# Properties

 Label 16-7e16-1.1-c5e8-0-0 Degree $16$ Conductor $3.323\times 10^{13}$ Sign $1$ Analytic cond. $1.45496\times 10^{7}$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 10·2-s + 109·4-s + 790·8-s + 376·9-s + 1.95e3·11-s + 5.91e3·16-s + 3.76e3·18-s + 1.95e4·22-s + 7.13e3·23-s + 4.86e3·25-s − 6.70e3·29-s + 2.91e4·32-s + 4.09e4·36-s + 9.20e3·37-s + 4.08e4·43-s + 2.12e5·44-s + 7.13e4·46-s + 4.86e4·50-s + 1.02e5·53-s − 6.70e4·58-s + 1.54e5·64-s + 2.28e4·67-s − 3.07e5·71-s + 2.97e5·72-s + 9.20e4·74-s + 9.06e4·79-s + 8.83e4·81-s + ⋯
 L(s)  = 1 + 1.76·2-s + 3.40·4-s + 4.36·8-s + 1.54·9-s + 4.86·11-s + 5.77·16-s + 2.73·18-s + 8.59·22-s + 2.81·23-s + 1.55·25-s − 1.48·29-s + 5.02·32-s + 5.27·36-s + 1.10·37-s + 3.37·43-s + 16.5·44-s + 4.97·46-s + 2.75·50-s + 5.03·53-s − 2.61·58-s + 4.72·64-s + 0.623·67-s − 7.24·71-s + 6.75·72-s + 1.95·74-s + 1.63·79-s + 1.49·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$7^{16}$$ Sign: $1$ Analytic conductor: $$1.45496\times 10^{7}$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$141.4569029$$ $$L(\frac12)$$ $$\approx$$ $$141.4569029$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad7 $$1$$
good2 $$( 1 - 5 T - 17 T^{2} + 55 p T^{3} - 15 p^{2} T^{4} + 55 p^{6} T^{5} - 17 p^{10} T^{6} - 5 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
3 $$1 - 376 T^{2} + 17674 p T^{4} + 11183744 T^{6} - 4028466509 T^{8} + 11183744 p^{10} T^{10} + 17674 p^{21} T^{12} - 376 p^{30} T^{14} + p^{40} T^{16}$$
5 $$1 - 4868 T^{2} + 1134618 T^{4} - 14757614608 T^{6} + 182292626703011 T^{8} - 14757614608 p^{10} T^{10} + 1134618 p^{20} T^{12} - 4868 p^{30} T^{14} + p^{40} T^{16}$$
11 $$( 1 - 976 T + 396398 T^{2} - 228458176 T^{3} + 129983225707 T^{4} - 228458176 p^{5} T^{5} + 396398 p^{10} T^{6} - 976 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
13 $$( 1 + 224500 T^{2} - 32848089674 T^{4} + 224500 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
17 $$1 - 3065760 T^{2} + 3024788435134 T^{4} - 7180341456093511680 T^{6} +$$$$17\!\cdots\!55$$$$T^{8} - 7180341456093511680 p^{10} T^{10} + 3024788435134 p^{20} T^{12} - 3065760 p^{30} T^{14} + p^{40} T^{16}$$
19 $$1 - 8544504 T^{2} + 42956548559710 T^{4} -$$$$15\!\cdots\!16$$$$T^{6} +$$$$41\!\cdots\!19$$$$T^{8} -$$$$15\!\cdots\!16$$$$p^{10} T^{10} + 42956548559710 p^{20} T^{12} - 8544504 p^{30} T^{14} + p^{40} T^{16}$$
23 $$( 1 - 3568 T + 18274 T^{2} + 572078848 T^{3} + 38238668640819 T^{4} + 572078848 p^{5} T^{5} + 18274 p^{10} T^{6} - 3568 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
29 $$( 1 + 1676 T + 24360510 T^{2} + 1676 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
