Properties

Label 8-5e8-1.1-c1e4-0-0
Degree $8$
Conductor $390625$
Sign $1$
Analytic cond. $0.00158806$
Root an. cond. $0.446795$
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  − 2·2-s − 3-s + 2·4-s − 5·5-s + 2·6-s − 2·7-s − 5·8-s + 3·9-s + 10·10-s − 2·11-s − 2·12-s + 9·13-s + 4·14-s + 5·15-s + 5·16-s + 8·17-s − 6·18-s − 5·19-s − 10·20-s + 2·21-s + 4·22-s − 11·23-s + 5·24-s + 10·25-s − 18·26-s − 4·28-s + 5·29-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 2.23·5-s + 0.816·6-s − 0.755·7-s − 1.76·8-s + 9-s + 3.16·10-s − 0.603·11-s − 0.577·12-s + 2.49·13-s + 1.06·14-s + 1.29·15-s + 5/4·16-s + 1.94·17-s − 1.41·18-s − 1.14·19-s − 2.23·20-s + 0.436·21-s + 0.852·22-s − 2.29·23-s + 1.02·24-s + 2·25-s − 3.53·26-s − 0.755·28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(390625\)    =    \(5^{8}\)
Sign: $1$
Analytic conductor: \(0.00158806\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 390625,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07419878294\)
\(L(\frac12)\) \(\approx\) \(0.07419878294\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
good2$C_2^2:C_4$ \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
3$C_4\times C_2$ \( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 5 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
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$C_2^2:C_4$ \( 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 110 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2:C_4$ \( 1 + 11 T + 28 T^{2} - 245 T^{3} - 2259 T^{4} - 245 p T^{5} + 28 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 - 5 T - 19 T^{2} + 5 p T^{3} - 4 T^{4} + 5 p^{2} T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_4\times C_2$ \( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 145 p T^{5} - 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 - 8 T - 17 T^{2} + 254 T^{3} - 435 T^{4} + 254 p T^{5} - 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 2 T - 43 T^{2} + 50 T^{3} + 2351 T^{4} + 50 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 9 T + 8 T^{2} - 315 T^{3} + 5131 T^{4} - 315 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 + 31 T^{2} + 210 T^{3} + 2851 T^{4} + 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 410 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 - 8 T - 37 T^{2} + 694 T^{3} - 2425 T^{4} + 694 p T^{5} - 37 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_4\times C_2$ \( 1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 585 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 - 9 T - 52 T^{2} + 675 T^{3} + 121 T^{4} + 675 p T^{5} - 52 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
89$C_4\times C_2$ \( 1 + 20 T + 151 T^{2} + 1600 T^{3} + 21441 T^{4} + 1600 p T^{5} + 151 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 20 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−13.57080781772427432924421874687, −12.78745959025090485470809083746, −12.57691585280446067468854990988, −12.41568399753153466766064232053, −11.79657060833100328562822762174, −11.77994962840411386420439821725, −11.51714826958993390227905514135, −10.89473160089159248954574943642, −10.73364131395871901679656565556, −10.25003038188384685117228914797, −9.970660453849164957529609063316, −9.472056313954219443054734010489, −9.275463700894892957584831283175, −8.400323665832239832022596326221, −8.156378553166274970781992955722, −8.117834134407304095731906220012, −8.059710434113124745718597171927, −6.88842797735002891028014004884, −6.78819721404471653212800767011, −6.27578904102493561363949704096, −5.91719153272041345964305495670, −5.09978582189619516920936475269, −4.01305883699761872997695967334, −3.87632635392220914344699637256, −3.17592678118584122041376424892, 3.17592678118584122041376424892, 3.87632635392220914344699637256, 4.01305883699761872997695967334, 5.09978582189619516920936475269, 5.91719153272041345964305495670, 6.27578904102493561363949704096, 6.78819721404471653212800767011, 6.88842797735002891028014004884, 8.059710434113124745718597171927, 8.117834134407304095731906220012, 8.156378553166274970781992955722, 8.400323665832239832022596326221, 9.275463700894892957584831283175, 9.472056313954219443054734010489, 9.970660453849164957529609063316, 10.25003038188384685117228914797, 10.73364131395871901679656565556, 10.89473160089159248954574943642, 11.51714826958993390227905514135, 11.77994962840411386420439821725, 11.79657060833100328562822762174, 12.41568399753153466766064232053, 12.57691585280446067468854990988, 12.78745959025090485470809083746, 13.57080781772427432924421874687

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