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Properties

Label 8-5445e4-1.1-c1e4-0-2
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  + 3·2-s + 4-s − 4·5-s − 6·7-s − 7·8-s − 12·10-s − 7·13-s − 18·14-s − 10·16-s + 10·17-s − 9·19-s − 4·20-s + 3·23-s + 10·25-s − 21·26-s − 6·28-s + 15·29-s − 13·31-s − 32-s + 30·34-s + 24·35-s − 3·37-s − 27·38-s + 28·40-s + 22·41-s + 9·46-s + 2·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 1/2·4-s − 1.78·5-s − 2.26·7-s − 2.47·8-s − 3.79·10-s − 1.94·13-s − 4.81·14-s − 5/2·16-s + 2.42·17-s − 2.06·19-s − 0.894·20-s + 0.625·23-s + 2·25-s − 4.11·26-s − 1.13·28-s + 2.78·29-s − 2.33·31-s − 0.176·32-s + 5.14·34-s + 4.05·35-s − 0.493·37-s − 4.37·38-s + 4.42·40-s + 3.43·41-s + 1.32·46-s + 0.291·47-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

\(L(1)\) \(\approx\) \(2.680518563\)
\(L(\frac12)\) \(\approx\) \(2.680518563\)
\(L(\frac{3}{2})\) 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$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + p^{3} T^{2} - 7 p T^{3} + 23 T^{4} - 7 p^{2} T^{5} + p^{5} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 38 T^{2} + 129 T^{3} + 433 T^{4} + 129 p T^{5} + 38 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 62 T^{2} + 267 T^{3} + 1263 T^{4} + 267 p T^{5} + 62 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 84 T^{2} - 500 T^{3} + 2277 T^{4} - 500 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 98 T^{2} + 529 T^{3} + 3003 T^{4} + 529 p T^{5} + 98 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 37 T^{2} - 78 T^{3} + 1093 T^{4} - 78 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 120 T^{2} - 895 T^{3} + 5777 T^{4} - 895 p T^{5} + 120 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 107 T^{2} + 602 T^{3} + 3463 T^{4} + 602 p T^{5} + 107 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 27 T^{2} + 120 T^{3} + 2801 T^{4} + 120 p T^{5} + 27 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 302 T^{2} - 2913 T^{3} + 523 p T^{4} - 2913 p T^{5} + 302 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 62 T^{2} + 375 T^{3} + 1909 T^{4} + 375 p T^{5} + 62 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 96 T^{2} - 7 p T^{3} + 4637 T^{4} - 7 p^{2} T^{5} + 96 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 128 T^{2} + 775 T^{3} + 7309 T^{4} + 775 p T^{5} + 128 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 353 T^{2} - 3746 T^{3} + 34103 T^{4} - 3746 p T^{5} + 353 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 156 T^{2} + 1147 T^{3} + 12741 T^{4} + 1147 p T^{5} + 156 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - T + 168 T^{2} - 89 T^{3} + 14153 T^{4} - 89 p T^{5} + 168 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 227 T^{2} - 2382 T^{3} + 22433 T^{4} - 2382 p T^{5} + 227 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + T + 218 T^{2} + 135 T^{3} + 22101 T^{4} + 135 p T^{5} + 218 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 227 T^{2} - 662 T^{3} + 23215 T^{4} - 662 p T^{5} + 227 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 255 T^{2} - 600 T^{3} + 28593 T^{4} - 600 p T^{5} + 255 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 3 p T^{2} - 2360 T^{3} + 31893 T^{4} - 2360 p T^{5} + 3 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 22 T + 476 T^{2} + 6179 T^{3} + 72367 T^{4} + 6179 p T^{5} + 476 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.81838524244242231129416032100, −5.29758328987668496556178315151, −5.22370336950938861215155568484, −5.12616424865540169856656501940, −4.97603691355528032482903619575, −4.62738810461890059264459019615, −4.61541700433339256510327857219, −4.37440684231329582000972757845, −4.27348318379266641738087207320, −4.03874939681459987594564604879, −3.69374592479664980794296282284, −3.66209741067202119529095632891, −3.60993208351145967217812960714, −3.19398742062003389408047053717, −3.10642955683042394951692401140, −2.94311656298813691108482038487, −2.86822422940152526205248874718, −2.41419810660417517615620660204, −2.28223239334179280661818481951, −1.89336923195852666457738531826, −1.57365290290787952423054453250, −0.986588212519492156897995969034, −0.55665835876674829324235420335, −0.54459351823411936705292415728, −0.32545693568164759740014700188, 0.32545693568164759740014700188, 0.54459351823411936705292415728, 0.55665835876674829324235420335, 0.986588212519492156897995969034, 1.57365290290787952423054453250, 1.89336923195852666457738531826, 2.28223239334179280661818481951, 2.41419810660417517615620660204, 2.86822422940152526205248874718, 2.94311656298813691108482038487, 3.10642955683042394951692401140, 3.19398742062003389408047053717, 3.60993208351145967217812960714, 3.66209741067202119529095632891, 3.69374592479664980794296282284, 4.03874939681459987594564604879, 4.27348318379266641738087207320, 4.37440684231329582000972757845, 4.61541700433339256510327857219, 4.62738810461890059264459019615, 4.97603691355528032482903619575, 5.12616424865540169856656501940, 5.22370336950938861215155568484, 5.29758328987668496556178315151, 5.81838524244242231129416032100

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