# Properties

 Label 8-48e4-1.1-c15e4-0-0 Degree $8$ Conductor $5308416$ Sign $1$ Analytic cond. $2.20080\times 10^{7}$ Root an. cond. $8.27604$ Motivic weight $15$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2.65e7·9-s − 6.65e8·13-s − 4.47e10·25-s − 1.25e12·37-s + 1.88e13·49-s + 8.62e11·61-s − 3.72e14·73-s + 5.01e14·81-s + 4.64e15·97-s − 1.21e16·109-s − 1.76e16·117-s − 6.92e15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.20e16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 1.85·9-s − 2.94·13-s − 1.46·25-s − 2.16·37-s + 3.96·49-s + 0.0351·61-s − 3.94·73-s + 2.43·81-s + 5.83·97-s − 6.37·109-s − 5.45·117-s − 1.65·121-s + 1.40·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5308416$$    =    $$2^{16} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$2.20080\times 10^{7}$$ Root analytic conductor: $$8.27604$$ Motivic weight: $$15$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5308416,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)$$

## Particular Values

 $$L(8)$$ $$\approx$$ $$0.8477034184$$ $$L(\frac12)$$ $$\approx$$ $$0.8477034184$$ $$L(\frac{17}{2})$$ 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$C_2^2$ $$1 - 12158 p^{7} T^{2} + p^{30} T^{4}$$
good5$C_2^2$ $$( 1 + 4472960494 p T^{2} + p^{30} T^{4} )^{2}$$
7$C_2^2$ $$( 1 - 192004967426 p^{2} T^{2} + p^{30} T^{4} )^{2}$$
11$C_2^2$ $$( 1 + 28628981943862 p^{2} T^{2} + p^{30} T^{4} )^{2}$$
13$C_2$ $$( 1 + 12798770 p T + p^{15} T^{2} )^{4}$$
17$C_2^2$ $$( 1 - 5621089645499631586 T^{2} + p^{30} T^{4} )^{2}$$
19$C_2^2$ $$( 1 + 12206075725639426102 T^{2} + p^{30} T^{4} )^{2}$$
23$C_2^2$ $$( 1 +$$$$28\!\cdots\!74$$$$T^{2} + p^{30} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 9767706874797348058 p^{2} T^{2} + p^{30} T^{4} )^{2}$$
31$C_2^2$ $$( 1 -$$$$17\!\cdots\!02$$$$T^{2} + p^{30} T^{4} )^{2}$$
37$C_2$ $$( 1 + 312995380930 T + p^{15} T^{2} )^{4}$$
41$C_2^2$ $$( 1 -$$$$14\!\cdots\!22$$$$T^{2} + p^{30} T^{4} )^{2}$$
43$C_2^2$ $$( 1 +$$$$20\!\cdots\!54$$$$T^{2} + p^{30} T^{4} )^{2}$$
47$C_2^2$ $$( 1 +$$$$23\!\cdots\!46$$$$T^{2} + p^{30} T^{4} )^{2}$$
53$C_2^2$ $$( 1 -$$$$47\!\cdots\!14$$$$T^{2} + p^{30} T^{4} )^{2}$$
59$C_2^2$ $$( 1 +$$$$17\!\cdots\!98$$$$T^{2} + p^{30} T^{4} )^{2}$$
61$C_2$ $$( 1 - 215703610022 T + p^{15} T^{2} )^{4}$$
67$C_2^2$ $$( 1 -$$$$50\!\cdots\!14$$$$T^{2} + p^{30} T^{4} )^{2}$$
71$C_2^2$ $$( 1 +$$$$27\!\cdots\!02$$$$T^{2} + p^{30} T^{4} )^{2}$$
73$C_2$ $$( 1 + 93057038642630 T + p^{15} T^{2} )^{4}$$
79$C_2^2$ $$( 1 +$$$$24\!\cdots\!02$$$$T^{2} + p^{30} T^{4} )^{2}$$
83$C_2^2$ $$( 1 +$$$$91\!\cdots\!74$$$$T^{2} + p^{30} T^{4} )^{2}$$
89$C_2^2$ $$( 1 +$$$$20\!\cdots\!82$$$$T^{2} + p^{30} T^{4} )^{2}$$
97$C_2$ $$( 1 - 1161599117677810 T + p^{15} T^{2} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$