Properties

Label 8-3040e4-1.1-c1e4-0-1
Degree $8$
Conductor $8.541\times 10^{13}$
Sign $1$
Analytic cond. $347218.$
Root an. cond. $4.92691$
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  − 3-s − 4·5-s + 5·7-s − 9-s + 6·11-s + 5·13-s + 4·15-s − 5·17-s − 4·19-s − 5·21-s + 21·23-s + 10·25-s + 2·27-s − 29-s + 4·31-s − 6·33-s − 20·35-s + 10·37-s − 5·39-s − 2·41-s − 6·43-s + 4·45-s + 24·47-s + 49-s + 5·51-s − 5·53-s − 24·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1.88·7-s − 1/3·9-s + 1.80·11-s + 1.38·13-s + 1.03·15-s − 1.21·17-s − 0.917·19-s − 1.09·21-s + 4.37·23-s + 2·25-s + 0.384·27-s − 0.185·29-s + 0.718·31-s − 1.04·33-s − 3.38·35-s + 1.64·37-s − 0.800·39-s − 0.312·41-s − 0.914·43-s + 0.596·45-s + 3.50·47-s + 1/7·49-s + 0.700·51-s − 0.686·53-s − 3.23·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(347218.\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.316154622\)
\(L(\frac12)\) \(\approx\) \(5.316154622\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + T + 2 T^{2} + T^{3} + 4 p T^{4} + p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 5 T + 24 T^{2} - 65 T^{3} + 30 p T^{4} - 65 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 6 T + 36 T^{2} - 170 T^{3} + 582 T^{4} - 170 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 5 T + 42 T^{2} - 9 p T^{3} + 668 T^{4} - 9 p^{2} T^{5} + 42 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 5 T + 42 T^{2} + 91 T^{3} + 682 T^{4} + 91 p T^{5} + 42 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 21 T + 244 T^{2} - 1873 T^{3} + 10490 T^{4} - 1873 p T^{5} + 244 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + T + 62 T^{2} + 11 T^{3} + 1986 T^{4} + 11 p T^{5} + 62 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 4 T + 44 T^{2} - 388 T^{3} + 1558 T^{4} - 388 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 10 T + 160 T^{2} - 1068 T^{3} + 9058 T^{4} - 1068 p T^{5} + 160 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 2 T + 28 T^{2} + 86 T^{3} + 2390 T^{4} + 86 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 6 T + 100 T^{2} + 554 T^{3} + 5950 T^{4} + 554 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 24 T + 328 T^{2} - 3244 T^{3} + 25038 T^{4} - 3244 p T^{5} + 328 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 5 T + 144 T^{2} + 733 T^{3} + 10372 T^{4} + 733 p T^{5} + 144 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 11 T + 106 T^{2} - 735 T^{3} + 7858 T^{4} - 735 p T^{5} + 106 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 6 T + 236 T^{2} + 1070 T^{3} + 21382 T^{4} + 1070 p T^{5} + 236 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 19 T + 280 T^{2} + 2883 T^{3} + 26764 T^{4} + 2883 p T^{5} + 280 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 2 T + 232 T^{2} - 258 T^{3} + 22926 T^{4} - 258 p T^{5} + 232 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 5 T + 130 T^{2} - 315 T^{3} + 7450 T^{4} - 315 p T^{5} + 130 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 4 T + 220 T^{2} + 564 T^{3} + 22294 T^{4} + 564 p T^{5} + 220 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 10 T + 284 T^{2} - 2102 T^{3} + 34014 T^{4} - 2102 p T^{5} + 284 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 20 T + 380 T^{2} - 4764 T^{3} + 52230 T^{4} - 4764 p T^{5} + 380 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 6 T + 84 T^{2} + 784 T^{3} + 586 T^{4} + 784 p T^{5} + 84 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.33418631087002467373581064950, −5.94969110076985092628436029925, −5.65027970997018349561560336760, −5.59014830806570694585308208695, −5.45864757697308698381448172102, −4.94939376608863781529137762204, −4.82074321250868044922724125153, −4.62326830523207137709662596325, −4.61781691988556818517367887222, −4.40351454765831493365977760856, −4.21283183523785510363162492553, −3.93416559270493762085704377476, −3.81818948258221125358358272104, −3.37265983826205096697019155906, −3.15245234625981408337246521820, −3.06171377895269886352287931993, −3.00834377558978269323652494966, −2.29912738106816343959111198444, −2.19805967985969834514527739630, −1.96939576158349609987188687536, −1.40120941377115658260478596404, −1.22434621387784133529060562152, −1.02552313335756724967161731382, −0.76040197716327222908052154906, −0.44583666389836350551576647011, 0.44583666389836350551576647011, 0.76040197716327222908052154906, 1.02552313335756724967161731382, 1.22434621387784133529060562152, 1.40120941377115658260478596404, 1.96939576158349609987188687536, 2.19805967985969834514527739630, 2.29912738106816343959111198444, 3.00834377558978269323652494966, 3.06171377895269886352287931993, 3.15245234625981408337246521820, 3.37265983826205096697019155906, 3.81818948258221125358358272104, 3.93416559270493762085704377476, 4.21283183523785510363162492553, 4.40351454765831493365977760856, 4.61781691988556818517367887222, 4.62326830523207137709662596325, 4.82074321250868044922724125153, 4.94939376608863781529137762204, 5.45864757697308698381448172102, 5.59014830806570694585308208695, 5.65027970997018349561560336760, 5.94969110076985092628436029925, 6.33418631087002467373581064950

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