# Properties

 Label 16-384e8-1.1-c8e8-0-0 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Analytic cond. $3.58620\times 10^{17}$ Root an. cond. $12.5073$ Motivic weight $8$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8.74e3·9-s + 9.30e4·17-s + 1.76e6·25-s + 2.97e6·41-s + 3.11e7·49-s + 1.91e8·73-s + 4.78e7·81-s + 4.22e8·89-s + 3.13e7·97-s − 2.27e8·113-s − 4.90e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.13e8·153-s + 157-s + 163-s + 167-s + 2.68e9·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
 L(s)  = 1 + 4/3·9-s + 1.11·17-s + 4.50·25-s + 1.05·41-s + 5.40·49-s + 6.73·73-s + 10/9·81-s + 6.73·89-s + 0.354·97-s − 1.39·113-s − 2.28·121-s + 1.48·153-s + 3.28·169-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$3.58620\times 10^{17}$$ Root analytic conductor: $$12.5073$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [4]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$36.14992703$$ $$L(\frac12)$$ $$\approx$$ $$36.14992703$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$( 1 - p^{7} T^{2} )^{4}$$
good5 $$( 1 - 880228 T^{2} + 18667806774 p^{2} T^{4} - 880228 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
7 $$( 1 - 158834 p^{2} T^{2} + p^{16} T^{4} )^{4}$$
11 $$( 1 + 245275684 T^{2} + 97275433541512902 T^{4} + 245275684 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
13 $$( 1 - 1340219908 T^{2} + 865655323360267398 T^{4} - 1340219908 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
17 $$( 1 - 23260 T + 6041535558 T^{2} - 23260 p^{8} T^{3} + p^{16} T^{4} )^{4}$$
19 $$( 1 + 65391715876 T^{2} +$$$$16\!\cdots\!70$$$$T^{4} + 65391715876 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
23 $$( 1 - 76045718404 T^{2} +$$$$89\!\cdots\!10$$$$T^{4} - 76045718404 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
29 $$( 1 - 510589441636 T^{2} +$$$$28\!\cdots\!50$$$$T^{4} - 510589441636 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
31 $$( 1 - 524615630500 T^{2} +$$$$95\!\cdots\!38$$$$T^{4} - 524615630500 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
37 $$( 1 - 6548545929604 T^{2} +$$$$27\!\cdots\!42$$$$T^{4} - 6548545929604 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
41 $$( 1 - 744644 T - 1663452688890 T^{2} - 744644 p^{8} T^{3} + p^{16} T^{4} )^{4}$$
43 $$( 1 + 34703104689700 T^{2} +$$$$56\!\cdots\!02$$$$T^{4} + 34703104689700 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
47 $$( 1 - 3500072077444 T^{2} +$$$$10\!\cdots\!10$$$$T^{4} - 3500072077444 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
53 $$( 1 - 160871699435620 T^{2} +$$$$13\!\cdots\!42$$$$T^{4} - 160871699435620 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
59 $$( 1 + 197954346294820 T^{2} +$$$$37\!\cdots\!82$$$$T^{4} + 197954346294820 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
61 $$( 1 - 303812279368324 T^{2} +$$$$49\!\cdots\!22$$$$T^{4} - 303812279368324 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
67 $$( 1 + 1573471302773284 T^{2} +$$$$94\!\cdots\!62$$$$T^{4} + 1573471302773284 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
71 $$( 1 - 1674075860991364 T^{2} +$$$$14\!\cdots\!90$$$$T^{4} - 1674075860991364 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
73 $$( 1 - 47847164 T + 1936048722224262 T^{2} - 47847164 p^{8} T^{3} + p^{16} T^{4} )^{4}$$
79 $$( 1 - 4210102515025060 T^{2} +$$$$81\!\cdots\!78$$$$T^{4} - 4210102515025060 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
83 $$( 1 + 7510101349972516 T^{2} +$$$$23\!\cdots\!50$$$$T^{4} + 7510101349972516 p^{16} T^{6} + p^{32} T^{8} )^{2}$$
89 $$( 1 - 105718108 T + 10045489164843462 T^{2} - 105718108 p^{8} T^{3} + p^{16} T^{4} )^{4}$$
97 $$( 1 - 7842140 T + 4976272175418438 T^{2} - 7842140 p^{8} T^{3} + p^{16} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$