Properties

Label 8-3072e4-1.1-c1e4-0-8
Degree $8$
Conductor $8.906\times 10^{13}$
Sign $1$
Analytic cond. $362070.$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 10·9-s + 8·11-s + 16·19-s − 4·25-s + 20·27-s + 32·33-s + 32·41-s − 16·49-s + 64·57-s + 32·67-s + 16·73-s − 16·75-s + 35·81-s + 8·83-s + 8·89-s + 32·97-s + 80·99-s − 16·107-s + 72·113-s + 20·121-s + 128·123-s + 127-s + 131-s + 137-s + 139-s − 64·147-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 2.41·11-s + 3.67·19-s − 4/5·25-s + 3.84·27-s + 5.57·33-s + 4.99·41-s − 2.28·49-s + 8.47·57-s + 3.90·67-s + 1.87·73-s − 1.84·75-s + 35/9·81-s + 0.878·83-s + 0.847·89-s + 3.24·97-s + 8.04·99-s − 1.54·107-s + 6.77·113-s + 1.81·121-s + 11.5·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.27·147-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(362070.\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.84986251\)
\(L(\frac12)\) \(\approx\) \(34.84986251\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
good5$D_4$ \( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
11$D_{4}$ \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 4 T^{2} + 1254 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 64 T^{2} + 2034 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 100 T^{2} + 6918 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 160 T^{2} + 12114 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 20 T^{2} - 2106 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 4 T + 158 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.40022793473364447458911177961, −5.86419575817852579177074794099, −5.79489898903480612273943841283, −5.74899536970872883666464557580, −5.27753602297995164610784448116, −5.03858634664365566238766702091, −4.79178558893748889434439273247, −4.72462497404040423643125040893, −4.57229136410099624432517569983, −3.96595193752890180410464905455, −3.92255310065725778516523564569, −3.86221845182884493905404584267, −3.76554254503579320989267947350, −3.31545313217385080417520618090, −3.16025019486193454359513104479, −3.07709379239301356467124176969, −2.93941496353539956998599925223, −2.21433665071209340817154489239, −2.17643494284857105043127354728, −2.16964616267818509208410897429, −1.86859272527778196603443171995, −1.13720292026675151979891718530, −1.11270505145013229578390795304, −0.963120409678928536648681214448, −0.72924079848520114127583645393, 0.72924079848520114127583645393, 0.963120409678928536648681214448, 1.11270505145013229578390795304, 1.13720292026675151979891718530, 1.86859272527778196603443171995, 2.16964616267818509208410897429, 2.17643494284857105043127354728, 2.21433665071209340817154489239, 2.93941496353539956998599925223, 3.07709379239301356467124176969, 3.16025019486193454359513104479, 3.31545313217385080417520618090, 3.76554254503579320989267947350, 3.86221845182884493905404584267, 3.92255310065725778516523564569, 3.96595193752890180410464905455, 4.57229136410099624432517569983, 4.72462497404040423643125040893, 4.79178558893748889434439273247, 5.03858634664365566238766702091, 5.27753602297995164610784448116, 5.74899536970872883666464557580, 5.79489898903480612273943841283, 5.86419575817852579177074794099, 6.40022793473364447458911177961

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