Properties

Degree $8$
Conductor $390625$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·4-s + 290·9-s − 392·11-s − 523·16-s + 6.36e3·19-s + 7.84e3·29-s − 2.19e3·31-s − 1.45e3·36-s + 5.55e4·41-s + 1.96e3·44-s + 4.53e4·49-s − 2.39e4·59-s − 4.87e4·61-s + 235·64-s − 1.74e5·71-s − 3.18e4·76-s − 1.30e5·79-s + 4.13e4·81-s + 1.45e5·89-s − 1.13e5·99-s + 1.46e5·101-s + 4.59e5·109-s − 3.92e4·116-s − 2.46e5·121-s + 1.09e4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.156·4-s + 1.19·9-s − 0.976·11-s − 0.510·16-s + 4.04·19-s + 1.73·29-s − 0.409·31-s − 0.186·36-s + 5.15·41-s + 0.152·44-s + 2.69·49-s − 0.894·59-s − 1.67·61-s + 0.00717·64-s − 4.11·71-s − 0.631·76-s − 2.36·79-s + 0.700·81-s + 1.94·89-s − 1.16·99-s + 1.43·101-s + 3.70·109-s − 0.270·116-s − 1.53·121-s + 0.0640·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(390625\)    =    \(5^{8}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 390625,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.19535\)
\(L(\frac12)\) \(\approx\) \(3.19535\)
\(L(\frac{7}{2})\) 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)$
bad5 \( 1 \)
good2$D_4\times C_2$ \( 1 + 5 T^{2} + 137 p^{2} T^{4} + 5 p^{10} T^{6} + p^{20} T^{8} \)
3$D_4\times C_2$ \( 1 - 290 T^{2} + 4747 p^{2} T^{4} - 290 p^{10} T^{6} + p^{20} T^{8} \)
7$D_4\times C_2$ \( 1 - 45300 T^{2} + 1039412998 T^{4} - 45300 p^{10} T^{6} + p^{20} T^{8} \)
11$D_{4}$ \( ( 1 + 196 T + 181081 T^{2} + 196 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 1296980 T^{2} + 688260462198 T^{4} - 1296980 p^{10} T^{6} + p^{20} T^{8} \)
17$D_4\times C_2$ \( 1 - 2340610 T^{2} + 2927557675523 T^{4} - 2340610 p^{10} T^{6} + p^{20} T^{8} \)
19$D_{4}$ \( ( 1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 24511220 T^{2} + 233031884985798 T^{4} - 24511220 p^{10} T^{6} + p^{20} T^{8} \)
29$D_{4}$ \( ( 1 - 3920 T + 44478298 T^{2} - 3920 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 1096 T - 15343894 T^{2} + 1096 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 257567180 T^{2} + 26166130610514198 T^{4} - 257567180 p^{10} T^{6} + p^{20} T^{8} \)
41$D_{4}$ \( ( 1 - 27754 T + 414643531 T^{2} - 27754 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 37863500 T^{2} + 41125859560630998 T^{4} - 37863500 p^{10} T^{6} + p^{20} T^{8} \)
47$D_4\times C_2$ \( 1 - 298326940 T^{2} + 32136931855726598 T^{4} - 298326940 p^{10} T^{6} + p^{20} T^{8} \)
53$D_4\times C_2$ \( 1 - 1240678540 T^{2} + 709794326287792598 T^{4} - 1240678540 p^{10} T^{6} + p^{20} T^{8} \)
59$D_{4}$ \( ( 1 + 11960 T + 1234152598 T^{2} + 11960 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 4294993410 T^{2} + 8014205534576787523 T^{4} - 4294993410 p^{10} T^{6} + p^{20} T^{8} \)
71$D_{4}$ \( ( 1 + 87296 T + 5453356606 T^{2} + 87296 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 5672940770 T^{2} + 16272726164284201923 T^{4} - 5672940770 p^{10} T^{6} + p^{20} T^{8} \)
79$D_{4}$ \( ( 1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 11381926610 T^{2} + 63038986097658341523 T^{4} - 11381926610 p^{10} T^{6} + p^{20} T^{8} \)
89$D_{4}$ \( ( 1 - 72810 T + 5926578523 T^{2} - 72810 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 11961289340 T^{2} + 68433641741002238598 T^{4} - 11961289340 p^{10} T^{6} + p^{20} T^{8} \)
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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

−12.14540939484042289857060710099, −11.90523044566214385587897925862, −11.49928751644936514407336830291, −11.33436925882232191672499237582, −10.57556128662513838306914408334, −10.39122002952179453712342248818, −10.20211745679126918043741198789, −9.672876579993069266008313221120, −9.258705529028324016959758688327, −9.036303311022754685227716307362, −8.764450992946443129523729325976, −7.72132452802252651792505504663, −7.48680807645462244794859444809, −7.48600928504577769048600575301, −7.26605538204975956496636512817, −6.13054596981145812317308703548, −5.96381000751118474725766750654, −5.44693637640829885635126766000, −4.76564772446388301793929903484, −4.50796753060133859929917454680, −3.84125762573755899005461780097, −2.84117052438160518418823621882, −2.73982120197850173732765105466, −1.26538541216834812006344965203, −0.830399145543979191803325939975, 0.830399145543979191803325939975, 1.26538541216834812006344965203, 2.73982120197850173732765105466, 2.84117052438160518418823621882, 3.84125762573755899005461780097, 4.50796753060133859929917454680, 4.76564772446388301793929903484, 5.44693637640829885635126766000, 5.96381000751118474725766750654, 6.13054596981145812317308703548, 7.26605538204975956496636512817, 7.48600928504577769048600575301, 7.48680807645462244794859444809, 7.72132452802252651792505504663, 8.764450992946443129523729325976, 9.036303311022754685227716307362, 9.258705529028324016959758688327, 9.672876579993069266008313221120, 10.20211745679126918043741198789, 10.39122002952179453712342248818, 10.57556128662513838306914408334, 11.33436925882232191672499237582, 11.49928751644936514407336830291, 11.90523044566214385587897925862, 12.14540939484042289857060710099

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