Properties

Label 8-4608e4-1.1-c1e4-0-4
Degree $8$
Conductor $4.509\times 10^{14}$
Sign $1$
Analytic cond. $1.83298\times 10^{6}$
Root an. cond. $6.06589$
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  − 8·17-s + 16·25-s + 24·41-s − 24·49-s + 32·73-s + 8·89-s − 56·97-s − 24·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 1.94·17-s + 16/5·25-s + 3.74·41-s − 3.42·49-s + 3.74·73-s + 0.847·89-s − 5.68·97-s − 2.25·113-s + 3.27·121-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 + 4·169-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 + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.83298\times 10^{6}\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.369213138\)
\(L(\frac12)\) \(\approx\) \(1.369213138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 140 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
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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

−5.81581248942971836272300801721, −5.76431547908572845294945901719, −5.34979043634966811847091014094, −5.25175197735978266990159926893, −5.08133374374905111586069591822, −4.90500264392061985260636658766, −4.71644448163048735560586080506, −4.44413664508296496714480593828, −4.31783802286267246989101793499, −4.11395722423283056664546112277, −4.03859908282393807157700676854, −3.67143712359787460018648099296, −3.35310689775266721256322981158, −3.26906844975400471772941219146, −2.89722773300279085305839051145, −2.75109823258849035998954484800, −2.53689249433176150367220977743, −2.49135906024492660545991812208, −2.03067494994519599869099220131, −1.88860240265936070572036459358, −1.47752378891968983728573047630, −1.24496234369651940898923662399, −0.854833749996929003678924908984, −0.75062795931874660358265460023, −0.16411757004543384290744746077, 0.16411757004543384290744746077, 0.75062795931874660358265460023, 0.854833749996929003678924908984, 1.24496234369651940898923662399, 1.47752378891968983728573047630, 1.88860240265936070572036459358, 2.03067494994519599869099220131, 2.49135906024492660545991812208, 2.53689249433176150367220977743, 2.75109823258849035998954484800, 2.89722773300279085305839051145, 3.26906844975400471772941219146, 3.35310689775266721256322981158, 3.67143712359787460018648099296, 4.03859908282393807157700676854, 4.11395722423283056664546112277, 4.31783802286267246989101793499, 4.44413664508296496714480593828, 4.71644448163048735560586080506, 4.90500264392061985260636658766, 5.08133374374905111586069591822, 5.25175197735978266990159926893, 5.34979043634966811847091014094, 5.76431547908572845294945901719, 5.81581248942971836272300801721

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