Properties

Label 8-75e4-1.1-c2e4-0-3
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $17.4415$
Root an. cond. $1.42954$
Motivic weight $2$
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  + 24·11-s + 16·16-s + 100·31-s − 240·41-s − 148·61-s + 528·71-s − 9·81-s + 384·101-s − 124·121-s + ⋯
L(s)  = 1  + 2.18·11-s + 16-s + 3.22·31-s − 5.85·41-s − 2.42·61-s + 7.43·71-s − 1/9·81-s + 3.80·101-s − 1.02·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(17.4415\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.164004310\)
\(L(\frac12)\) \(\approx\) \(2.164004310\)
\(L(2)\) 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)$
bad3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 - p^{4} T^{4} + p^{8} T^{8} \)
7$C_2^3$ \( 1 - 4273 T^{4} + p^{8} T^{8} \)
11$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{4} \)
13$C_2^3$ \( 1 + 39599 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 - 166942 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 193 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 14882 T^{4} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 1646 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 25 T + p^{2} T^{2} )^{4} \)
37$C_2^3$ \( 1 - 1382878 T^{4} + p^{8} T^{8} \)
41$C_2$ \( ( 1 + 60 T + p^{2} T^{2} )^{4} \)
43$C_2^3$ \( 1 + 5447423 T^{4} + p^{8} T^{8} \)
47$C_2^3$ \( 1 + 8816738 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 3737762 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 - 6638 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 37 T + p^{2} T^{2} )^{4} \)
67$C_2^3$ \( 1 + 18296783 T^{4} + p^{8} T^{8} \)
71$C_2$ \( ( 1 - 132 T + p^{2} T^{2} )^{4} \)
73$C_2^3$ \( 1 + 32657282 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 12382 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 94586114 T^{4} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 137471663 T^{4} + p^{8} T^{8} \)
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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

−10.73313285649708825216194191616, −9.979312106933349297940642928424, −9.854452544617375886750276873513, −9.825520015002470066158079617471, −9.571373820714294532874240338461, −8.834241635868928579751834673959, −8.792159796464316481280703112612, −8.286301967998567619072060482068, −8.286212671758363353112453966722, −7.933750501455744754928093117542, −7.35522672622542769213856283456, −6.98403542956305800646220986393, −6.45366319561090326663556086140, −6.43638066083236128175001783863, −6.43472920017098589228149225837, −5.70361512322432404007507444070, −5.03268717009551690126860067223, −4.89626252124949471795528775813, −4.64396468834940077900665989565, −3.66387028279258430218307520044, −3.64230797300429977544416017103, −3.34001919396738248977279837510, −2.40820235898044724990435222190, −1.65734021545161170150571757133, −1.00593379324106127071407779195, 1.00593379324106127071407779195, 1.65734021545161170150571757133, 2.40820235898044724990435222190, 3.34001919396738248977279837510, 3.64230797300429977544416017103, 3.66387028279258430218307520044, 4.64396468834940077900665989565, 4.89626252124949471795528775813, 5.03268717009551690126860067223, 5.70361512322432404007507444070, 6.43472920017098589228149225837, 6.43638066083236128175001783863, 6.45366319561090326663556086140, 6.98403542956305800646220986393, 7.35522672622542769213856283456, 7.933750501455744754928093117542, 8.286212671758363353112453966722, 8.286301967998567619072060482068, 8.792159796464316481280703112612, 8.834241635868928579751834673959, 9.571373820714294532874240338461, 9.825520015002470066158079617471, 9.854452544617375886750276873513, 9.979312106933349297940642928424, 10.73313285649708825216194191616

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