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Properties

Label 8-9680e4-1.1-c1e4-0-6
Degree $8$
Conductor $8.780\times 10^{15}$
Sign $1$
Analytic cond. $3.56952\times 10^{7}$
Root an. cond. $8.79176$
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  − 4·5-s − 3·7-s − 6·9-s + 13-s − 17-s − 20·19-s − 5·23-s + 10·25-s + 5·27-s + 12·29-s + 5·31-s + 12·35-s + 7·37-s + 11·41-s − 19·43-s + 24·45-s − 5·47-s − 8·49-s − 11·53-s − 9·59-s + 12·61-s + 18·63-s − 4·65-s + 19·67-s − 5·71-s + 11·73-s − 34·79-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.13·7-s − 2·9-s + 0.277·13-s − 0.242·17-s − 4.58·19-s − 1.04·23-s + 2·25-s + 0.962·27-s + 2.22·29-s + 0.898·31-s + 2.02·35-s + 1.15·37-s + 1.71·41-s − 2.89·43-s + 3.57·45-s − 0.729·47-s − 8/7·49-s − 1.51·53-s − 1.17·59-s + 1.53·61-s + 2.26·63-s − 0.496·65-s + 2.32·67-s − 0.593·71-s + 1.28·73-s − 3.82·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.56952\times 10^{7}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} - 5 T^{3} + 17 T^{4} - 5 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
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

−5.95430897082631411497463374025, −5.54341895124594354299225961742, −5.34892083625859737365914785781, −5.18126176242549560505948788649, −4.77715266943125648306374186284, −4.71951377786776438824138885119, −4.59068618643107331610683139042, −4.58627376177203728531021127092, −4.40013582664552973478468409648, −3.98813183156905657674051357630, −3.84578704054518799661381113033, −3.72557943477329449842928239051, −3.67302069571452946633645718245, −3.24870669307758951432644738762, −3.00010097561422136249223346851, −2.98836653134635728271275514334, −2.92935112798719844106227151936, −2.42135371470593800247695561892, −2.30465590426029365971219000151, −2.29296500855155191772923758559, −2.03211207697037822270869088900, −1.59292011444179445847895636792, −1.25254771231527296724415334825, −0.914044305019185699156916939627, −0.77670354111748107666248555826, 0, 0, 0, 0, 0.77670354111748107666248555826, 0.914044305019185699156916939627, 1.25254771231527296724415334825, 1.59292011444179445847895636792, 2.03211207697037822270869088900, 2.29296500855155191772923758559, 2.30465590426029365971219000151, 2.42135371470593800247695561892, 2.92935112798719844106227151936, 2.98836653134635728271275514334, 3.00010097561422136249223346851, 3.24870669307758951432644738762, 3.67302069571452946633645718245, 3.72557943477329449842928239051, 3.84578704054518799661381113033, 3.98813183156905657674051357630, 4.40013582664552973478468409648, 4.58627376177203728531021127092, 4.59068618643107331610683139042, 4.71951377786776438824138885119, 4.77715266943125648306374186284, 5.18126176242549560505948788649, 5.34892083625859737365914785781, 5.54341895124594354299225961742, 5.95430897082631411497463374025

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