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Properties

Label 8-31e4-1.1-c1e4-0-0
Degree $8$
Conductor $923521$
Sign $1$
Analytic cond. $0.00375452$
Root an. cond. $0.497530$
Motivic weight $1$
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 + 2·3-s + 6·4-s − 2·5-s − 8·6-s − 2·7-s − 4·8-s + 5·9-s + 8·10-s + 2·11-s + 12·12-s + 2·13-s + 8·14-s − 4·15-s + 3·16-s + 6·17-s − 20·18-s − 6·19-s − 12·20-s − 4·21-s − 8·22-s − 16·23-s − 8·24-s + 11·25-s − 8·26-s + 10·27-s − 12·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 1.15·3-s + 3·4-s − 0.894·5-s − 3.26·6-s − 0.755·7-s − 1.41·8-s + 5/3·9-s + 2.52·10-s + 0.603·11-s + 3.46·12-s + 0.554·13-s + 2.13·14-s − 1.03·15-s + 3/4·16-s + 1.45·17-s − 4.71·18-s − 1.37·19-s − 2.68·20-s − 0.872·21-s − 1.70·22-s − 3.33·23-s − 1.63·24-s + 11/5·25-s − 1.56·26-s + 1.92·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 923521 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 923521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(923521\)    =    \(31^{4}\)
Sign: $1$
Analytic conductor: \(0.00375452\)
Root analytic conductor: \(0.497530\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 923521,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08031418475\)
\(L(\frac12)\) \(\approx\) \(0.08031418475\)
\(L(\frac{3}{2})\) 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)$
bad31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
good2$D_{4}$ \( ( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \)
3$D_4\times C_2$ \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^2$ \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 2 T - 9 T^{2} - 2 T^{3} + 92 T^{4} - 2 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 2 T - T^{2} + 34 T^{3} - 140 T^{4} + 34 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 2 T - 15 T^{2} + 14 T^{3} + 140 T^{4} + 14 p T^{5} - 15 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T - 9 T^{2} + 42 T^{3} + 980 T^{4} + 42 p T^{5} - 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
29$D_{4}$ \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
41$D_4\times C_2$ \( 1 + 2 T - 7 T^{2} - 142 T^{3} - 1724 T^{4} - 142 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 - 2 T + 15 T^{2} + 194 T^{3} - 1900 T^{4} + 194 p T^{5} + 15 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 6 T - 71 T^{2} + 6 T^{3} + 6732 T^{4} + 6 p T^{5} - 71 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 6 T - 41 T^{2} + 246 T^{3} + 324 T^{4} + 246 p T^{5} - 41 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 34 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 14 T + 55 T^{2} + 14 T^{3} + 924 T^{4} + 14 p T^{5} + 55 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 2 T - 135 T^{2} - 14 T^{3} + 13700 T^{4} - 14 p T^{5} - 135 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 22 T + 223 T^{2} - 2266 T^{3} + 23644 T^{4} - 2266 p T^{5} + 223 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 6 T - 89 T^{2} + 246 T^{3} + 5748 T^{4} + 246 p T^{5} - 89 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−12.75642799665712825399666513073, −12.45864769530597376429176992395, −12.23484494524968300585607223052, −11.93735652217521342755533904634, −11.35507817901893618363075489079, −10.67766489343954453639141019555, −10.63266447894344353369695100852, −10.40389187153593045943570773306, −9.928252672436318742225145012911, −9.407388892625902277829897186403, −9.248566860357097575608400920380, −9.210521158862579387229930737215, −8.839704871685090155405517474736, −8.464004914666357168084196747531, −7.88257003994256340065368716256, −7.81249002575234326240777543890, −7.65023331655886740009580965906, −6.86466203132722501192569275906, −6.69249135921681364561766067139, −5.77236974894511654073662100791, −5.52882929282229684328658185625, −4.10575882823759228164850980824, −3.87027652297521950499404573403, −3.56314932660478550441116226111, −2.04312140270729444007225676723, 2.04312140270729444007225676723, 3.56314932660478550441116226111, 3.87027652297521950499404573403, 4.10575882823759228164850980824, 5.52882929282229684328658185625, 5.77236974894511654073662100791, 6.69249135921681364561766067139, 6.86466203132722501192569275906, 7.65023331655886740009580965906, 7.81249002575234326240777543890, 7.88257003994256340065368716256, 8.464004914666357168084196747531, 8.839704871685090155405517474736, 9.210521158862579387229930737215, 9.248566860357097575608400920380, 9.407388892625902277829897186403, 9.928252672436318742225145012911, 10.40389187153593045943570773306, 10.63266447894344353369695100852, 10.67766489343954453639141019555, 11.35507817901893618363075489079, 11.93735652217521342755533904634, 12.23484494524968300585607223052, 12.45864769530597376429176992395, 12.75642799665712825399666513073

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