Properties

Label 8-320e4-1.1-c3e4-0-3
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $127076.$
Root an. cond. $4.34518$
Motivic weight $3$
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  + 12·5-s + 76·9-s − 142·25-s − 632·29-s − 680·41-s + 912·45-s + 1.34e3·49-s − 328·61-s + 2.87e3·81-s − 24·89-s − 4.05e3·101-s + 8.05e3·109-s − 1.61e3·121-s − 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 7.58e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.86e3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.07·5-s + 2.81·9-s − 1.13·25-s − 4.04·29-s − 2.59·41-s + 3.02·45-s + 3.90·49-s − 0.688·61-s + 3.94·81-s − 0.0285·89-s − 3.99·101-s + 7.07·109-s − 1.21·121-s − 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 4.34·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.57·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(127076.\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(5.153709569\)
\(L(\frac12)\) \(\approx\) \(5.153709569\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - 6 T + p^{3} T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 38 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 670 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 806 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 3930 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 7970 T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 2986 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 21630 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 158 T + p^{3} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 29886 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 22890 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 170 T + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 59158 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 148110 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 52298 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6842 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 82 T + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 122662 T^{2} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 182482 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 592434 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 867294 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 259974 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p^{3} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 665346 T^{2} + p^{6} T^{4} )^{2} \)
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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

−7.929878206461381410977859231740, −7.55492349048832832107277239078, −7.42493460130815172522521486026, −7.17571904774894764788738321438, −7.14162005363644173125536426614, −6.71465616967036987419443743749, −6.65524479306107812473410029939, −5.98266567921707765269299099833, −5.86555403235997451632619295975, −5.64946981802233357453117174886, −5.52634749362905251221890284059, −5.01346302371628956398508410082, −4.83900153161497861385355887015, −4.32877952344997045420504685331, −4.13948253143165837337601582924, −3.91067714977194054741124886494, −3.64395668175804502055556935066, −3.36492477554099515696462360290, −2.76986430481996602751039945105, −2.21437807986573636610887871890, −1.83517951677057620329067047007, −1.74938816787478721775664371872, −1.61640797696914413981665877773, −0.842347184336042013070015873529, −0.36941537318091857752422503598, 0.36941537318091857752422503598, 0.842347184336042013070015873529, 1.61640797696914413981665877773, 1.74938816787478721775664371872, 1.83517951677057620329067047007, 2.21437807986573636610887871890, 2.76986430481996602751039945105, 3.36492477554099515696462360290, 3.64395668175804502055556935066, 3.91067714977194054741124886494, 4.13948253143165837337601582924, 4.32877952344997045420504685331, 4.83900153161497861385355887015, 5.01346302371628956398508410082, 5.52634749362905251221890284059, 5.64946981802233357453117174886, 5.86555403235997451632619295975, 5.98266567921707765269299099833, 6.65524479306107812473410029939, 6.71465616967036987419443743749, 7.14162005363644173125536426614, 7.17571904774894764788738321438, 7.42493460130815172522521486026, 7.55492349048832832107277239078, 7.929878206461381410977859231740

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