Properties

Label 8-48e4-1.1-c10e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $865041.$
Root an. cond. $5.52242$
Motivic weight $10$
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  − 7.56e3·5-s − 3.93e4·9-s − 5.36e5·13-s − 3.85e6·17-s + 2.86e7·25-s + 1.12e7·29-s + 3.05e7·37-s − 2.37e8·41-s + 2.97e8·45-s + 1.07e8·49-s + 8.35e8·53-s + 2.61e9·61-s + 4.05e9·65-s − 2.37e8·73-s + 1.16e9·81-s + 2.91e10·85-s − 1.11e10·89-s + 9.66e9·97-s − 3.73e10·101-s + 1.10e10·109-s − 7.13e10·113-s + 2.11e10·117-s + 6.79e10·121-s − 1.28e11·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2.41·5-s − 2/3·9-s − 1.44·13-s − 2.71·17-s + 2.93·25-s + 0.549·29-s + 0.439·37-s − 2.05·41-s + 1.61·45-s + 0.382·49-s + 1.99·53-s + 3.09·61-s + 3.49·65-s − 0.114·73-s + 1/3·81-s + 6.57·85-s − 2.00·89-s + 1.12·97-s − 3.55·101-s + 0.717·109-s − 3.87·113-s + 0.962·117-s + 2.62·121-s − 4.20·125-s − 1.33·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(865041.\)
Root analytic conductor: \(5.52242\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :5, 5, 5, 5),\ 1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.01266944810\)
\(L(\frac12)\) \(\approx\) \(0.01266944810\)
\(L(6)\) 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$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 + 756 p T + 1424446 p T^{2} + 756 p^{11} T^{3} + p^{20} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 107909092 T^{2} + 515752461638502 p^{2} T^{4} - 107909092 p^{20} T^{6} + p^{40} T^{8} \)
11$D_4\times C_2$ \( 1 - 67987804420 T^{2} + \)\(23\!\cdots\!82\)\( T^{4} - 67987804420 p^{20} T^{6} + p^{40} T^{8} \)
13$D_{4}$ \( ( 1 + 268156 T + 247092947862 T^{2} + 268156 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 1928556 T + 4674862871462 T^{2} + 1928556 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 11396009075908 T^{2} + \)\(67\!\cdots\!98\)\( T^{4} - 11396009075908 p^{20} T^{6} + p^{40} T^{8} \)
23$D_4\times C_2$ \( 1 - 57560728155460 T^{2} + \)\(42\!\cdots\!22\)\( T^{4} - 57560728155460 p^{20} T^{6} + p^{40} T^{8} \)
29$D_{4}$ \( ( 1 - 5639436 T + 551257283207606 T^{2} - 5639436 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1785509547336484 T^{2} + \)\(18\!\cdots\!66\)\( T^{4} - 1785509547336484 p^{20} T^{6} + p^{40} T^{8} \)
37$D_{4}$ \( ( 1 - 15255220 T + 9227793694532118 T^{2} - 15255220 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 118854540 T + 28110027532085702 T^{2} + 118854540 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3337921746848260 T^{2} + \)\(68\!\cdots\!22\)\( T^{4} - 3337921746848260 p^{20} T^{6} + p^{40} T^{8} \)
47$D_4\times C_2$ \( 1 - 22958275505088772 T^{2} - \)\(31\!\cdots\!82\)\( T^{4} - 22958275505088772 p^{20} T^{6} + p^{40} T^{8} \)
53$D_{4}$ \( ( 1 - 417942828 T + 380158218111049814 T^{2} - 417942828 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 1717674283247984260 T^{2} + \)\(12\!\cdots\!82\)\( T^{4} - 1717674283247984260 p^{20} T^{6} + p^{40} T^{8} \)
61$D_{4}$ \( ( 1 - 1306619188 T + 1842940593015440118 T^{2} - 1306619188 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 3067696319988122692 T^{2} + \)\(51\!\cdots\!98\)\( T^{4} - 3067696319988122692 p^{20} T^{6} + p^{40} T^{8} \)
71$D_4\times C_2$ \( 1 - 9494363384388213700 T^{2} + \)\(43\!\cdots\!22\)\( T^{4} - 9494363384388213700 p^{20} T^{6} + p^{40} T^{8} \)
73$D_{4}$ \( ( 1 + 118687324 T + 7332861243362857062 T^{2} + 118687324 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 15414374288754462628 T^{2} + \)\(19\!\cdots\!78\)\( T^{4} - 15414374288754462628 p^{20} T^{6} + p^{40} T^{8} \)
83$D_4\times C_2$ \( 1 - 27382880992584417220 T^{2} + \)\(47\!\cdots\!22\)\( T^{4} - 27382880992584417220 p^{20} T^{6} + p^{40} T^{8} \)
89$D_{4}$ \( ( 1 + 5592224988 T + 68730968898102971558 T^{2} + 5592224988 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 4834774532 T - 44713135597567105146 T^{2} - 4834774532 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
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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

−9.315370407868701870115926681555, −8.987628238820809734504908422878, −8.449185741521678697687193877080, −8.426933886975524637037265772011, −8.424686238122973697426421459989, −7.66955842773256931159394737737, −7.52532551532498228260503895036, −7.07823047332585690892729128613, −6.83039671506433051047785194352, −6.61978848703264835036107973620, −6.22928066882322428606268240190, −5.38371245421200411327623246558, −5.17362309786627251978527813839, −4.96475961004433933033473590597, −4.31034018875139257800680829458, −4.01479208203771715343971313912, −4.01139306594274148171715101194, −3.45353512436770603254777078516, −2.75300346806793427728377037284, −2.52678091650525696125288300052, −2.33437980719693062003971717686, −1.56491873190058335773866940602, −0.937439873612220542062592148374, −0.40177968077556841812839031985, −0.03420738398587589474792939913, 0.03420738398587589474792939913, 0.40177968077556841812839031985, 0.937439873612220542062592148374, 1.56491873190058335773866940602, 2.33437980719693062003971717686, 2.52678091650525696125288300052, 2.75300346806793427728377037284, 3.45353512436770603254777078516, 4.01139306594274148171715101194, 4.01479208203771715343971313912, 4.31034018875139257800680829458, 4.96475961004433933033473590597, 5.17362309786627251978527813839, 5.38371245421200411327623246558, 6.22928066882322428606268240190, 6.61978848703264835036107973620, 6.83039671506433051047785194352, 7.07823047332585690892729128613, 7.52532551532498228260503895036, 7.66955842773256931159394737737, 8.424686238122973697426421459989, 8.426933886975524637037265772011, 8.449185741521678697687193877080, 8.987628238820809734504908422878, 9.315370407868701870115926681555

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