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Properties

Label 8-1200e4-1.1-c0e4-0-1
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $0.128633$
Root an. cond. $0.773872$
Motivic weight $0$
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  − 16-s + 4·19-s + 4·49-s + 4·61-s − 8·79-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 16-s + 4·19-s + 4·49-s + 4·61-s − 8·79-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.128633\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188821036\)
\(L(\frac12)\) \(\approx\) \(1.188821036\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
3$C_2^2$ \( 1 + T^{4} \)
5 \( 1 \)
good7$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^2$ \( ( 1 + T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + T^{4} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
79$C_1$ \( ( 1 + T )^{8} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−7.17941249348476041711943969770, −7.05560212437867572219302770829, −6.90586631075897811206785935729, −6.78944905782335106903409069493, −6.16401463795883761344139401552, −5.82519521958052877072932764571, −5.82370807001801801500211143230, −5.61352216969456553572159878063, −5.57226206177328113976871753949, −5.25136674907686538888713125825, −4.99501247829726113877168130486, −4.61261543005880542674168139020, −4.37948971945085264098297565671, −4.27611281541011731829627389402, −4.08158034082665512924573218032, −3.52839064144937048498977292805, −3.35123632032944131225093329490, −3.24688955565716854955685822371, −2.97651785823872670969237959653, −2.44930427984677945439635248465, −2.32311407062823274440728446077, −2.18684375825663080811168665870, −1.41653257128971909545445554546, −1.06212636651580308915662667725, −1.01483283962229245839022808768, 1.01483283962229245839022808768, 1.06212636651580308915662667725, 1.41653257128971909545445554546, 2.18684375825663080811168665870, 2.32311407062823274440728446077, 2.44930427984677945439635248465, 2.97651785823872670969237959653, 3.24688955565716854955685822371, 3.35123632032944131225093329490, 3.52839064144937048498977292805, 4.08158034082665512924573218032, 4.27611281541011731829627389402, 4.37948971945085264098297565671, 4.61261543005880542674168139020, 4.99501247829726113877168130486, 5.25136674907686538888713125825, 5.57226206177328113976871753949, 5.61352216969456553572159878063, 5.82370807001801801500211143230, 5.82519521958052877072932764571, 6.16401463795883761344139401552, 6.78944905782335106903409069493, 6.90586631075897811206785935729, 7.05560212437867572219302770829, 7.17941249348476041711943969770

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