Properties

 Label 8-384e4-1.1-c2e4-0-2 Degree $8$ Conductor $21743271936$ Sign $1$ Analytic cond. $11985.7$ Root an. cond. $3.23469$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

Origins of factors

Dirichlet series

 L(s)  = 1 + 6·9-s − 72·17-s + 68·25-s + 72·41-s + 100·49-s − 328·73-s + 27·81-s + 504·89-s + 440·97-s − 504·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 432·153-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 2/3·9-s − 4.23·17-s + 2.71·25-s + 1.75·41-s + 2.04·49-s − 4.49·73-s + 1/3·81-s + 5.66·89-s + 4.53·97-s − 4.46·113-s − 3.20·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 2.82·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$11985.7$$ Root analytic conductor: $$3.23469$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.877496290$$ $$L(\frac12)$$ $$\approx$$ $$2.877496290$$ $$L(2)$$ 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$C_2$ $$( 1 - p T^{2} )^{2}$$
good5$C_2^2$ $$( 1 - 34 T^{2} + p^{4} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 50 T^{2} + p^{4} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 194 T^{2} + p^{4} T^{4} )^{2}$$
13$C_1$$\times$$C_1$ $$( 1 - p T )^{4}( 1 + p T )^{4}$$
17$C_2$ $$( 1 + 18 T + p^{2} T^{2} )^{4}$$
19$C_2^2$ $$( 1 + 290 T^{2} + p^{4} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + 670 T^{2} + p^{4} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 1666 T^{2} + p^{4} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 430 T^{2} + p^{4} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 2446 T^{2} + p^{4} T^{4} )^{2}$$
41$C_2$ $$( 1 - 18 T + p^{2} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 190 T^{2} + p^{4} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 2690 T^{2} + p^{4} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 3682 T^{2} + p^{4} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 3074 T^{2} + p^{4} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 2258 T^{2} + p^{4} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 8546 T^{2} + p^{4} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 8354 T^{2} + p^{4} T^{4} )^{2}$$
73$C_2$ $$( 1 + 82 T + p^{2} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 8594 T^{2} + p^{4} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 3550 T^{2} + p^{4} T^{4} )^{2}$$
89$C_2$ $$( 1 - 126 T + p^{2} T^{2} )^{4}$$
97$C_2$ $$( 1 - 110 T + p^{2} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$