Properties

Degree $8$
Conductor $4.669\times 10^{12}$
Sign $1$
Motivic weight $1$
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 + 4·5-s + 9-s + 10·11-s − 14·19-s + 4·20-s + 5·25-s − 12·31-s + 36-s + 36·41-s + 10·44-s + 4·45-s + 40·55-s − 8·59-s − 4·61-s − 64-s − 8·71-s − 14·76-s − 28·79-s − 20·89-s − 56·95-s + 10·99-s + 5·100-s + 16·101-s − 36·109-s + 47·121-s − 12·124-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s + 1/3·9-s + 3.01·11-s − 3.21·19-s + 0.894·20-s + 25-s − 2.15·31-s + 1/6·36-s + 5.62·41-s + 1.50·44-s + 0.596·45-s + 5.39·55-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 0.949·71-s − 1.60·76-s − 3.15·79-s − 2.11·89-s − 5.74·95-s + 1.00·99-s + 1/2·100-s + 1.59·101-s − 3.44·109-s + 4.27·121-s − 1.07·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.836592647\)
\(L(\frac12)\) \(\approx\) \(4.836592647\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good11$C_2^2$ \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 75 T^{2} + 3416 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
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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

−6.70961437116929632648492208728, −6.35626794314856406408041272069, −6.29448405768742356315335945177, −6.24123521958903892584338073862, −6.17206546255510874341110078811, −5.62939234406027827604453509770, −5.57220493575768833728711896508, −5.46874296862761588396016630138, −5.25479737088151368307019615635, −4.48580288724954478520821387002, −4.31376024040273172655938024966, −4.25250627929462847079527339121, −4.25231144002982700438526931217, −4.06546125420022607425354934876, −3.73455847310880372740679756228, −3.17778809265516785632724721791, −3.07274155075159331636336727263, −2.64420450594449491502242355548, −2.31797635134604043994652416418, −2.22934482189743452760916840784, −1.90965459849557283694028087086, −1.55372784150797293839212904482, −1.25883331463203513692296584288, −1.25792023263463398658579996005, −0.35103269051662378624537017799, 0.35103269051662378624537017799, 1.25792023263463398658579996005, 1.25883331463203513692296584288, 1.55372784150797293839212904482, 1.90965459849557283694028087086, 2.22934482189743452760916840784, 2.31797635134604043994652416418, 2.64420450594449491502242355548, 3.07274155075159331636336727263, 3.17778809265516785632724721791, 3.73455847310880372740679756228, 4.06546125420022607425354934876, 4.25231144002982700438526931217, 4.25250627929462847079527339121, 4.31376024040273172655938024966, 4.48580288724954478520821387002, 5.25479737088151368307019615635, 5.46874296862761588396016630138, 5.57220493575768833728711896508, 5.62939234406027827604453509770, 6.17206546255510874341110078811, 6.24123521958903892584338073862, 6.29448405768742356315335945177, 6.35626794314856406408041272069, 6.70961437116929632648492208728

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