Properties

Label 8-2240e4-1.1-c1e4-0-9
Degree $8$
Conductor $2.518\times 10^{13}$
Sign $1$
Analytic cond. $102352.$
Root an. cond. $4.22924$
Motivic weight $1$
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  + 10·9-s − 10·25-s − 4·29-s − 2·49-s + 40·61-s + 57·81-s + 56·89-s − 8·101-s − 60·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 10/3·9-s − 2·25-s − 0.742·29-s − 2/7·49-s + 5.12·61-s + 19/3·81-s + 5.93·89-s − 0.796·101-s − 5.74·109-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(102352.\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.871401728\)
\(L(\frac12)\) \(\approx\) \(4.871401728\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 87 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 189 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.38857888190956179183808445893, −6.38748240772806625723192131078, −6.30787715043698031376826514963, −5.50876031497222174143493923568, −5.46695701416177221568204135411, −5.42853753794673142791958022815, −5.27324172503916018765511062727, −4.99067705464786828848853516993, −4.55959488596816723376912827017, −4.55512528583073675220146868242, −4.08866089473425745504222418192, −4.04173861688421788630604594692, −3.93196490711746188289785885559, −3.71087713920433097833165598756, −3.59883459846303452851210336219, −3.19015667997840440688566057518, −2.78933448044841655034375021694, −2.49382488499642862704336093923, −2.13190167974315958213565080951, −2.03164135567164852220885381282, −1.80515070423288133806582461702, −1.49244610810978637579401269319, −1.02840239041770209080745946756, −0.943062176170527011755228258069, −0.36284780573880206912029201834, 0.36284780573880206912029201834, 0.943062176170527011755228258069, 1.02840239041770209080745946756, 1.49244610810978637579401269319, 1.80515070423288133806582461702, 2.03164135567164852220885381282, 2.13190167974315958213565080951, 2.49382488499642862704336093923, 2.78933448044841655034375021694, 3.19015667997840440688566057518, 3.59883459846303452851210336219, 3.71087713920433097833165598756, 3.93196490711746188289785885559, 4.04173861688421788630604594692, 4.08866089473425745504222418192, 4.55512528583073675220146868242, 4.55959488596816723376912827017, 4.99067705464786828848853516993, 5.27324172503916018765511062727, 5.42853753794673142791958022815, 5.46695701416177221568204135411, 5.50876031497222174143493923568, 6.30787715043698031376826514963, 6.38748240772806625723192131078, 6.38857888190956179183808445893

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