# Properties

 Label 16-21e16-1.1-c3e8-0-0 Degree $16$ Conductor $1.431\times 10^{21}$ Sign $1$ Analytic cond. $2.10105\times 10^{11}$ Root an. cond. $5.10096$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4-s − 34·8-s + 100·11-s + 44·16-s + 200·22-s + 352·23-s + 314·25-s − 520·29-s + 78·32-s − 212·37-s + 1.08e3·43-s + 100·44-s + 704·46-s + 628·50-s + 16·53-s − 1.04e3·58-s + 371·64-s + 1.94e3·67-s − 4.49e3·71-s − 424·74-s + 1.04e3·79-s + 2.16e3·86-s − 3.40e3·88-s + 352·92-s + 314·100-s + 32·106-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1/8·4-s − 1.50·8-s + 2.74·11-s + 0.687·16-s + 1.93·22-s + 3.19·23-s + 2.51·25-s − 3.32·29-s + 0.430·32-s − 0.941·37-s + 3.83·43-s + 0.342·44-s + 2.25·46-s + 1.77·50-s + 0.0414·53-s − 2.35·58-s + 0.724·64-s + 3.54·67-s − 7.51·71-s − 0.666·74-s + 1.49·79-s + 2.70·86-s − 4.11·88-s + 0.398·92-s + 0.313·100-s + 0.0293·106-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$3^{16} \cdot 7^{16}$$ Sign: $1$ Analytic conductor: $$2.10105\times 10^{11}$$ Root analytic conductor: $$5.10096$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{441} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.314023444$$ $$L(\frac12)$$ $$\approx$$ $$1.314023444$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$( 1 - T + T^{2} + p^{4} T^{3} - 9 p^{3} T^{4} + p^{7} T^{5} + p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} )^{2}$$
5 $$1 - 314 T^{2} + 50562 T^{4} - 5270176 T^{6} + 522562031 T^{8} - 5270176 p^{6} T^{10} + 50562 p^{12} T^{12} - 314 p^{18} T^{14} + p^{24} T^{16}$$
11 $$( 1 - 50 T - 202 T^{2} - 2000 T^{3} + 2201743 T^{4} - 2000 p^{3} T^{5} - 202 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
13 $$( 1 + 5554 T^{2} + 17209282 T^{4} + 5554 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
17 $$1 - 5088 T^{2} + 14421310 T^{4} + 187282685952 T^{6} - 1076010335607741 T^{8} + 187282685952 p^{6} T^{10} + 14421310 p^{12} T^{12} - 5088 p^{18} T^{14} + p^{24} T^{16}$$
19 $$1 - 12690 T^{2} + 28454938 T^{4} - 488430486000 T^{6} + 7788364400899983 T^{8} - 488430486000 p^{6} T^{10} + 28454938 p^{12} T^{12} - 12690 p^{18} T^{14} + p^{24} T^{16}$$
23 $$( 1 - 176 T - 62 T^{2} - 1179904 T^{3} + 438436563 T^{4} - 1179904 p^{3} T^{5} - 62 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
29 $$( 1 + 130 T + 49818 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
31 $$1 - 19060 T^{2} + 373250298 T^{4} + 34021605583600 T^{6} - 1109978980966458157 T^{8} + 34021605583600 p^{6} T^{10} + 373250298 p^{12} T^{12} - 19060 p^{18} T^{14} + p^{24} T^{16}$$
37 $$( 1 + 106 T - 92294 T^{2} + 235744 T^{3} + 7583597383 T^{4} + 235744 p^{3} T^{5} - 92294 p^{6} T^{6} + 106 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
41 $$( 1 + 208848 T^{2} + 19595539298 T^{4} + 208848 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
43 $$( 1 - 270 T + 97614 T^{2} - 270 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
47 $$1 - 228052 T^{2} + 17880021370 T^{4} - 2866445491787152 T^{6} +$$$$48\!\cdots\!99$$$$T^{8} - 2866445491787152 p^{6} T^{10} + 17880021370 p^{12} T^{12} - 228052 p^{18} T^{14} + p^{24} T^{16}$$
53 $$( 1 - 8 T - 276646 T^{2} + 168352 T^{3} + 54394534843 T^{4} + 168352 p^{3} T^{5} - 276646 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
59 $$1 - 632162 T^{2} + 222723302586 T^{4} - 58503068402380912 T^{6} +$$$$12\!\cdots\!79$$$$T^{8} - 58503068402380912 p^{6} T^{10} + 222723302586 p^{12} T^{12} - 632162 p^{18} T^{14} + p^{24} T^{16}$$
61 $$1 - 8978 p T^{2} + 149331839266 T^{4} - 426964025450528 p T^{6} +$$$$45\!\cdots\!79$$$$T^{8} - 426964025450528 p^{7} T^{10} + 149331839266 p^{12} T^{12} - 8978 p^{19} T^{14} + p^{24} T^{16}$$
67 $$( 1 - 972 T + 200922 T^{2} - 138350592 T^{3} + 178716222683 T^{4} - 138350592 p^{3} T^{5} + 200922 p^{6} T^{6} - 972 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
71 $$( 1 + 1124 T + 1018926 T^{2} + 1124 p^{3} T^{3} + p^{6} T^{4} )^{4}$$
73 $$1 - 1279312 T^{2} + 931531176990 T^{4} - 514845763213402112 T^{6} +$$$$22\!\cdots\!79$$$$T^{8} - 514845763213402112 p^{6} T^{10} + 931531176990 p^{12} T^{12} - 1279312 p^{18} T^{14} + p^{24} T^{16}$$
79 $$( 1 - 524 T - 205806 T^{2} + 264984704 T^{3} - 147697266061 T^{4} + 264984704 p^{3} T^{5} - 205806 p^{6} T^{6} - 524 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
83 $$( 1 + 2275682 T^{2} + 1948536513034 T^{4} + 2275682 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
89 $$1 + 804000 T^{2} - 70725036962 T^{4} - 222564522147840000 T^{6} -$$$$53\!\cdots\!77$$$$T^{8} - 222564522147840000 p^{6} T^{10} - 70725036962 p^{12} T^{12} + 804000 p^{18} T^{14} + p^{24} T^{16}$$
97 $$( 1 + 567808 T^{2} + 211724872834 T^{4} + 567808 p^{6} T^{6} + p^{12} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$