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Properties

Label 8-31e4-1.1-c1e4-0-1
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  − 3·2-s + 3-s + 7·4-s − 6·5-s − 3·6-s − 3·7-s − 15·8-s + 3·9-s + 18·10-s − 2·11-s + 7·12-s + 6·13-s + 9·14-s − 6·15-s + 30·16-s − 3·17-s − 9·18-s − 5·19-s − 42·20-s − 3·21-s + 6·22-s + 11·23-s − 15·24-s + 5·25-s − 18·26-s − 21·28-s + 5·29-s + ⋯
L(s)  = 1  − 2.12·2-s + 0.577·3-s + 7/2·4-s − 2.68·5-s − 1.22·6-s − 1.13·7-s − 5.30·8-s + 9-s + 5.69·10-s − 0.603·11-s + 2.02·12-s + 1.66·13-s + 2.40·14-s − 1.54·15-s + 15/2·16-s − 0.727·17-s − 2.12·18-s − 1.14·19-s − 9.39·20-s − 0.654·21-s + 1.27·22-s + 2.29·23-s − 3.06·24-s + 25-s − 3.53·26-s − 3.96·28-s + 0.928·29-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

\(L(1)\) \(\approx\) \(0.1047280925\)
\(L(\frac12)\) \(\approx\) \(0.1047280925\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 + 11 T + 61 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
good2$C_2^2:C_4$ \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
3$C_4\times C_2$ \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_4\times C_2$ \( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 15 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_4\times C_2$ \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 3 T + 2 T^{2} + 75 T^{3} + 511 T^{4} + 75 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 65 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2:C_4$ \( 1 - 11 T + 38 T^{2} - 125 T^{3} + 821 T^{4} - 125 p T^{5} + 38 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 115 p T^{5} + 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2:C_4$ \( 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 356 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2^2:C_4$ \( 1 - T - 27 T^{2} + 235 T^{3} + 1196 T^{4} + 235 p T^{5} - 27 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^2:C_4$ \( 1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 385 p T^{5} - 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 21 T + 118 T^{2} + 555 T^{3} - 9989 T^{4} + 555 p T^{5} + 118 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 - 5 T + 26 T^{2} - 515 T^{3} + 6161 T^{4} - 515 p T^{5} + 26 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 + 7 T + 53 T^{2} + 799 T^{3} + 10580 T^{4} + 799 p T^{5} + 53 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 2595 p T^{5} + 233 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
79$C_4\times C_2$ \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 14 T + 13 T^{2} + 70 T^{3} + 6651 T^{4} + 70 p T^{5} + 13 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 - 5 T - 29 T^{2} + 785 T^{3} - 624 T^{4} + 785 p T^{5} - 29 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 345 p T^{5} + 182 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
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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

−12.91858372072629820218699870260, −12.22917086864965877804078583878, −12.10350767476457834285170163722, −11.76843876927165282474894896132, −11.42547298108203992927067657943, −11.08247801506541672351027421811, −10.68767631280942441817826275505, −10.58749408409296169775567131946, −10.34083536217138989203461251263, −9.546495903272986379630782905837, −9.305256134931976375631630499151, −8.989830554457769108652548569150, −8.593274253077195400660673104268, −8.391174888530887729454457221271, −7.949373621221349143315831768295, −7.62832989512876994316291626328, −7.19042644833126543563050771444, −6.86513297011710528604701685740, −6.82636523187518682935201420961, −5.78436243846332643293479433510, −5.69417246556555821462298156022, −4.10831542512091253099542963216, −3.76198785283307356381246442880, −3.46493639857579719593935929257, −2.48296375840074006334440528554, 2.48296375840074006334440528554, 3.46493639857579719593935929257, 3.76198785283307356381246442880, 4.10831542512091253099542963216, 5.69417246556555821462298156022, 5.78436243846332643293479433510, 6.82636523187518682935201420961, 6.86513297011710528604701685740, 7.19042644833126543563050771444, 7.62832989512876994316291626328, 7.949373621221349143315831768295, 8.391174888530887729454457221271, 8.593274253077195400660673104268, 8.989830554457769108652548569150, 9.305256134931976375631630499151, 9.546495903272986379630782905837, 10.34083536217138989203461251263, 10.58749408409296169775567131946, 10.68767631280942441817826275505, 11.08247801506541672351027421811, 11.42547298108203992927067657943, 11.76843876927165282474894896132, 12.10350767476457834285170163722, 12.22917086864965877804078583878, 12.91858372072629820218699870260

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