Properties

Degree $8$
Conductor $21743271936$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 32·23-s − 4·25-s + 12·49-s + 32·71-s − 40·73-s − 5·81-s − 24·97-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 64·207-s + ⋯
L(s)  = 1  + 2/3·9-s + 6.67·23-s − 4/5·25-s + 12/7·49-s + 3.79·71-s − 4.68·73-s − 5/9·81-s − 2.43·97-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 4.44·207-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.96512\)
\(L(\frac12)\) \(\approx\) \(2.96512\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−8.207351237007476044536365432384, −8.120392734803400601703102035981, −7.51722692481107305037871800976, −7.30018491953753815298702269467, −7.15802566357469675757158148371, −7.08832167594432746985729948718, −6.93173108089170759841363258284, −6.56216569056071767764736280298, −6.27151545042709666373914242059, −5.73925680093140389530464301458, −5.71176362597283345245879739875, −5.38132132023293670107484485591, −4.94690964170703788921002533785, −4.90262128472470923969857010055, −4.62376623683424415542603396228, −4.35263455807574320277553178882, −3.83511327565108663264206937062, −3.61995776869795313834644009391, −3.17345858220063849604074900420, −2.89369658756497020588932642867, −2.73251475822297227269246547105, −2.27558610678440678205869454755, −1.54725662438713011073558818112, −1.15882525896736858807680651954, −0.832003487129236713181405303910, 0.832003487129236713181405303910, 1.15882525896736858807680651954, 1.54725662438713011073558818112, 2.27558610678440678205869454755, 2.73251475822297227269246547105, 2.89369658756497020588932642867, 3.17345858220063849604074900420, 3.61995776869795313834644009391, 3.83511327565108663264206937062, 4.35263455807574320277553178882, 4.62376623683424415542603396228, 4.90262128472470923969857010055, 4.94690964170703788921002533785, 5.38132132023293670107484485591, 5.71176362597283345245879739875, 5.73925680093140389530464301458, 6.27151545042709666373914242059, 6.56216569056071767764736280298, 6.93173108089170759841363258284, 7.08832167594432746985729948718, 7.15802566357469675757158148371, 7.30018491953753815298702269467, 7.51722692481107305037871800976, 8.120392734803400601703102035981, 8.207351237007476044536365432384

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