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Properties

Label 8-72e4-1.1-c6e4-0-1
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $75274.8$
Root an. cond. $4.06987$
Motivic weight $6$
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  + 16·2-s − 46·3-s + 64·4-s − 736·6-s − 1.02e3·8-s + 729·9-s − 2.33e3·11-s − 2.94e3·12-s − 1.63e4·16-s + 3.45e3·17-s + 1.16e4·18-s + 4.96e3·19-s − 3.74e4·22-s + 4.71e4·24-s − 3.12e4·25-s − 3.26e3·27-s − 6.55e4·32-s + 1.07e5·33-s + 5.52e4·34-s + 4.66e4·36-s + 7.94e4·38-s + 1.34e5·41-s − 7.49e4·43-s − 1.49e5·44-s + 7.53e5·48-s − 2.35e5·49-s − 5.00e5·50-s + ⋯
L(s)  = 1  + 2·2-s − 1.70·3-s + 4-s − 3.40·6-s − 2·8-s + 9-s − 1.75·11-s − 1.70·12-s − 4·16-s + 0.702·17-s + 2·18-s + 0.723·19-s − 3.51·22-s + 3.40·24-s − 2·25-s − 0.165·27-s − 2·32-s + 2.99·33-s + 1.40·34-s + 36-s + 1.44·38-s + 1.95·41-s − 0.942·43-s − 1.75·44-s + 6.81·48-s − 2·49-s − 4·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(75274.8\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :3, 3, 3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.05564874536\)
\(L(\frac12)\) \(\approx\) \(0.05564874536\)
\(L(4)\) 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$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \)
3$C_2^2$ \( 1 + 46 T + 1387 T^{2} + 46 p^{6} T^{3} + p^{12} T^{4} \)
good5$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + 2338 T + p^{6} T^{2} )^{2}( 1 - 2338 T + 3694683 T^{2} - 2338 p^{6} T^{3} + p^{12} T^{4} ) \)
13$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 - 1726 T - 21158493 T^{2} - 1726 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2482 T - 40885557 T^{2} - 2482 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
23$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
31$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \)
41$C_2$$\times$$C_2^2$ \( ( 1 - 134642 T + p^{6} T^{2} )^{2}( 1 + 134642 T + 13378363923 T^{2} + 134642 p^{6} T^{3} + p^{12} T^{4} ) \)
43$C_2$$\times$$C_2^2$ \( ( 1 + 74914 T + p^{6} T^{2} )^{2}( 1 - 74914 T - 709255653 T^{2} - 74914 p^{6} T^{3} + p^{12} T^{4} ) \)
47$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 304958 T + p^{6} T^{2} )^{2}( 1 + 304958 T + 50818848123 T^{2} + 304958 p^{6} T^{3} + p^{12} T^{4} ) \)
61$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 596626 T + p^{6} T^{2} )^{2}( 1 - 596626 T + 265504201707 T^{2} - 596626 p^{6} T^{3} + p^{12} T^{4} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \)
73$C_2^2$ \( ( 1 - 593134 T + 200473715667 T^{2} - 593134 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
79$C_2$ \( ( 1 - p^{3} T + p^{6} T^{2} )^{2}( 1 + p^{3} T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 678926 T + 134000140107 T^{2} + 678926 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 357262 T + p^{6} T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 1822754 T + p^{6} T^{2} )^{2}( 1 + 1822754 T + 2489460139587 T^{2} + 1822754 p^{6} T^{3} + p^{12} T^{4} ) \)
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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

−9.755590052095375907945503014750, −9.555913917026413845914249653998, −8.996203398777102256388928144726, −8.598654226236895243676773884189, −8.214874147951765205036725822843, −8.076983003410990563783208768612, −7.44382206039485962873858142031, −7.36665355460694209543425651532, −6.84144496556257648251471475281, −6.26984899395387005699752589136, −5.99299757296368637615936061798, −5.86491277149181523997300276094, −5.64229649764124377959026473301, −5.27386285978128633471717914636, −4.85546909803141734030723560388, −4.84016480205265311225534951083, −4.32598277945314832681280965462, −3.83720199693787020556173602748, −3.29443577496651659506057226117, −3.29260740100081068882816981840, −2.41625288535633742893240538869, −2.28344916346026899468383408901, −1.25218933788460665569657769865, −0.58607238921042014671026758117, −0.05215035899606472063758235683, 0.05215035899606472063758235683, 0.58607238921042014671026758117, 1.25218933788460665569657769865, 2.28344916346026899468383408901, 2.41625288535633742893240538869, 3.29260740100081068882816981840, 3.29443577496651659506057226117, 3.83720199693787020556173602748, 4.32598277945314832681280965462, 4.84016480205265311225534951083, 4.85546909803141734030723560388, 5.27386285978128633471717914636, 5.64229649764124377959026473301, 5.86491277149181523997300276094, 5.99299757296368637615936061798, 6.26984899395387005699752589136, 6.84144496556257648251471475281, 7.36665355460694209543425651532, 7.44382206039485962873858142031, 8.076983003410990563783208768612, 8.214874147951765205036725822843, 8.598654226236895243676773884189, 8.996203398777102256388928144726, 9.555913917026413845914249653998, 9.755590052095375907945503014750

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