# Properties

 Label 16-12e24-1.1-c2e8-0-3 Degree $16$ Conductor $7.950\times 10^{25}$ Sign $1$ Analytic cond. $2.41562\times 10^{13}$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·5-s − 8·13-s − 24·17-s − 56·25-s − 128·29-s − 24·37-s − 160·41-s + 124·49-s − 48·53-s + 136·61-s − 64·65-s + 72·73-s − 192·85-s + 168·89-s + 104·97-s − 352·101-s + 48·109-s − 728·113-s + 568·121-s − 744·125-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·145-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 8/5·5-s − 0.615·13-s − 1.41·17-s − 2.23·25-s − 4.41·29-s − 0.648·37-s − 3.90·41-s + 2.53·49-s − 0.905·53-s + 2.22·61-s − 0.984·65-s + 0.986·73-s − 2.25·85-s + 1.88·89-s + 1.07·97-s − 3.48·101-s + 0.440·109-s − 6.44·113-s + 4.69·121-s − 5.95·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.06·145-s + 0.00671·149-s + 0.00662·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{48} \cdot 3^{24}$$ Sign: $1$ Analytic conductor: $$2.41562\times 10^{13}$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1728} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.06015887585$$ $$L(\frac12)$$ $$\approx$$ $$0.06015887585$$ $$L(2)$$ 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$$
good5 $$( 1 - 4 T + 52 T^{2} - 124 T^{3} + 1558 T^{4} - 124 p^{2} T^{5} + 52 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
7 $$1 - 124 T^{2} + 5002 T^{4} + 178832 T^{6} - 21308909 T^{8} + 178832 p^{4} T^{10} + 5002 p^{8} T^{12} - 124 p^{12} T^{14} + p^{16} T^{16}$$
11 $$1 - 568 T^{2} + 14804 p T^{4} - 31597576 T^{6} + 4462882054 T^{8} - 31597576 p^{4} T^{10} + 14804 p^{9} T^{12} - 568 p^{12} T^{14} + p^{16} T^{16}$$
13 $$( 1 + 4 T + 394 T^{2} - 272 T^{3} + 72883 T^{4} - 272 p^{2} T^{5} + 394 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
17 $$( 1 + 12 T + 436 T^{2} + 7380 T^{3} + 90294 T^{4} + 7380 p^{2} T^{5} + 436 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
19 $$1 - 1340 T^{2} + 1130122 T^{4} - 629721776 T^{6} + 266027004115 T^{8} - 629721776 p^{4} T^{10} + 1130122 p^{8} T^{12} - 1340 p^{12} T^{14} + p^{16} T^{16}$$
23 $$1 - 2872 T^{2} + 3949276 T^{4} - 3455093896 T^{6} + 2139369590086 T^{8} - 3455093896 p^{4} T^{10} + 3949276 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16}$$
29 $$( 1 + 64 T + 2596 T^{2} + 67264 T^{3} + 1985254 T^{4} + 67264 p^{2} T^{5} + 2596 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
31 $$1 - 184 p T^{2} + 15692188 T^{4} - 26845223416 T^{6} + 31031256945478 T^{8} - 26845223416 p^{4} T^{10} + 15692188 p^{8} T^{12} - 184 p^{13} T^{14} + p^{16} T^{16}$$
37 $$( 1 + 12 T + 2650 T^{2} - 36432 T^{3} + 2846211 T^{4} - 36432 p^{2} T^{5} + 2650 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
41 $$( 1 + 80 T + 5956 T^{2} + 285680 T^{3} + 13442758 T^{4} + 285680 p^{2} T^{5} + 5956 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
43 $$1 - 12424 T^{2} + 71116252 T^{4} - 244515946552 T^{6} + 551660618273158 T^{8} - 244515946552 p^{4} T^{10} + 71116252 p^{8} T^{12} - 12424 p^{12} T^{14} + p^{16} T^{16}$$
47 $$1 - 2872 T^{2} + 15358300 T^{4} - 35747158408 T^{6} + 102474993504454 T^{8} - 35747158408 p^{4} T^{10} + 15358300 p^{8} T^{12} - 2872 p^{12} T^{14} + p^{16} T^{16}$$
53 $$( 1 + 24 T + 7204 T^{2} + 159624 T^{3} + 28315302 T^{4} + 159624 p^{2} T^{5} + 7204 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
59 $$1 - 11320 T^{2} + 50642716 T^{4} - 127137029128 T^{6} + 320662122219526 T^{8} - 127137029128 p^{4} T^{10} + 50642716 p^{8} T^{12} - 11320 p^{12} T^{14} + p^{16} T^{16}$$
61 $$( 1 - 68 T + 8266 T^{2} - 137072 T^{3} + 22167667 T^{4} - 137072 p^{2} T^{5} + 8266 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
67 $$1 - 27292 T^{2} + 345823978 T^{4} - 2700513303280 T^{6} + 14433873700614067 T^{8} - 2700513303280 p^{4} T^{10} + 345823978 p^{8} T^{12} - 27292 p^{12} T^{14} + p^{16} T^{16}$$
71 $$1 - 11080 T^{2} + 30779164 T^{4} + 199057866248 T^{6} - 1902994100012090 T^{8} + 199057866248 p^{4} T^{10} + 30779164 p^{8} T^{12} - 11080 p^{12} T^{14} + p^{16} T^{16}$$
73 $$( 1 - 36 T + 9130 T^{2} - 318096 T^{3} + 41836659 T^{4} - 318096 p^{2} T^{5} + 9130 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
79 $$1 - 41308 T^{2} + 786693994 T^{4} - 9045860371504 T^{6} + 68675576167144435 T^{8} - 9045860371504 p^{4} T^{10} + 786693994 p^{8} T^{12} - 41308 p^{12} T^{14} + p^{16} T^{16}$$
83 $$1 - 8200 T^{2} + 140721244 T^{4} - 1081724015032 T^{6} + 8821781005832710 T^{8} - 1081724015032 p^{4} T^{10} + 140721244 p^{8} T^{12} - 8200 p^{12} T^{14} + p^{16} T^{16}$$
89 $$( 1 - 84 T + 22900 T^{2} - 1903500 T^{3} + 237150582 T^{4} - 1903500 p^{2} T^{5} + 22900 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
97 $$( 1 - 52 T + 22522 T^{2} - 1223152 T^{3} + 279148675 T^{4} - 1223152 p^{2} T^{5} + 22522 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$