# Properties

 Label 8-384e4-1.1-c1e4-0-7 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $88.3961$ Root an. cond. $1.75107$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## 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: $1$ Analytic conductor: $$88.3961$$ Root analytic conductor: $$1.75107$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes 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

 $$L(1)$$ $$\approx$$ $$2.965121222$$ $$L(\frac12)$$ $$\approx$$ $$2.965121222$$ $$L(\frac{3}{2})$$ 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