Properties

Label 8-3e16-1.1-c1e4-0-0
Degree $8$
Conductor $43046721$
Sign $1$
Analytic cond. $0.175004$
Root an. cond. $0.804231$
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  + 4-s − 4·7-s + 2·13-s + 4·16-s + 8·19-s + 7·25-s − 4·28-s − 16·31-s − 28·37-s − 4·43-s + 18·49-s + 2·52-s + 14·61-s + 11·64-s + 20·67-s − 28·73-s + 8·76-s − 4·79-s − 8·91-s − 4·97-s + 7·100-s − 16·103-s + 44·109-s − 16·112-s + 10·121-s − 16·124-s + 127-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s + 0.554·13-s + 16-s + 1.83·19-s + 7/5·25-s − 0.755·28-s − 2.87·31-s − 4.60·37-s − 0.609·43-s + 18/7·49-s + 0.277·52-s + 1.79·61-s + 11/8·64-s + 2.44·67-s − 3.27·73-s + 0.917·76-s − 0.450·79-s − 0.838·91-s − 0.406·97-s + 7/10·100-s − 1.57·103-s + 4.21·109-s − 1.51·112-s + 0.909·121-s − 1.43·124-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(43046721\)    =    \(3^{16}\)
Sign: $1$
Analytic conductor: \(0.175004\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 43046721,\ (\ :1/2, 1/2, 1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8010889404\)
\(L(\frac12)\) \(\approx\) \(0.8010889404\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
good2$C_2^3$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^3$ \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
23$C_2^3$ \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 55 T^{2} + 2184 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2$ \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
41$C_2^3$ \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2^3$ \( 1 + 74 T^{2} + 1995 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 26 T^{2} - 6213 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−10.49906119444550916472184614542, −10.44545798280249175447112710406, −9.936034701283110710966056203504, −9.905815052488909485105400398566, −9.778329325714545354326930716590, −8.946766933064434179955492562636, −8.861804788918030230872677075765, −8.710000172223182455253992341055, −8.559284665351692289005324853885, −7.80685317461636902016303167721, −7.25675471668951485737925185870, −7.22811549961406415405763942158, −7.20981090805211652553070547516, −6.51394804337638602176312893508, −6.49233329593592705810758932099, −5.76005416609235568050894748916, −5.49826976682736586324062303365, −5.23834673929669910147529011183, −5.07166294607354757338284475118, −4.03569429808329983165749692300, −3.55736496586257136563323663316, −3.46636983875570748824133002690, −3.13565787939421405824023771537, −2.30223289907924439389455355950, −1.45663024891825141509450155705, 1.45663024891825141509450155705, 2.30223289907924439389455355950, 3.13565787939421405824023771537, 3.46636983875570748824133002690, 3.55736496586257136563323663316, 4.03569429808329983165749692300, 5.07166294607354757338284475118, 5.23834673929669910147529011183, 5.49826976682736586324062303365, 5.76005416609235568050894748916, 6.49233329593592705810758932099, 6.51394804337638602176312893508, 7.20981090805211652553070547516, 7.22811549961406415405763942158, 7.25675471668951485737925185870, 7.80685317461636902016303167721, 8.559284665351692289005324853885, 8.710000172223182455253992341055, 8.861804788918030230872677075765, 8.946766933064434179955492562636, 9.778329325714545354326930716590, 9.905815052488909485105400398566, 9.936034701283110710966056203504, 10.44545798280249175447112710406, 10.49906119444550916472184614542

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