Properties

Label 8-63e8-1.1-c0e4-0-6
Degree $8$
Conductor $2.482\times 10^{14}$
Sign $1$
Analytic cond. $15.3940$
Root an. cond. $1.40740$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 16-s + 4·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-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 + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 16-s + 4·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-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 + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(15.3940\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3969} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 7^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.385762966\)
\(L(\frac12)\) \(\approx\) \(2.385762966\)
\(L(1)\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^3$ \( 1 - T^{4} + T^{8} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T^{4} )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
53$C_2^3$ \( 1 - T^{4} + T^{8} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_2$ \( ( 1 - T + T^{2} )^{4} \)
71$C_2^3$ \( 1 - T^{4} + T^{8} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.13133776019401321388510169051, −5.92555473152344411233792404851, −5.78296006982442827819035302821, −5.55704033591448449378196093489, −5.42421416350708734240397328349, −5.12433118170940068414114745194, −4.87394311557662320559463637253, −4.86811669000795365775407743210, −4.68668345871981838052957328624, −4.43821875281113136516768523565, −4.04203104024629401648405164716, −3.93528692086243315427971413839, −3.88407554631482228822492648406, −3.38336366409589532500310773252, −3.11028620019379152938681199562, −3.09662523196564915871590242491, −3.03253884671891232351926797042, −2.60679355840232943341171392509, −2.40539446366987347553481715892, −2.08183176202678648937278717595, −1.87254659880856011903497559680, −1.44785938348991474911451193446, −1.15120527358256937590115026030, −0.951767437125475136559348898996, −0.65498837616697276601980948779, 0.65498837616697276601980948779, 0.951767437125475136559348898996, 1.15120527358256937590115026030, 1.44785938348991474911451193446, 1.87254659880856011903497559680, 2.08183176202678648937278717595, 2.40539446366987347553481715892, 2.60679355840232943341171392509, 3.03253884671891232351926797042, 3.09662523196564915871590242491, 3.11028620019379152938681199562, 3.38336366409589532500310773252, 3.88407554631482228822492648406, 3.93528692086243315427971413839, 4.04203104024629401648405164716, 4.43821875281113136516768523565, 4.68668345871981838052957328624, 4.86811669000795365775407743210, 4.87394311557662320559463637253, 5.12433118170940068414114745194, 5.42421416350708734240397328349, 5.55704033591448449378196093489, 5.78296006982442827819035302821, 5.92555473152344411233792404851, 6.13133776019401321388510169051

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