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  + 4·13-s − 2·19-s − 2·31-s − 2·37-s − 4·43-s + 2·67-s + 2·73-s − 2·79-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 4·13-s − 2·19-s − 2·31-s − 2·37-s − 4·43-s + 2·67-s + 2·73-s − 2·79-s − 2·103-s + 2·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-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

\(L(\frac{1}{2})\) \(\approx\) \(1.035408875\)
\(L(\frac12)\) \(\approx\) \(1.035408875\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 - T^{4} + T^{8} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_2$ \( ( 1 - T + T^{2} )^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{4} \)
47$C_2^3$ \( 1 - T^{4} + T^{8} \)
53$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + T^{4} )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
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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

−6.35029311160322581954581122817, −6.05464373439526974414179292244, −5.84169771051646778767065579917, −5.74284353428467623929114088432, −5.56070801828993697873976343942, −5.21209064831300390378905128625, −5.05481438898857624395573915695, −4.93073349739182697894477832075, −4.71808483062269148402882622766, −4.26081058188177777243893952061, −4.22209425775612428703850884425, −3.89622480377167573362255900124, −3.67350046840134224942105899317, −3.64116853052654115677327253668, −3.38874110237557171302395999255, −3.26495586219787160874483694853, −3.17671985002054231632029758597, −2.54349786270367631372483893525, −2.16898883701288896554781900893, −2.15193637122187343174648184778, −1.89520071877361127533978749223, −1.38735657387229775615645375447, −1.32584255438721064839670523976, −1.26395969672897782674459178465, −0.35933448142704463429978873713, 0.35933448142704463429978873713, 1.26395969672897782674459178465, 1.32584255438721064839670523976, 1.38735657387229775615645375447, 1.89520071877361127533978749223, 2.15193637122187343174648184778, 2.16898883701288896554781900893, 2.54349786270367631372483893525, 3.17671985002054231632029758597, 3.26495586219787160874483694853, 3.38874110237557171302395999255, 3.64116853052654115677327253668, 3.67350046840134224942105899317, 3.89622480377167573362255900124, 4.22209425775612428703850884425, 4.26081058188177777243893952061, 4.71808483062269148402882622766, 4.93073349739182697894477832075, 5.05481438898857624395573915695, 5.21209064831300390378905128625, 5.56070801828993697873976343942, 5.74284353428467623929114088432, 5.84169771051646778767065579917, 6.05464373439526974414179292244, 6.35029311160322581954581122817

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