Properties

Label 8-96e4-1.1-c2e4-0-3
Degree $8$
Conductor $84934656$
Sign $1$
Analytic cond. $46.8193$
Root an. cond. $1.61734$
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  + 6·9-s + 32·11-s − 8·17-s − 32·19-s + 28·25-s + 40·41-s − 32·43-s + 76·49-s + 128·59-s + 256·67-s + 200·73-s + 27·81-s − 160·83-s − 200·89-s + 56·97-s + 192·99-s + 320·107-s − 344·113-s + 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.90·11-s − 0.470·17-s − 1.68·19-s + 1.11·25-s + 0.975·41-s − 0.744·43-s + 1.55·49-s + 2.16·59-s + 3.82·67-s + 2.73·73-s + 1/3·81-s − 1.92·83-s − 2.24·89-s + 0.577·97-s + 1.93·99-s + 2.99·107-s − 3.04·113-s + 1.28·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.313·153-s + 0.00636·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(84934656\)    =    \(2^{20} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(46.8193\)
Root analytic conductor: \(1.61734\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 84934656,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.683083744\)
\(L(\frac12)\) \(\approx\) \(2.683083744\)
\(L(2)\) 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)$
bad2 \( 1 \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2 \wr C_2$ \( 1 - 28 T^{2} + 678 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{4} \)
13$C_2^2 \wr C_2$ \( 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} \)
17$D_{4}$ \( ( 1 + 4 T + 390 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 16 T + 738 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8} \)
31$C_2^2 \wr C_2$ \( 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8} \)
37$C_2^2 \wr C_2$ \( 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8} \)
41$D_{4}$ \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 16 T + 3330 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8} \)
59$D_{4}$ \( ( 1 - 64 T + 7554 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 128 T + 12642 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 100 T + 10086 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8} \)
83$D_{4}$ \( ( 1 + 80 T + 14610 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 28 T + 12102 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−10.07658144442517021937209412550, −9.650031991667628483914487083457, −9.525964134755084538091811670113, −9.193307244819598524491510999011, −8.984623839582391828825584813326, −8.629283470338112893205699526480, −8.353235085482889908631035353596, −8.201039569535917050627134093300, −7.72527875843249215604856316257, −7.09762586723807530275583481679, −6.81659379983574007725336817223, −6.81294755735736500508846832778, −6.64185458853050468309583150079, −6.02012144720189482322314123805, −5.95283299149645828436300591544, −5.10212536718697014132861994491, −5.06533523333496423088131741856, −4.44289649370149009860206090090, −3.89574298449500398082031596806, −3.83767617505855486641527890487, −3.76758436922444267332777499929, −2.52380985762095774734926340682, −2.35503825140727610443945900475, −1.46272874626522415542166720358, −0.929413234716795791901505677024, 0.929413234716795791901505677024, 1.46272874626522415542166720358, 2.35503825140727610443945900475, 2.52380985762095774734926340682, 3.76758436922444267332777499929, 3.83767617505855486641527890487, 3.89574298449500398082031596806, 4.44289649370149009860206090090, 5.06533523333496423088131741856, 5.10212536718697014132861994491, 5.95283299149645828436300591544, 6.02012144720189482322314123805, 6.64185458853050468309583150079, 6.81294755735736500508846832778, 6.81659379983574007725336817223, 7.09762586723807530275583481679, 7.72527875843249215604856316257, 8.201039569535917050627134093300, 8.353235085482889908631035353596, 8.629283470338112893205699526480, 8.984623839582391828825584813326, 9.193307244819598524491510999011, 9.525964134755084538091811670113, 9.650031991667628483914487083457, 10.07658144442517021937209412550

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