↑

Properties

Label 8-575e4-1.1-c1e4-0-2
Degree $8$
Conductor $109312890625$
Sign $1$
Analytic cond. $444.405$
Root an. cond. $2.14275$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s − 4·6-s + 3·7-s + 8-s − 9-s + 4·11-s − 6·14-s + 2·16-s + 17-s + 2·18-s − 4·19-s + 6·21-s − 8·22-s + 4·23-s + 2·24-s − 2·27-s + 19·29-s − 31-s − 2·32-s + 8·33-s − 2·34-s + 3·37-s + 8·38-s + 13·41-s − 12·42-s + 6·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1.63·6-s + 1.13·7-s + 0.353·8-s − 1/3·9-s + 1.20·11-s − 1.60·14-s + 1/2·16-s + 0.242·17-s + 0.471·18-s − 0.917·19-s + 1.30·21-s − 1.70·22-s + 0.834·23-s + 0.408·24-s − 0.384·27-s + 3.52·29-s − 0.179·31-s − 0.353·32-s + 1.39·33-s − 0.342·34-s + 0.493·37-s + 1.29·38-s + 2.03·41-s − 1.85·42-s + 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(444.405\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.885948596\)
\(L(\frac12)\) \(\approx\) \(1.885948596\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + p^{2} T^{2} + 7 T^{3} + 5 p T^{4} + 7 p T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$D_{4}$ \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 2 p T^{2} - 11 T^{3} + 66 T^{4} - 11 p T^{5} + 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 28 T^{2} - 92 T^{3} + 406 T^{4} - 92 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 11 T^{2} + 160 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - T + 50 T^{2} - 27 T^{3} + 1154 T^{4} - 27 p T^{5} + 50 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 60 T^{2} + 188 T^{3} + 1590 T^{4} + 188 p T^{5} + 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 233 T^{2} - 1922 T^{3} + 12034 T^{4} - 1922 p T^{5} + 233 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + T + 23 T^{2} + 104 T^{3} + 1648 T^{4} + 104 p T^{5} + 23 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 32 T^{2} - 349 T^{3} + 1638 T^{4} - 349 p T^{5} + 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 209 T^{2} - 1602 T^{3} + 13682 T^{4} - 1602 p T^{5} + 209 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 136 T^{2} - 758 T^{3} + 8126 T^{4} - 758 p T^{5} + 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 105 T^{2} + 298 T^{3} + 5324 T^{4} + 298 p T^{5} + 105 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 178 T^{2} + 929 T^{3} + 4474 T^{4} + 929 p T^{5} + 178 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 23 T + 336 T^{2} - 3511 T^{3} + 29550 T^{4} - 3511 p T^{5} + 336 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 188 T^{2} + 136 T^{3} + 15462 T^{4} + 136 p T^{5} + 188 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 170 T^{2} - 391 T^{3} + 15834 T^{4} - 391 p T^{5} + 170 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 135 T^{2} + 104 T^{3} + 9080 T^{4} + 104 p T^{5} + 135 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 635 T^{2} - 8400 T^{3} + 83736 T^{4} - 8400 p T^{5} + 635 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 176 T^{2} - 826 T^{3} + 15838 T^{4} - 826 p T^{5} + 176 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 428 T^{2} - 5005 T^{3} + 56054 T^{4} - 5005 p T^{5} + 428 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 140 T^{2} - 1496 T^{3} + 6326 T^{4} - 1496 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 460 T^{2} - 5038 T^{3} + 69350 T^{4} - 5038 p T^{5} + 460 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
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

−7.963417746135657912745761459046, −7.73165095935106642287185348720, −7.54633314367246065431864967000, −6.95766523957726088913756993468, −6.88505590445169359957037214029, −6.65501434051028953912884227501, −6.20464532483750848170133489440, −6.19793482056054653519540181052, −6.12938815990328371463142812601, −5.34868729542929459468453791655, −5.33734943185786415814925965610, −4.89821639022827424842606205994, −4.65846765624495401587858561159, −4.62873757401045513689162271667, −4.12332864172076314820056943170, −3.97486853439401001304470510294, −3.38203602047130816303858143486, −3.28900299351073401635048515195, −3.06292148989148608128267343911, −2.40532099323058439947713651133, −2.39591454939295799992968904885, −2.03076493003610732027814470346, −1.33629019094940189757918261626, −0.803561033654616124973485372450, −0.77528697202795990716319501340, 0.77528697202795990716319501340, 0.803561033654616124973485372450, 1.33629019094940189757918261626, 2.03076493003610732027814470346, 2.39591454939295799992968904885, 2.40532099323058439947713651133, 3.06292148989148608128267343911, 3.28900299351073401635048515195, 3.38203602047130816303858143486, 3.97486853439401001304470510294, 4.12332864172076314820056943170, 4.62873757401045513689162271667, 4.65846765624495401587858561159, 4.89821639022827424842606205994, 5.33734943185786415814925965610, 5.34868729542929459468453791655, 6.12938815990328371463142812601, 6.19793482056054653519540181052, 6.20464532483750848170133489440, 6.65501434051028953912884227501, 6.88505590445169359957037214029, 6.95766523957726088913756993468, 7.54633314367246065431864967000, 7.73165095935106642287185348720, 7.963417746135657912745761459046

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