Properties

Degree $8$
Conductor $1.549\times 10^{14}$
Sign $1$
Motivic weight $0$
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 + 4-s + 2·8-s − 4·16-s − 2·25-s + 2·32-s + 4·50-s + 3·64-s + 4·67-s − 2·100-s + 4·107-s + 2·121-s + 127-s − 6·128-s + 131-s − 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2·2-s + 4-s + 2·8-s − 4·16-s − 2·25-s + 2·32-s + 4·50-s + 3·64-s + 4·67-s − 2·100-s + 4·107-s + 2·121-s + 127-s − 6·128-s + 131-s − 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4003211977\)
\(L(\frac12)\) \(\approx\) \(0.4003211977\)
\(L(1)\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2^3$ \( 1 - T^{4} + T^{8} \)
19$C_2^3$ \( 1 - T^{4} + T^{8} \)
23$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
59$C_2^3$ \( 1 - T^{4} + T^{8} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
73$C_2^3$ \( 1 - T^{4} + T^{8} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2^3$ \( 1 - T^{4} + T^{8} \)
97$C_2^2$ \( ( 1 + T^{4} )^{2} \)
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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

−6.45279465060330201168052791981, −5.92067638445988745052820920810, −5.77763032657000889302325771798, −5.75633845321950685573796082268, −5.60627014802211170006788264246, −5.27352421058608895990154443443, −4.96070999035228669709231458711, −4.86681238347315266997285581112, −4.63522485560747612613519669813, −4.41086108454520350672420838128, −4.34608589508292863984545091000, −3.87247669782138582780345235465, −3.82458300370464019627451903443, −3.65534635771493383146997571848, −3.37248104038385488029820420339, −3.10589249017634638689089847983, −2.85220503842300141855344897917, −2.29823547647708984712685316879, −2.14987089495500068427035132349, −1.99380360492751140790703285030, −1.98523321486003270273318274431, −1.37156773131665540528566931529, −1.21473151227846289722687204000, −0.67591441734767280284797769524, −0.53661693225407695761098432685, 0.53661693225407695761098432685, 0.67591441734767280284797769524, 1.21473151227846289722687204000, 1.37156773131665540528566931529, 1.98523321486003270273318274431, 1.99380360492751140790703285030, 2.14987089495500068427035132349, 2.29823547647708984712685316879, 2.85220503842300141855344897917, 3.10589249017634638689089847983, 3.37248104038385488029820420339, 3.65534635771493383146997571848, 3.82458300370464019627451903443, 3.87247669782138582780345235465, 4.34608589508292863984545091000, 4.41086108454520350672420838128, 4.63522485560747612613519669813, 4.86681238347315266997285581112, 4.96070999035228669709231458711, 5.27352421058608895990154443443, 5.60627014802211170006788264246, 5.75633845321950685573796082268, 5.77763032657000889302325771798, 5.92067638445988745052820920810, 6.45279465060330201168052791981

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