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Properties

Label 8-51e8-1.1-c1e4-0-2
Degree $8$
Conductor $4.577\times 10^{13}$
Sign $1$
Analytic cond. $186067.$
Root an. cond. $4.55731$
Motivic weight $1$
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  + 2·2-s + 4-s + 6·5-s − 4·7-s + 12·10-s + 6·11-s + 2·13-s − 8·14-s − 16-s + 10·19-s + 6·20-s + 12·22-s − 6·23-s + 9·25-s + 4·26-s − 4·28-s + 16·29-s − 4·31-s − 2·32-s − 24·35-s − 12·37-s + 20·38-s − 14·41-s + 14·43-s + 6·44-s − 12·46-s − 4·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 2.68·5-s − 1.51·7-s + 3.79·10-s + 1.80·11-s + 0.554·13-s − 2.13·14-s − 1/4·16-s + 2.29·19-s + 1.34·20-s + 2.55·22-s − 1.25·23-s + 9/5·25-s + 0.784·26-s − 0.755·28-s + 2.97·29-s − 0.718·31-s − 0.353·32-s − 4.05·35-s − 1.97·37-s + 3.24·38-s − 2.18·41-s + 2.13·43-s + 0.904·44-s − 1.76·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(186067.\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 17^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(16.87808189\)
\(L(\frac12)\) \(\approx\) \(16.87808189\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
17 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + 3 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 27 T^{2} - 86 T^{3} + 218 T^{4} - 86 p T^{5} + 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 4 T + 22 T^{2} + 68 T^{3} + 226 T^{4} + 68 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 45 T^{2} - 174 T^{3} + 732 T^{4} - 174 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 29 T^{2} + 2 T^{3} + 352 T^{4} + 2 p T^{5} + 29 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 101 T^{2} - 30 p T^{3} + 3084 T^{4} - 30 p^{2} T^{5} + 101 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 77 T^{2} + 278 T^{3} + 100 p T^{4} + 278 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 186 T^{2} - 1472 T^{3} + 9090 T^{4} - 1472 p T^{5} + 186 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 104 T^{2} + 308 T^{3} + 4558 T^{4} + 308 p T^{5} + 104 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 4 p T^{2} + 988 T^{3} + 7442 T^{4} + 988 p T^{5} + 4 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 147 T^{2} + 1078 T^{3} + 7026 T^{4} + 1078 p T^{5} + 147 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 189 T^{2} - 1414 T^{3} + 11388 T^{4} - 1414 p T^{5} + 189 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 136 T^{2} + 404 T^{3} + 8302 T^{4} + 404 p T^{5} + 136 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 336 T^{2} - 3436 T^{3} + 30126 T^{4} - 3436 p T^{5} + 336 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 362 T^{2} - 3640 T^{3} + 31082 T^{4} - 3640 p T^{5} + 362 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 260 T^{2} + 2044 T^{3} + 23922 T^{4} + 2044 p T^{5} + 260 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 72 T^{2} + 220 T^{3} + 2718 T^{4} + 220 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 232 T^{2} - 788 T^{3} + 22894 T^{4} - 788 p T^{5} + 232 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 406 T^{2} + 4500 T^{3} + 48290 T^{4} + 4500 p T^{5} + 406 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 274 T^{2} + 1944 T^{3} + 30794 T^{4} + 1944 p T^{5} + 274 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 168 T^{2} - 868 T^{3} + 19358 T^{4} - 868 p T^{5} + 168 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 246 T^{2} + 716 T^{3} + 29034 T^{4} + 716 p T^{5} + 246 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 554 T^{2} + 7416 T^{3} + 89026 T^{4} + 7416 p T^{5} + 554 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.20618673110573439074245775753, −6.10351376711060952553768321756, −5.76392217517162998528855607608, −5.69838797761608838345579845659, −5.65022514189239177336155748539, −5.12352418114157760156083326668, −5.09365283102548073813281347185, −5.08964045102987032793588760322, −4.79255277283826474732865208162, −4.16762106151696017488692815626, −4.13006313016347211238080755343, −4.07063207948840865248369735412, −3.88055636929149595098882972955, −3.50496914994272462627180519163, −3.27507538201181152769641449474, −3.10063771355666350978177850962, −2.85903743046351064886108439926, −2.69200973308664024927522101805, −2.25190010893232849696383206063, −1.92897111751193044753882890818, −1.85083586631776684060365560938, −1.44947981851302267642509160819, −1.33025802781400670820512301819, −0.841696080020897582673071797337, −0.44537720398239109674726359902, 0.44537720398239109674726359902, 0.841696080020897582673071797337, 1.33025802781400670820512301819, 1.44947981851302267642509160819, 1.85083586631776684060365560938, 1.92897111751193044753882890818, 2.25190010893232849696383206063, 2.69200973308664024927522101805, 2.85903743046351064886108439926, 3.10063771355666350978177850962, 3.27507538201181152769641449474, 3.50496914994272462627180519163, 3.88055636929149595098882972955, 4.07063207948840865248369735412, 4.13006313016347211238080755343, 4.16762106151696017488692815626, 4.79255277283826474732865208162, 5.08964045102987032793588760322, 5.09365283102548073813281347185, 5.12352418114157760156083326668, 5.65022514189239177336155748539, 5.69838797761608838345579845659, 5.76392217517162998528855607608, 6.10351376711060952553768321756, 6.20618673110573439074245775753

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