31 $$1 - 29395756 T^{2} + 205881674857002 T^{4} +$$$$28\!\cdots\!08$$$$T^{6} -$$$$94\!\cdots\!81$$$$T^{8} +$$$$28\!\cdots\!08$$$$p^{10} T^{10} + 205881674857002 p^{20} T^{12} - 29395756 p^{30} T^{14} + p^{40} T^{16}$$
37 $$( 1 - 4604 T - 122725214 T^{2} - 24097870064 T^{3} + 14435095249662139 T^{4} - 24097870064 p^{5} T^{5} - 122725214 p^{10} T^{6} - 4604 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
41 $$( 1 + 212331360 T^{2} + 36724946155085474 T^{4} + 212331360 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
43 $$( 1 - 10224 T + 293018130 T^{2} - 10224 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
47 $$1 - 568199404 T^{2} + 150307861516914922 T^{4} -$$$$38\!\cdots\!84$$$$T^{6} +$$$$10\!\cdots\!51$$$$T^{8} -$$$$38\!\cdots\!84$$$$p^{10} T^{10} + 150307861516914922 p^{20} T^{12} - 568199404 p^{30} T^{14} + p^{40} T^{16}$$
53 $$( 1 - 51460 T + 1339504322 T^{2} - 24301279586320 T^{3} + 430181182369297435 T^{4} - 24301279586320 p^{5} T^{5} + 1339504322 p^{10} T^{6} - 51460 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
59 $$1 - 1609944152 T^{2} + 1200185847501479934 T^{4} -$$$$59\!\cdots\!36$$$$T^{6} +$$$$35\!\cdots\!95$$$$T^{8} -$$$$59\!\cdots\!36$$$$p^{10} T^{10} + 1200185847501479934 p^{20} T^{12} - 1609944152 p^{30} T^{14} + p^{40} T^{16}$$
61 $$1 - 1094138468 T^{2} - 412207137807081734 T^{4} -$$$$19\!\cdots\!08$$$$T^{6} +$$$$12\!\cdots\!39$$$$T^{8} -$$$$19\!\cdots\!08$$$$p^{10} T^{10} - 412207137807081734 p^{20} T^{12} - 1094138468 p^{30} T^{14} + p^{40} T^{16}$$
67 $$( 1 - 11448 T + 1220364042 T^{2} + 43382854855296 T^{3} - 813190274442182533 T^{4} + 43382854855296 p^{5} T^{5} + 1220364042 p^{10} T^{6} - 11448 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
71 $$( 1 + 76912 T + 4562060670 T^{2} + 76912 p^{5} T^{3} + p^{10} T^{4} )^{4}$$
73 $$1 - 2170565248 T^{2} + 4286181938510097630 T^{4} +$$$$17\!\cdots\!52$$$$T^{6} -$$$$38\!\cdots\!21$$$$T^{8} +$$$$17\!\cdots\!52$$$$p^{10} T^{10} + 4286181938510097630 p^{20} T^{12} - 2170565248 p^{30} T^{14} + p^{40} T^{16}$$
79 $$( 1 - 45344 T - 4098421134 T^{2} - 17533255168 T^{3} + 22082918642969361635 T^{4} - 17533255168 p^{5} T^{5} - 4098421134 p^{10} T^{6} - 45344 p^{15} T^{7} + p^{20} T^{8} )^{2}$$
83 $$( 1 + 6721792472 T^{2} + 21939991980485194594 T^{4} + 6721792472 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
89 $$1 - 14718300768 T^{2} +$$$$10\!\cdots\!66$$$$T^{4} -$$$$78\!\cdots\!08$$$$T^{6} +$$$$56\!\cdots\!39$$$$T^{8} -$$$$78\!\cdots\!08$$$$p^{10} T^{10} +$$$$10\!\cdots\!66$$$$p^{20} T^{12} - 14718300768 p^{30} T^{14} + p^{40} T^{16}$$
97 $$( 1 + 22489794400 T^{2} + 25388976518139394 p^{2} T^{4} + 22489794400 p^{10} T^{6} + p^{20} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$