Properties

Label 8-60e4-1.1-c1e4-0-1
Degree $8$
Conductor $12960000$
Sign $1$
Analytic cond. $0.0526882$
Root an. cond. $0.692172$
Motivic weight $1$
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  + 4-s − 6·9-s − 3·16-s + 10·25-s − 6·36-s − 28·49-s − 8·61-s − 7·64-s + 27·81-s + 10·100-s + 56·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 18·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1/2·4-s − 2·9-s − 3/4·16-s + 2·25-s − 36-s − 4·49-s − 1.02·61-s − 7/8·64-s + 3·81-s + 100-s + 5.36·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.0526882\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 12960000,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5814071280\)
\(L(\frac12)\) \(\approx\) \(0.5814071280\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−11.26835583302356152021837947426, −11.21786645245511680209390523208, −10.66466370514893032831262775356, −10.50611649443584399182254907936, −10.23193054429604855466787590971, −9.652140086505478420732511583620, −9.384786331947788898180251109218, −9.031933621810800486731780558847, −8.710604288240186882615011007122, −8.680362817376058769151202987593, −7.997428208638764822405330422004, −7.84751197970826154652720172425, −7.62130075998172126003502019318, −6.75590916740167205077218524161, −6.64970921095770118350862487943, −6.55054750427678082702588075641, −5.91230804043502411888542382903, −5.61553303794249346366429241746, −5.11991821723020958559461200269, −4.71240468314306366469445297338, −4.44720620051296434479123125516, −3.34562197376954970251902214197, −3.21612165045091494677779403163, −2.71428527524336496712653239371, −1.91753482664858860475141963054, 1.91753482664858860475141963054, 2.71428527524336496712653239371, 3.21612165045091494677779403163, 3.34562197376954970251902214197, 4.44720620051296434479123125516, 4.71240468314306366469445297338, 5.11991821723020958559461200269, 5.61553303794249346366429241746, 5.91230804043502411888542382903, 6.55054750427678082702588075641, 6.64970921095770118350862487943, 6.75590916740167205077218524161, 7.62130075998172126003502019318, 7.84751197970826154652720172425, 7.997428208638764822405330422004, 8.680362817376058769151202987593, 8.710604288240186882615011007122, 9.031933621810800486731780558847, 9.384786331947788898180251109218, 9.652140086505478420732511583620, 10.23193054429604855466787590971, 10.50611649443584399182254907936, 10.66466370514893032831262775356, 11.21786645245511680209390523208, 11.26835583302356152021837947426

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