Properties

Label 8-3040e4-1.1-c1e4-0-5
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 $4$

Origins

Origins of factors

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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: \(4\)
Selberg data: \((8,\ 2^{20} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−6.55681383502581379241536657821, −6.28979584302188614158725544481, −6.12962529523963639421332292126, −6.11598071931275420850188347579, −6.02656234366650591509893354852, −5.53235337870454220830269275731, −5.33798853004363415490341309574, −5.06666762529730812252551471033, −5.05577143445902499195788382008, −4.74572253576747119414717233593, −4.34643707014948884884533216622, −4.20121716683316481391955119021, −3.95934216913525218572669779748, −3.86720483455137807043800473674, −3.59874423279411195369117667436, −3.49772456487021535195278252685, −3.38286339968352686063976977862, −2.87580278148223007771841052522, −2.63252602184875401793574193522, −2.62608282104647283928607884271, −2.57388503909319832812387332922, −1.83274378701399444687965125692, −1.79921411454893221575727194163, −1.36310213805739471874167821405, −1.00478088833368203970031020972, 0, 0, 0, 0, 1.00478088833368203970031020972, 1.36310213805739471874167821405, 1.79921411454893221575727194163, 1.83274378701399444687965125692, 2.57388503909319832812387332922, 2.62608282104647283928607884271, 2.63252602184875401793574193522, 2.87580278148223007771841052522, 3.38286339968352686063976977862, 3.49772456487021535195278252685, 3.59874423279411195369117667436, 3.86720483455137807043800473674, 3.95934216913525218572669779748, 4.20121716683316481391955119021, 4.34643707014948884884533216622, 4.74572253576747119414717233593, 5.05577143445902499195788382008, 5.06666762529730812252551471033, 5.33798853004363415490341309574, 5.53235337870454220830269275731, 6.02656234366650591509893354852, 6.11598071931275420850188347579, 6.12962529523963639421332292126, 6.28979584302188614158725544481, 6.55681383502581379241536657821

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