# Properties

 Label 16-384e8-1.1-c6e8-0-1 Degree $16$ Conductor $4.728\times 10^{20}$ Sign $1$ Analytic cond. $3.70927\times 10^{15}$ Root an. cond. $9.39897$ Motivic weight $6$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 72·3-s + 1.76e3·9-s − 1.64e3·11-s − 6.26e4·25-s − 9.72e3·27-s + 1.18e5·33-s − 4.17e5·49-s − 8.36e5·59-s − 1.96e6·73-s + 4.50e6·75-s − 1.28e5·81-s + 5.87e5·83-s − 1.47e6·97-s − 2.90e6·99-s + 7.65e5·107-s − 1.21e7·121-s + 127-s + 131-s + 137-s + 139-s + 3.00e7·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.38e7·169-s + ⋯
 L(s)  = 1 − 8/3·3-s + 2.41·9-s − 1.23·11-s − 4.00·25-s − 0.493·27-s + 3.30·33-s − 3.55·49-s − 4.07·59-s − 5.05·73-s + 10.6·75-s − 0.242·81-s + 1.02·83-s − 1.61·97-s − 2.99·99-s + 0.624·107-s − 6.86·121-s + 9.47·147-s + 2.87·169-s + 10.8·177-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(7-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{56} \cdot 3^{8}$$ Sign: $1$ Analytic conductor: $$3.70927\times 10^{15}$$ Root analytic conductor: $$9.39897$$ Motivic weight: $$6$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{384} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{56} \cdot 3^{8} ,\ ( \ : [3]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.002476320356$$ $$L(\frac12)$$ $$\approx$$ $$0.002476320356$$ $$L(4)$$ 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 + 4 p^{2} T + 118 p^{2} T^{2} + 4 p^{8} T^{3} + p^{12} T^{4} )^{2}$$
good5 $$( 1 + 1252 p^{2} T^{2} + 5863326 p^{3} T^{4} + 1252 p^{14} T^{6} + p^{24} T^{8} )^{2}$$
7 $$( 1 + 208996 T^{2} + 30837454086 T^{4} + 208996 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
11 $$( 1 + 412 T + 314898 p T^{2} + 412 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
13 $$( 1 - 6934756 T^{2} + 21911820809766 T^{4} - 6934756 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
17 $$( 1 - 64896772 T^{2} + 2198931719582598 T^{4} - 64896772 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
19 $$( 1 - 177090628 T^{2} + 12242150590475238 T^{4} - 177090628 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
23 $$( 1 - 1858756 T^{2} + 43746394215781446 T^{4} - 1858756 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
29 $$( 1 + 384071044 T^{2} + 262631660267478246 T^{4} + 384071044 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
31 $$( 1 + 2960240164 T^{2} + 3713060857169843526 T^{4} + 2960240164 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
37 $$( 1 - 3235129636 T^{2} + 7977254229970771686 T^{4} - 3235129636 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
41 $$( 1 - 13543268548 T^{2} + 89790600389496386118 T^{4} - 13543268548 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
43 $$( 1 - 6754505476 T^{2} + 39481734937898737446 T^{4} - 6754505476 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
47 $$( 1 - 13978081156 T^{2} +$$$$16\!\cdots\!86$$$$T^{4} - 13978081156 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
53 $$( 1 + 52183699396 T^{2} +$$$$13\!\cdots\!66$$$$T^{4} + 52183699396 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
59 $$( 1 + 209156 T + 54505500486 T^{2} + 209156 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
61 $$( 1 - 97087344484 T^{2} +$$$$75\!\cdots\!26$$$$T^{4} - 97087344484 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
67 $$( 1 - 172396628932 T^{2} +$$$$20\!\cdots\!98$$$$T^{4} - 172396628932 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
71 $$( 1 - 254155821124 T^{2} +$$$$48\!\cdots\!46$$$$T^{4} - 254155821124 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
73 $$( 1 + 491236 T + 352157486502 T^{2} + 491236 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
79 $$( 1 - 65015134556 T^{2} +$$$$11\!\cdots\!46$$$$T^{4} - 65015134556 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
83 $$( 1 - 146756 T + 654996035622 T^{2} - 146756 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
89 $$( 1 - 329765563588 T^{2} +$$$$84\!\cdots\!58$$$$T^{4} - 329765563588 p^{12} T^{6} + p^{24} T^{8} )^{2}$$
97 $$( 1 + 369308 T - 106588984506 T^{2} + 369308 p^{6} T^{3} + p^{12} T^{4} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$