# Properties

 Label 8-192e4-1.1-c3e4-0-3 Degree $8$ Conductor $1358954496$ Sign $1$ Analytic cond. $16469.0$ Root an. cond. $3.36576$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 18·9-s + 216·17-s + 476·25-s − 1.17e3·41-s − 196·49-s − 4.02e3·73-s + 243·81-s − 5.92e3·89-s + 7.28e3·97-s − 1.56e3·113-s + 716·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.88e3·153-s + 157-s + 163-s + 167-s + 5.33e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 − 2/3·9-s + 3.08·17-s + 3.80·25-s − 4.47·41-s − 4/7·49-s − 6.45·73-s + 1/3·81-s − 7.06·89-s + 7.62·97-s − 1.29·113-s + 0.537·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 2.05·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.42·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{24} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$16469.0$$ Root analytic conductor: $$3.36576$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{24} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.712290147$$ $$L(\frac12)$$ $$\approx$$ $$1.712290147$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 + p^{2} T^{2} )^{2}$$
good5$C_2^2$ $$( 1 - 238 T^{2} + p^{6} T^{4} )^{2}$$
7$C_2^2$ $$( 1 + 2 p^{2} T^{2} + p^{6} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 358 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 2666 T^{2} + p^{6} T^{4} )^{2}$$
17$C_2$ $$( 1 - 54 T + p^{3} T^{2} )^{4}$$
19$C_2^2$ $$( 1 - 13702 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2^2$ $$( 1 - 5666 T^{2} + p^{6} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 22270 T^{2} + p^{6} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 56114 T^{2} + p^{6} T^{4} )^{2}$$
37$C_2^2$ $$( 1 + 4726 T^{2} + p^{6} T^{4} )^{2}$$
41$C_2$ $$( 1 + 294 T + p^{3} T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 123670 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2^2$ $$( 1 - 48146 T^{2} + p^{6} T^{4} )^{2}$$
53$C_2^2$ $$( 1 + 256946 T^{2} + p^{6} T^{4} )^{2}$$
59$C_2^2$ $$( 1 - 347254 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 445850 T^{2} + p^{6} T^{4} )^{2}$$
67$C_2^2$ $$( 1 - 207142 T^{2} + p^{6} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 715774 T^{2} + p^{6} T^{4} )^{2}$$
73$C_2$ $$( 1 + 1006 T + p^{3} T^{2} )^{4}$$
79$C_2^2$ $$( 1 - 811150 T^{2} + p^{6} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 625174 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2$ $$( 1 + 1482 T + p^{3} T^{2} )^{4}$$
97$C_2$ $$( 1 - 1822 T + p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$