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Properties

Label 8-5445e4-1.1-c1e4-0-5
Degree $8$
Conductor $8.790\times 10^{14}$
Sign $1$
Analytic cond. $3.57355\times 10^{6}$
Root an. cond. $6.59382$
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-s − 4·4-s + 4·5-s + 3·7-s + 4·8-s − 4·10-s + 13-s − 3·14-s + 8·16-s + 17-s + 20·19-s − 16·20-s − 5·23-s + 10·25-s − 26-s − 12·28-s − 12·29-s − 5·31-s − 5·32-s − 34-s + 12·35-s + 7·37-s − 20·38-s + 16·40-s − 11·41-s + 19·43-s + 5·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 2·4-s + 1.78·5-s + 1.13·7-s + 1.41·8-s − 1.26·10-s + 0.277·13-s − 0.801·14-s + 2·16-s + 0.242·17-s + 4.58·19-s − 3.57·20-s − 1.04·23-s + 2·25-s − 0.196·26-s − 2.26·28-s − 2.22·29-s − 0.898·31-s − 0.883·32-s − 0.171·34-s + 2.02·35-s + 1.15·37-s − 3.24·38-s + 2.52·40-s − 1.71·41-s + 2.89·43-s + 0.737·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.57355\times 10^{6}\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.161225935\)
\(L(\frac12)\) \(\approx\) \(7.161225935\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + T + 5 T^{2} + 5 T^{3} + 13 T^{4} + 5 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 17 T^{2} - 40 T^{3} + 171 T^{4} - 40 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - T + 27 T^{2} - 32 T^{3} + 503 T^{4} - 32 p T^{5} + 27 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - T + 48 T^{2} - 19 T^{3} + 1073 T^{4} - 19 p T^{5} + 48 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 206 T^{2} - 1415 T^{3} + 7131 T^{4} - 1415 p T^{5} + 206 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 96 T^{2} + 335 T^{3} + 3347 T^{4} + 335 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 136 T^{2} + 873 T^{3} + 5755 T^{4} + 873 p T^{5} + 136 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 49 T^{2} + 340 T^{3} + 1741 T^{4} + 340 p T^{5} + 49 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 48 T^{2} + 49 T^{3} - 337 T^{4} + 49 p T^{5} + 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 170 T^{2} + 1179 T^{3} + 10259 T^{4} + 1179 p T^{5} + 170 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 293 T^{2} - 2740 T^{3} + 21711 T^{4} - 2740 p T^{5} + 293 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 167 T^{2} + 640 T^{3} + 11449 T^{4} + 640 p T^{5} + 167 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 169 T^{2} - 1438 T^{3} + 13237 T^{4} - 1438 p T^{5} + 169 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 163 T^{2} + 1044 T^{3} + 11443 T^{4} + 1044 p T^{5} + 163 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 267 T^{2} - 2118 T^{3} + 24963 T^{4} - 2118 p T^{5} + 267 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 290 T^{2} + 2805 T^{3} + 25803 T^{4} + 2805 p T^{5} + 290 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 238 T^{2} + 895 T^{3} + 23583 T^{4} + 895 p T^{5} + 238 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 302 T^{2} - 2397 T^{3} + 33423 T^{4} - 2397 p T^{5} + 302 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 34 T + 652 T^{2} - 8357 T^{3} + 83755 T^{4} - 8357 p T^{5} + 652 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 233 T^{2} - 1500 T^{3} + 23201 T^{4} - 1500 p T^{5} + 233 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 254 T^{2} - 1664 T^{3} + 31231 T^{4} - 1664 p T^{5} + 254 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 598 T^{2} - 8416 T^{3} + 94183 T^{4} - 8416 p T^{5} + 598 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.78551499587026475692977543134, −5.41175250378560885017779075745, −5.31675978205562957668617287447, −5.24782715806797978630574137435, −5.11663234621609151601260680330, −4.85420496750639406846481785444, −4.54325117930890060491180517969, −4.53822302524968779606946948733, −4.46707862373855602004093978590, −3.88232241749436607814590223850, −3.68045586047959694982603565099, −3.61492691997818064242133730460, −3.48321136399821809845693734481, −3.15941053721900801697488218029, −2.99701728721222685117736591446, −2.80824504424917121707065571149, −2.34560232016506572514376134312, −1.98386665933140871145108068990, −1.89119075147286308815426900944, −1.67896075687322598330625708984, −1.65553492254295174200017440835, −0.977631091136392721414969935023, −0.75785672374658652774994530033, −0.62735264682104969961957670911, −0.61511304454935169058903052120, 0.61511304454935169058903052120, 0.62735264682104969961957670911, 0.75785672374658652774994530033, 0.977631091136392721414969935023, 1.65553492254295174200017440835, 1.67896075687322598330625708984, 1.89119075147286308815426900944, 1.98386665933140871145108068990, 2.34560232016506572514376134312, 2.80824504424917121707065571149, 2.99701728721222685117736591446, 3.15941053721900801697488218029, 3.48321136399821809845693734481, 3.61492691997818064242133730460, 3.68045586047959694982603565099, 3.88232241749436607814590223850, 4.46707862373855602004093978590, 4.53822302524968779606946948733, 4.54325117930890060491180517969, 4.85420496750639406846481785444, 5.11663234621609151601260680330, 5.24782715806797978630574137435, 5.31675978205562957668617287447, 5.41175250378560885017779075745, 5.78551499587026475692977543134

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