Properties

Label 8-63e8-1.1-c0e4-0-5
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  − 4-s + 16-s − 2·25-s − 2·37-s + 4·43-s − 2·64-s + 2·67-s + 2·79-s + 2·100-s + 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 16-s − 2·25-s − 2·37-s + 4·43-s − 2·64-s + 2·67-s + 2·79-s + 2·100-s + 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-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

\(L(\frac{1}{2})\) \(\approx\) \(1.275093541\)
\(L(\frac12)\) \(\approx\) \(1.275093541\)
\(L(1)\) 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$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
5$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
11$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
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_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
43$C_2$ \( ( 1 - T + T^{2} )^{4} \)
47$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
53$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \)
59$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
71$C_2^2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_2$ \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.09959669603606085305955191550, −6.01645291934499231668423535754, −5.71474397797089338983141582548, −5.44738114227180034468637125991, −5.41058071613123887342595419718, −5.25723751028695610758884285003, −5.03739806561117057335019659410, −4.69596103101586496822016335487, −4.66770034701869856115863797515, −4.22865586431004958468657100943, −4.11597796690524910150070249510, −3.92430679539376434561600110665, −3.89967735877040381077737769350, −3.60597465185793021626671310744, −3.35897459107896017395468529004, −3.09847668027719573965562988434, −2.81875103999286883267258364009, −2.63599727659766599596236320618, −2.35829523238620202553345877404, −1.93930073926760297653388806448, −1.87665003720305507883228671364, −1.66033805818253175682919739822, −1.17667399312569516433395050868, −0.65547176465598487825833996391, −0.62284213264159484691769096461, 0.62284213264159484691769096461, 0.65547176465598487825833996391, 1.17667399312569516433395050868, 1.66033805818253175682919739822, 1.87665003720305507883228671364, 1.93930073926760297653388806448, 2.35829523238620202553345877404, 2.63599727659766599596236320618, 2.81875103999286883267258364009, 3.09847668027719573965562988434, 3.35897459107896017395468529004, 3.60597465185793021626671310744, 3.89967735877040381077737769350, 3.92430679539376434561600110665, 4.11597796690524910150070249510, 4.22865586431004958468657100943, 4.66770034701869856115863797515, 4.69596103101586496822016335487, 5.03739806561117057335019659410, 5.25723751028695610758884285003, 5.41058071613123887342595419718, 5.44738114227180034468637125991, 5.71474397797089338983141582548, 6.01645291934499231668423535754, 6.09959669603606085305955191550

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