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Properties

Label 8-600e4-1.1-c0e4-0-2
Degree $8$
Conductor $129600000000$
Sign $1$
Analytic cond. $0.00803958$
Root an. cond. $0.547210$
Motivic weight $0$
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  − 16-s − 81-s + 4·121-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 + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 16-s − 81-s + 4·121-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 + 233-s + 239-s + 241-s + 251-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(0.00803958\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5955645218\)
\(L(\frac12)\) \(\approx\) \(0.5955645218\)
\(L(1)\) 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$C_2^2$ \( ( 1 + T^{4} )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
13$C_2^2$ \( ( 1 + T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + T^{4} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−7.891029457169636612642187342351, −7.71542836175261784312940339271, −7.29711525702640386291335044664, −7.24158237298587336364111639163, −7.13116825587942048008052427899, −6.58737435379548800679241011965, −6.52609314592678570021388224505, −6.37184887812364498097616006368, −6.05759419280903329947891325220, −5.61989522060658386020253946963, −5.45707401313710585920192779646, −5.44616737155818135769791162795, −4.80417957660723987035738135581, −4.80159844471133660201479248304, −4.35853059717107534012039461718, −4.21470765862868866762739786483, −4.04886870152949275112666470514, −3.39863170052532323827461680497, −3.32480159013121887546173704780, −3.05179862368931192218855114284, −2.60587304832788196123186125509, −2.16481294294981664926714320973, −2.05537942047708543105388562806, −1.52492142328152317322830966531, −0.934443192434912603597736463747, 0.934443192434912603597736463747, 1.52492142328152317322830966531, 2.05537942047708543105388562806, 2.16481294294981664926714320973, 2.60587304832788196123186125509, 3.05179862368931192218855114284, 3.32480159013121887546173704780, 3.39863170052532323827461680497, 4.04886870152949275112666470514, 4.21470765862868866762739786483, 4.35853059717107534012039461718, 4.80159844471133660201479248304, 4.80417957660723987035738135581, 5.44616737155818135769791162795, 5.45707401313710585920192779646, 5.61989522060658386020253946963, 6.05759419280903329947891325220, 6.37184887812364498097616006368, 6.52609314592678570021388224505, 6.58737435379548800679241011965, 7.13116825587942048008052427899, 7.24158237298587336364111639163, 7.29711525702640386291335044664, 7.71542836175261784312940339271, 7.891029457169636612642187342351

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