Properties

Degree $8$
Conductor $466948881$
Sign $1$
Motivic weight $1$
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 − 2·3-s + 3·4-s − 4·5-s − 4·6-s + 2·8-s + 9-s − 8·10-s + 4·11-s − 6·12-s + 16·13-s + 8·15-s − 4·17-s + 2·18-s − 12·20-s + 8·22-s + 4·23-s − 4·24-s + 12·25-s + 32·26-s + 2·27-s − 16·29-s + 16·30-s + 8·31-s − 6·32-s − 8·33-s − 8·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 2.52·10-s + 1.20·11-s − 1.73·12-s + 4.43·13-s + 2.06·15-s − 0.970·17-s + 0.471·18-s − 2.68·20-s + 1.70·22-s + 0.834·23-s − 0.816·24-s + 12/5·25-s + 6.27·26-s + 0.384·27-s − 2.97·29-s + 2.92·30-s + 1.43·31-s − 1.06·32-s − 1.39·33-s − 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{147} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.75799\)
\(L(\frac12)\) \(\approx\) \(1.75799\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 4 T - 4 T^{2} - 56 T^{3} - 161 T^{4} - 56 p T^{5} - 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 4 T - 2 T^{2} + 112 T^{3} - 573 T^{4} + 112 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 8 T - 6 T^{2} - 64 T^{3} + 1955 T^{4} - 64 p T^{5} - 6 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 86 T^{2} + 5187 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 8 T - 62 T^{2} - 64 T^{3} + 8619 T^{4} - 64 p T^{5} - 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 16 T + 88 T^{2} + 736 T^{3} + 8887 T^{4} + 736 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 102 T^{2} + 5915 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 8 T - 656 T^{3} - 5905 T^{4} - 656 p T^{5} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 1640 p T^{5} + 140 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−9.623971350396168809469681098609, −8.974662692645919751940224408015, −8.897148800663628897534960467783, −8.804703681731128566029989309013, −8.734890741513374116950316770026, −8.248126502192753607369006305068, −7.66455013513348093780612950244, −7.65256890414441409205250270468, −7.10883485010791613238355851647, −7.06756730521689905533640818552, −6.51096084429568302594207450597, −6.15976444849922651813775743876, −6.04602224659697085485975387550, −5.97371222579920045515647232408, −5.77188774024050862575162382151, −5.07039082709885831225379431904, −4.61891468562083640729430301363, −4.45176021106498281617327033521, −3.95109035381896103845163269446, −3.75065186391036688153790662991, −3.63845703396218886885321011549, −3.16040508286271385642267829764, −2.67266815677563568556195034027, −1.55290516222439578912540029753, −1.07459413278504544289171169784, 1.07459413278504544289171169784, 1.55290516222439578912540029753, 2.67266815677563568556195034027, 3.16040508286271385642267829764, 3.63845703396218886885321011549, 3.75065186391036688153790662991, 3.95109035381896103845163269446, 4.45176021106498281617327033521, 4.61891468562083640729430301363, 5.07039082709885831225379431904, 5.77188774024050862575162382151, 5.97371222579920045515647232408, 6.04602224659697085485975387550, 6.15976444849922651813775743876, 6.51096084429568302594207450597, 7.06756730521689905533640818552, 7.10883485010791613238355851647, 7.65256890414441409205250270468, 7.66455013513348093780612950244, 8.248126502192753607369006305068, 8.734890741513374116950316770026, 8.804703681731128566029989309013, 8.897148800663628897534960467783, 8.974662692645919751940224408015, 9.623971350396168809469681098609

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