Properties

Label 8-462e4-1.1-c1e4-0-10
Degree $8$
Conductor $45558341136$
Sign $1$
Analytic cond. $185.215$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 3·5-s + 6-s − 7-s − 3·10-s − 11-s − 2·13-s − 14-s − 3·15-s − 4·17-s + 9·19-s − 21-s − 22-s − 6·23-s + 10·25-s − 2·26-s + 8·29-s − 3·30-s − 31-s − 32-s − 33-s − 4·34-s + 3·35-s − 8·37-s + 9·38-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.948·10-s − 0.301·11-s − 0.554·13-s − 0.267·14-s − 0.774·15-s − 0.970·17-s + 2.06·19-s − 0.218·21-s − 0.213·22-s − 1.25·23-s + 2·25-s − 0.392·26-s + 1.48·29-s − 0.547·30-s − 0.179·31-s − 0.176·32-s − 0.174·33-s − 0.685·34-s + 0.507·35-s − 1.31·37-s + 1.45·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(185.215\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.959586289\)
\(L(\frac12)\) \(\approx\) \(2.959586289\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \)
good5$C_2^2:C_4$ \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 3 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_4\times C_2$ \( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 44 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 4 T - 11 T^{2} - 52 T^{3} + 69 T^{4} - 52 p T^{5} - 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 - 9 T + 17 T^{2} + 3 T^{3} + 100 T^{4} + 3 p T^{5} + 17 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 8 T + 35 T^{2} - 308 T^{3} + 2489 T^{4} - 308 p T^{5} + 35 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 + T - 15 T^{2} - 151 T^{3} + 524 T^{4} - 151 p T^{5} - 15 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 8 T - 3 T^{2} + 40 T^{3} + 1601 T^{4} + 40 p T^{5} - 3 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 4 T - 35 T^{2} - 124 T^{3} + 1149 T^{4} - 124 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
47$C_2^2:C_4$ \( 1 + 12 T + 17 T^{2} - 60 T^{3} + 961 T^{4} - 60 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 14 T + 23 T^{2} + 400 T^{3} - 2899 T^{4} + 400 p T^{5} + 23 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 8 T - 35 T^{2} + 362 T^{3} + 429 T^{4} + 362 p T^{5} - 35 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 24 T + 315 T^{2} - 3346 T^{3} + 29589 T^{4} - 3346 p T^{5} + 315 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - 9 T + 11 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} ) \)
73$C_2^2:C_4$ \( 1 - 24 T + 303 T^{2} - 3370 T^{3} + 33261 T^{4} - 3370 p T^{5} + 303 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 + 81 T^{2} + 20 T^{3} + 6321 T^{4} + 20 p T^{5} + 81 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 10 T - 43 T^{2} - 10 p T^{3} - 2671 T^{4} - 10 p^{2} T^{5} - 43 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
89$C_4$ \( ( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 + 26 T + 279 T^{2} + 2782 T^{3} + 31769 T^{4} + 2782 p T^{5} + 279 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.902604287268975596664115099304, −7.66596569593905994692533182505, −7.53608366864868532736802266152, −7.38927428209024765781830184714, −6.91148099580451337456724469555, −6.87278809737402159254171160802, −6.66147855206499859122448230872, −6.37371731216958144508431610859, −5.86073361327136568668475428090, −5.58476438239570726864493347485, −5.48738064838559483134581045076, −5.19322122455632027597117591744, −4.80530822766002509728142264104, −4.75112110472995897211315561201, −4.12107557411252908726188532367, −4.07167411250115941029182570773, −3.79050837996209070029185658774, −3.74973675352364082519574199288, −3.07554581297700805941662662986, −2.88855135146281238619127113332, −2.51500954380065041470914203025, −2.41950158009148225673486235856, −1.80471619089478651305007458642, −0.864116749979770617001202473838, −0.70270736443764762160574086859, 0.70270736443764762160574086859, 0.864116749979770617001202473838, 1.80471619089478651305007458642, 2.41950158009148225673486235856, 2.51500954380065041470914203025, 2.88855135146281238619127113332, 3.07554581297700805941662662986, 3.74973675352364082519574199288, 3.79050837996209070029185658774, 4.07167411250115941029182570773, 4.12107557411252908726188532367, 4.75112110472995897211315561201, 4.80530822766002509728142264104, 5.19322122455632027597117591744, 5.48738064838559483134581045076, 5.58476438239570726864493347485, 5.86073361327136568668475428090, 6.37371731216958144508431610859, 6.66147855206499859122448230872, 6.87278809737402159254171160802, 6.91148099580451337456724469555, 7.38927428209024765781830184714, 7.53608366864868532736802266152, 7.66596569593905994692533182505, 7.902604287268975596664115099304

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