# Properties

 Label 8-768e4-1.1-c3e4-0-9 Degree $8$ Conductor $347892350976$ Sign $1$ Analytic cond. $4.21608\times 10^{6}$ Root an. cond. $6.73152$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 46·9-s + 500·25-s + 1.37e3·49-s + 1.72e3·73-s + 1.38e3·81-s + 7.64e3·97-s + 4.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 2.30e4·225-s + ⋯
 L(s)  = 1 − 1.70·9-s + 4·25-s + 4·49-s + 2.75·73-s + 1.90·81-s + 7.99·97-s + 3.51·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 6.81·225-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \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^{32} \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: $$2^{32} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$4.21608\times 10^{6}$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$6.202457961$$ $$L(\frac12)$$ $$\approx$$ $$6.202457961$$ $$L(\frac{5}{2})$$ 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$C_2^2$ $$1 + 46 T^{2} + p^{6} T^{4}$$
good5$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
7$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
11$C_2^2$ $$( 1 - 2338 T^{2} + p^{6} T^{4} )^{2}$$
13$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
17$C_2$ $$( 1 - 90 T + p^{3} T^{2} )^{2}( 1 + 90 T + p^{3} T^{2} )^{2}$$
19$C_2^2$ $$( 1 - 2482 T^{2} + p^{6} T^{4} )^{2}$$
23$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
29$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
31$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
37$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
41$C_2$ $$( 1 - 522 T + p^{3} T^{2} )^{2}( 1 + 522 T + p^{3} T^{2} )^{2}$$
43$C_2^2$ $$( 1 - 74914 T^{2} + p^{6} T^{4} )^{2}$$
47$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
53$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
59$C_2^2$ $$( 1 + 304958 T^{2} + p^{6} T^{4} )^{2}$$
61$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
67$C_2^2$ $$( 1 - 596626 T^{2} + p^{6} T^{4} )^{2}$$
71$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
73$C_2$ $$( 1 - 430 T + p^{3} T^{2} )^{4}$$
79$C_2$ $$( 1 - p^{3} T^{2} )^{4}$$
83$C_2^2$ $$( 1 + 678926 T^{2} + p^{6} T^{4} )^{2}$$
89$C_2$ $$( 1 - 1026 T + p^{3} T^{2} )^{2}( 1 + 1026 T + p^{3} T^{2} )^{2}$$
97$C_2$ $$( 1 - 1910 T + p^{3} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$