Properties

Label 8-63e8-1.1-c0e4-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 16-s − 2·25-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 + 2·169-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 − 2·25-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 + 2·169-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

\(L(\frac{1}{2})\) \(\approx\) \(0.8691068670\)
\(L(\frac12)\) \(\approx\) \(0.8691068670\)
\(L(1)\) 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$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2^3$ \( 1 - T^{4} + T^{8} \)
13$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_2$ \( ( 1 + T^{2} )^{4} \)
23$C_2^3$ \( 1 - T^{4} + T^{8} \)
29$C_2^3$ \( 1 - T^{4} + T^{8} \)
31$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{4} \)
41$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + T^{4} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{4} \)
79$C_2$ \( ( 1 + T + T^{2} )^{4} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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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

−6.11043012879546794792683810892, −5.94829828174593325504806339157, −5.74277810231563740501728061195, −5.62118330665145066102425839919, −5.38080501808663299775779592343, −5.20633441695031926336165519034, −5.01174706373990138068642246131, −4.83046088358181510458449318977, −4.67162470091720355594567084568, −4.20968621606679618683479706587, −4.09762569699919628649442285259, −3.87956755178082820975356621848, −3.72032912777597919451117200989, −3.69995649107587175395581497799, −3.40225787720413641132118707418, −2.96212317648039465925110848689, −2.78466678808260653226199162778, −2.65480485602388637514979764527, −2.26912644634595336716858864484, −2.20917855984096276975244082780, −1.85598555304591033844870426382, −1.35139518479425626988002304017, −1.29770229535788770028691411624, −1.12187769223755682377893871922, −0.32288522951787767770156516364, 0.32288522951787767770156516364, 1.12187769223755682377893871922, 1.29770229535788770028691411624, 1.35139518479425626988002304017, 1.85598555304591033844870426382, 2.20917855984096276975244082780, 2.26912644634595336716858864484, 2.65480485602388637514979764527, 2.78466678808260653226199162778, 2.96212317648039465925110848689, 3.40225787720413641132118707418, 3.69995649107587175395581497799, 3.72032912777597919451117200989, 3.87956755178082820975356621848, 4.09762569699919628649442285259, 4.20968621606679618683479706587, 4.67162470091720355594567084568, 4.83046088358181510458449318977, 5.01174706373990138068642246131, 5.20633441695031926336165519034, 5.38080501808663299775779592343, 5.62118330665145066102425839919, 5.74277810231563740501728061195, 5.94829828174593325504806339157, 6.11043012879546794792683810892

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