, 6.67558110156609001274842377349, 6.72617146178320099045695754933, 6.95557565334194861427303394131, 6.98505464745093035386424058786

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