↑

Properties

Label 8-55e8-1.1-c1e4-0-7
Degree $8$
Conductor $8.373\times 10^{13}$
Sign $1$
Analytic cond. $340415.$
Root an. cond. $4.91474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·4-s + 3·7-s − 4·8-s − 6·9-s + 13-s + 3·14-s + 8·16-s − 17-s − 6·18-s − 20·19-s − 5·23-s + 26-s + 5·27-s − 12·28-s − 12·29-s − 5·31-s + 5·32-s − 34-s + 24·36-s − 7·37-s − 20·38-s − 11·41-s + 19·43-s − 5·46-s − 5·47-s − 8·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 2·4-s + 1.13·7-s − 1.41·8-s − 2·9-s + 0.277·13-s + 0.801·14-s + 2·16-s − 0.242·17-s − 1.41·18-s − 4.58·19-s − 1.04·23-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 + 4·36-s − 1.15·37-s − 3.24·38-s − 1.71·41-s + 2.89·43-s − 0.737·46-s − 0.729·47-s − 8/7·49-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(340415.\)
Root analytic conductor: \(4.91474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
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$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 598 T^{2} + 8416 T^{3} + 94183 T^{4} + 8416 p T^{5} + 598 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.57542678083022551073034917482, −6.10289680875602727868513214333, −6.03276093344065126175558895784, −5.93058801268673502156611849330, −5.74517768885541364992499140189, −5.53308939571691545605495331820, −5.18981435596052271098285748558, −5.17874514942422088074278133482, −5.01769134382177687201347553825, −4.64738874371400138588473654489, −4.62461086469371540094269045015, −4.26013875604725393084786097850, −4.09922146912808679316099545214, −3.96130282452095343879973662773, −3.72505608898561272047572671689, −3.64554049099246509150483798809, −3.61140089197373069154789462230, −2.72102883113919554852788625064, −2.70967446493084728628464806604, −2.66327820602116730324132191094, −2.16481757682081729486721054801, −2.01452882997318130861634647602, −1.76827699614898496038185961902, −1.31181507281753667769297886294, −1.13925083867890145318362143588, 0, 0, 0, 0, 1.13925083867890145318362143588, 1.31181507281753667769297886294, 1.76827699614898496038185961902, 2.01452882997318130861634647602, 2.16481757682081729486721054801, 2.66327820602116730324132191094, 2.70967446493084728628464806604, 2.72102883113919554852788625064, 3.61140089197373069154789462230, 3.64554049099246509150483798809, 3.72505608898561272047572671689, 3.96130282452095343879973662773, 4.09922146912808679316099545214, 4.26013875604725393084786097850, 4.62461086469371540094269045015, 4.64738874371400138588473654489, 5.01769134382177687201347553825, 5.17874514942422088074278133482, 5.18981435596052271098285748558, 5.53308939571691545605495331820, 5.74517768885541364992499140189, 5.93058801268673502156611849330, 6.03276093344065126175558895784, 6.10289680875602727868513214333, 6.57542678083022551073034917482

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