Properties

Degree $8$
Conductor $9.683\times 10^{12}$
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-s + 2·25-s − 4·37-s − 64-s + 2·100-s − 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 4-s + 2·25-s − 4·37-s − 64-s + 2·100-s − 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352577022\)
\(L(\frac12)\) \(\approx\) \(1.352577022\)
\(L(1)\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + T^{2} )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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

−6.87157246585498388548340881002, −6.78912918638547266293829705291, −6.39962531471180739800779995331, −6.27756029136800055621069338263, −6.11240860645178291181051337650, −5.68903376698362869304282153135, −5.42966471062023955239344779615, −5.42616194856615565403755394025, −5.20454618175549509549389498152, −4.78997946875933757265319941348, −4.71714491947273489038642057406, −4.61706481461209074363314089289, −4.10544195877832560964655738112, −3.82527172320038699897556483198, −3.79825535494140145397411078902, −3.34078547643171601110707041047, −3.13222330033251035775996215260, −3.07882727823879067072400770206, −2.61889142530479702165042925502, −2.34089575548995430852041619483, −2.29251402830539862297653218458, −1.65927051902591981804467826021, −1.45875696382827159344789518132, −1.44193440244662016617745613553, −0.60097015769081178863827817118, 0.60097015769081178863827817118, 1.44193440244662016617745613553, 1.45875696382827159344789518132, 1.65927051902591981804467826021, 2.29251402830539862297653218458, 2.34089575548995430852041619483, 2.61889142530479702165042925502, 3.07882727823879067072400770206, 3.13222330033251035775996215260, 3.34078547643171601110707041047, 3.79825535494140145397411078902, 3.82527172320038699897556483198, 4.10544195877832560964655738112, 4.61706481461209074363314089289, 4.71714491947273489038642057406, 4.78997946875933757265319941348, 5.20454618175549509549389498152, 5.42616194856615565403755394025, 5.42966471062023955239344779615, 5.68903376698362869304282153135, 6.11240860645178291181051337650, 6.27756029136800055621069338263, 6.39962531471180739800779995331, 6.78912918638547266293829705291, 6.87157246585498388548340881002

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