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Properties

Label 8-605e4-1.1-c1e4-0-4
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
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  + 2-s − 4·4-s − 4·5-s + 3·7-s − 4·8-s − 6·9-s − 4·10-s + 13-s + 3·14-s + 8·16-s − 17-s − 6·18-s + 20·19-s + 16·20-s + 5·23-s + 10·25-s + 26-s − 5·27-s − 12·28-s + 12·29-s − 5·31-s + 5·32-s − 34-s − 12·35-s + 24·36-s + 7·37-s + 20·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 2·4-s − 1.78·5-s + 1.13·7-s − 1.41·8-s − 2·9-s − 1.26·10-s + 0.277·13-s + 0.801·14-s + 2·16-s − 0.242·17-s − 1.41·18-s + 4.58·19-s + 3.57·20-s + 1.04·23-s + 2·25-s + 0.196·26-s − 0.962·27-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 + 4·36-s + 1.15·37-s + 3.24·38-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.958381285\)
\(L(\frac12)\) \(\approx\) \(1.958381285\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - T + 5 T^{2} - 5 T^{3} + 13 T^{4} - 5 p T^{5} + 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} + 5 T^{3} + 17 T^{4} + 5 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
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

−7.71442780400619689241060220696, −7.62658616698664626668176241454, −7.31456276770968938883249095825, −7.27822439335855927412177174593, −6.82163698092168120350873667973, −6.28378395144707040694834602043, −6.04102516397187895854466916531, −5.88521181159174531679615875841, −5.67013066239056324865369256834, −5.19029008891484340722498173788, −5.04383778001913608768518182240, −4.98754922558056189803189294065, −4.91671366030455785039037208773, −4.47081105832619136421584044285, −4.15358336999075292429009242956, −3.94342670419475847079348860797, −3.74048541888976094103557258536, −3.14893485823379414840832848208, −3.14635319896418929007149145221, −3.12761496644223067044287698684, −2.51616992306310626144715593287, −2.03825751663853670094581721175, −1.01759306538832874510871088712, −0.853182081451121548450904398732, −0.66691259212154053818966833566, 0.66691259212154053818966833566, 0.853182081451121548450904398732, 1.01759306538832874510871088712, 2.03825751663853670094581721175, 2.51616992306310626144715593287, 3.12761496644223067044287698684, 3.14635319896418929007149145221, 3.14893485823379414840832848208, 3.74048541888976094103557258536, 3.94342670419475847079348860797, 4.15358336999075292429009242956, 4.47081105832619136421584044285, 4.91671366030455785039037208773, 4.98754922558056189803189294065, 5.04383778001913608768518182240, 5.19029008891484340722498173788, 5.67013066239056324865369256834, 5.88521181159174531679615875841, 6.04102516397187895854466916531, 6.28378395144707040694834602043, 6.82163698092168120350873667973, 7.27822439335855927412177174593, 7.31456276770968938883249095825, 7.62658616698664626668176241454, 7.71442780400619689241060220696

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