Properties

Label 8-1014e4-1.1-c2e4-0-0
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $582763.$
Root an. cond. $5.25637$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 6·5-s − 8·7-s + 8·8-s + 6·9-s − 24·10-s − 24·11-s − 32·14-s − 4·16-s + 24·18-s − 8·19-s − 48·20-s − 96·22-s + 18·25-s − 64·28-s + 120·29-s − 88·31-s − 32·32-s + 48·35-s + 48·36-s + 26·37-s − 32·38-s − 48·40-s − 90·41-s − 192·44-s − 36·45-s + ⋯
L(s)  = 1  + 2·2-s + 2·4-s − 6/5·5-s − 8/7·7-s + 8-s + 2/3·9-s − 2.39·10-s − 2.18·11-s − 2.28·14-s − 1/4·16-s + 4/3·18-s − 0.421·19-s − 2.39·20-s − 4.36·22-s + 0.719·25-s − 2.28·28-s + 4.13·29-s − 2.83·31-s − 32-s + 1.37·35-s + 4/3·36-s + 0.702·37-s − 0.842·38-s − 6/5·40-s − 2.19·41-s − 4.36·44-s − 4/5·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(582763.\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.03694158796\)
\(L(\frac12)\) \(\approx\) \(0.03694158796\)
\(L(2)\) 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$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
13 \( 1 \)
good5$D_4\times C_2$ \( 1 + 6 T + 18 T^{2} + 168 T^{3} + 1559 T^{4} + 168 p^{2} T^{5} + 18 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \)
7$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} - 312 T^{3} - 4702 T^{4} - 312 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 18242 T^{4} + 2328 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 250 T^{2} + 161499 T^{4} - 250 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 8 T + 32 T^{2} - 120 T^{3} - 140926 T^{4} - 120 p^{2} T^{5} + 32 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1444 T^{2} + 1065414 T^{4} - 1444 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 60 T + 2507 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 88 T + 3872 T^{2} + 167640 T^{3} + 6366914 T^{4} + 167640 p^{2} T^{5} + 3872 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 - 26 T + 338 T^{2} - 36816 T^{3} + 4007903 T^{4} - 36816 p^{2} T^{5} + 338 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 + 90 T + 4050 T^{2} + 242280 T^{3} + 13471607 T^{4} + 242280 p^{2} T^{5} + 4050 p^{4} T^{6} + 90 p^{6} T^{7} + p^{8} T^{8} \)
43$D_4\times C_2$ \( 1 - 2692 T^{2} + 4500390 T^{4} - 2692 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2$ \( ( 1 - 60 T + 1800 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 30 T + 4391 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 120 T + 7200 T^{2} - 147000 T^{3} - 2088286 T^{4} - 147000 p^{2} T^{5} + 7200 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 126 T + 10979 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 32 T + 512 T^{2} + 85536 T^{3} + 10992002 T^{4} + 85536 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \)
71$C_2^3$ \( 1 + 28493474 T^{4} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 - 178 T + 15842 T^{2} - 1362768 T^{3} + 111813743 T^{4} - 1362768 p^{2} T^{5} + 15842 p^{4} T^{6} - 178 p^{6} T^{7} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 96 T + 13058 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 168 T + 14112 T^{2} + 717864 T^{3} + 29673602 T^{4} + 717864 p^{2} T^{5} + 14112 p^{4} T^{6} + 168 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 156 T + 12168 T^{2} - 1257204 T^{3} + 129875918 T^{4} - 1257204 p^{2} T^{5} + 12168 p^{4} T^{6} - 156 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 188 T + 17672 T^{2} - 564 T^{3} - 88472818 T^{4} - 564 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \)
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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

−6.97236487050822454436195374400, −6.45621434263364239137643289526, −6.38167803900346287551292019257, −6.35880011778437718803036150440, −5.97023297744317439633698101935, −5.63318136810876190771208817218, −5.46248864758579750713073799180, −5.40485837531980685835591178600, −4.98607337411219621459742785892, −4.61734370005718245373081177233, −4.56568558946197672385850074976, −4.50430976149312668293561302338, −4.32081003945188792847137372666, −3.60411369761935716413419708594, −3.54600795581631440485112704465, −3.54405364256803348035199833125, −3.21013076597697179959943960167, −2.91958668180051298461978337798, −2.47575322071156700812472904290, −2.31027611111496156799377378910, −2.30801506097297483052236661185, −1.55951373450482076870453324227, −1.01574894694329601069910335189, −0.66527753692379398370953724551, −0.02357475730637485274802559116, 0.02357475730637485274802559116, 0.66527753692379398370953724551, 1.01574894694329601069910335189, 1.55951373450482076870453324227, 2.30801506097297483052236661185, 2.31027611111496156799377378910, 2.47575322071156700812472904290, 2.91958668180051298461978337798, 3.21013076597697179959943960167, 3.54405364256803348035199833125, 3.54600795581631440485112704465, 3.60411369761935716413419708594, 4.32081003945188792847137372666, 4.50430976149312668293561302338, 4.56568558946197672385850074976, 4.61734370005718245373081177233, 4.98607337411219621459742785892, 5.40485837531980685835591178600, 5.46248864758579750713073799180, 5.63318136810876190771208817218, 5.97023297744317439633698101935, 6.35880011778437718803036150440, 6.38167803900346287551292019257, 6.45621434263364239137643289526, 6.97236487050822454436195374400

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