# Properties

 Label 16-378e8-1.1-c4e8-0-1 Degree $16$ Conductor $4.168\times 10^{20}$ Sign $1$ Analytic cond. $5.43362\times 10^{12}$ Root an. cond. $6.25090$ Motivic weight $4$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·4-s − 376·13-s + 640·16-s + 1.12e3·19-s + 2.89e3·25-s − 880·31-s + 1.57e3·37-s − 5.76e3·43-s + 1.37e3·49-s + 1.20e4·52-s − 2.56e3·61-s − 1.02e4·64-s + 2.37e4·67-s − 1.31e4·73-s − 3.58e4·76-s − 9.59e3·79-s − 1.40e4·97-s − 9.26e4·100-s − 5.74e3·103-s + 7.61e4·109-s + 4.53e4·121-s + 2.81e4·124-s + 127-s + 131-s + 137-s + 139-s − 5.04e4·148-s + ⋯
 L(s)  = 1 − 2·4-s − 2.22·13-s + 5/2·16-s + 3.10·19-s + 4.63·25-s − 0.915·31-s + 1.15·37-s − 3.11·43-s + 4/7·49-s + 4.44·52-s − 0.687·61-s − 5/2·64-s + 5.29·67-s − 2.47·73-s − 6.20·76-s − 1.53·79-s − 1.48·97-s − 9.26·100-s − 0.541·103-s + 6.41·109-s + 3.09·121-s + 1.83·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 2.30·148-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 3^{24} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$5.43362\times 10^{12}$$ Root analytic conductor: $$6.25090$$ Motivic weight: $$4$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [2]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.141054502$$ $$L(\frac12)$$ $$\approx$$ $$1.141054502$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 + p^{3} T^{2} )^{4}$$
3 $$1$$
7 $$( 1 - p^{3} T^{2} )^{4}$$
good5 $$1 - 2896 T^{2} + 3638054 T^{4} - 2780750784 T^{6} + 1742925696491 T^{8} - 2780750784 p^{8} T^{10} + 3638054 p^{16} T^{12} - 2896 p^{24} T^{14} + p^{32} T^{16}$$
11 $$1 - 45376 T^{2} + 1125096230 T^{4} - 1811353663872 p T^{6} + 310095746834536235 T^{8} - 1811353663872 p^{9} T^{10} + 1125096230 p^{16} T^{12} - 45376 p^{24} T^{14} + p^{32} T^{16}$$
13 $$( 1 + 188 T + 74156 T^{2} + 8050092 T^{3} + 2287381274 T^{4} + 8050092 p^{4} T^{5} + 74156 p^{8} T^{6} + 188 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
17 $$1 - 158704 T^{2} + 29778019076 T^{4} - 3081760811859408 T^{6} +$$$$31\!\cdots\!86$$$$T^{8} - 3081760811859408 p^{8} T^{10} + 29778019076 p^{16} T^{12} - 158704 p^{24} T^{14} + p^{32} T^{16}$$
19 $$( 1 - 560 T + 377398 T^{2} - 156336320 T^{3} + 75881622811 T^{4} - 156336320 p^{4} T^{5} + 377398 p^{8} T^{6} - 560 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
23 $$1 - 1281040 T^{2} + 888819235094 T^{4} - 405553787863355712 T^{6} +$$$$13\!\cdots\!15$$$$T^{8} - 405553787863355712 p^{8} T^{10} + 888819235094 p^{16} T^{12} - 1281040 p^{24} T^{14} + p^{32} T^{16}$$
29 $$1 - 2268016 T^{2} + 3273567985028 T^{4} - 3388632307546760784 T^{6} +$$$$27\!\cdots\!98$$$$T^{8} - 3388632307546760784 p^{8} T^{10} + 3273567985028 p^{16} T^{12} - 2268016 p^{24} T^{14} + p^{32} T^{16}$$
31 $$( 1 + 440 T + 2095934 T^{2} + 1329203904 T^{3} + 2254233834035 T^{4} + 1329203904 p^{4} T^{5} + 2095934 p^{8} T^{6} + 440 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
37 $$( 1 - 788 T + 2100034 T^{2} - 2401740416 T^{3} + 2338205615659 T^{4} - 2401740416 p^{4} T^{5} + 2100034 p^{8} T^{6} - 788 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
41 $$1 - 6887696 T^{2} + 28777618813030 T^{4} - 91098876450878501312 T^{6} +$$$$23\!\cdots\!95$$$$T^{8} - 91098876450878501312 p^{8} T^{10} + 28777618813030 p^{16} T^{12} - 6887696 p^{24} T^{14} + p^{32} T^{16}$$
43 $$( 1 + 2884 T + 7001404 T^{2} + 5531256052 T^{3} + 7968256367770 T^{4} + 5531256052 p^{4} T^{5} + 7001404 p^{8} T^{6} + 2884 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
47 $$1 - 25995728 T^{2} + 331798032373252 T^{4} -$$$$27\!\cdots\!00$$$$T^{6} +$$$$15\!\cdots\!86$$$$T^{8} -$$$$27\!\cdots\!00$$$$p^{8} T^{10} + 331798032373252 p^{16} T^{12} - 25995728 p^{24} T^{14} + p^{32} T^{16}$$
53 $$1 - 18145448 T^{2} + 321230002760956 T^{4} -$$$$33\!\cdots\!24$$$$T^{6} +$$$$32\!\cdots\!86$$$$T^{8} -$$$$33\!\cdots\!24$$$$p^{8} T^{10} + 321230002760956 p^{16} T^{12} - 18145448 p^{24} T^{14} + p^{32} T^{16}$$
59 $$1 - 44142112 T^{2} + 1088926691812772 T^{4} -$$$$18\!\cdots\!68$$$$T^{6} +$$$$25\!\cdots\!18$$$$T^{8} -$$$$18\!\cdots\!68$$$$p^{8} T^{10} + 1088926691812772 p^{16} T^{12} - 44142112 p^{24} T^{14} + p^{32} T^{16}$$
61 $$( 1 + 1280 T + 13865420 T^{2} + 17676181248 T^{3} + 395149923191270 T^{4} + 17676181248 p^{4} T^{5} + 13865420 p^{8} T^{6} + 1280 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
67 $$( 1 - 11892 T + 79780940 T^{2} - 378965460804 T^{3} + 1752487431659994 T^{4} - 378965460804 p^{4} T^{5} + 79780940 p^{8} T^{6} - 11892 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
71 $$1 + 13087616 T^{2} + 992211923650454 T^{4} -$$$$11\!\cdots\!04$$$$T^{6} +$$$$31\!\cdots\!75$$$$T^{8} -$$$$11\!\cdots\!04$$$$p^{8} T^{10} + 992211923650454 p^{16} T^{12} + 13087616 p^{24} T^{14} + p^{32} T^{16}$$
73 $$( 1 + 6588 T + 111876188 T^{2} + 533549603340 T^{3} + 4720448405156442 T^{4} + 533549603340 p^{4} T^{5} + 111876188 p^{8} T^{6} + 6588 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
79 $$( 1 + 4796 T + 124292444 T^{2} + 471046796940 T^{3} + 6993960834109274 T^{4} + 471046796940 p^{4} T^{5} + 124292444 p^{8} T^{6} + 4796 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
83 $$1 - 329875648 T^{2} + 49399645099250468 T^{4} -$$$$43\!\cdots\!88$$$$T^{6} +$$$$25\!\cdots\!78$$$$T^{8} -$$$$43\!\cdots\!88$$$$p^{8} T^{10} + 49399645099250468 p^{16} T^{12} - 329875648 p^{24} T^{14} + p^{32} T^{16}$$
89 $$1 - 49420160 T^{2} + 5451257749574998 T^{4} -$$$$47\!\cdots\!04$$$$T^{6} +$$$$31\!\cdots\!31$$$$T^{8} -$$$$47\!\cdots\!04$$$$p^{8} T^{10} + 5451257749574998 p^{16} T^{12} - 49420160 p^{24} T^{14} + p^{32} T^{16}$$
97 $$( 1 + 7008 T + 168353996 T^{2} + 1429649031072 T^{3} + 18573365118818982 T^{4} + 1429649031072 p^{4} T^{5} + 168353996 p^{8} T^{6} + 7008 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$