# Properties

 Label 16-378e8-1.1-c3e8-0-4 Degree $16$ Conductor $4.168\times 10^{20}$ Sign $1$ Analytic cond. $6.12157\times 10^{10}$ Root an. cond. $4.72257$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 8·2-s + 24·4-s − 5-s + 28·7-s − 8·10-s − 5·11-s + 21·13-s + 224·14-s − 240·16-s + 46·17-s − 188·19-s − 24·20-s − 40·22-s − 374·23-s + 230·25-s + 168·26-s + 672·28-s − 271·29-s + 243·31-s − 768·32-s + 368·34-s − 28·35-s − 362·37-s − 1.50e3·38-s + 213·41-s + 238·43-s − 120·44-s + ⋯
 L(s)  = 1 + 2.82·2-s + 3·4-s − 0.0894·5-s + 1.51·7-s − 0.252·10-s − 0.137·11-s + 0.448·13-s + 4.27·14-s − 3.75·16-s + 0.656·17-s − 2.27·19-s − 0.268·20-s − 0.387·22-s − 3.39·23-s + 1.83·25-s + 1.26·26-s + 4.53·28-s − 1.73·29-s + 1.40·31-s − 4.24·32-s + 1.85·34-s − 0.135·35-s − 1.60·37-s − 6.42·38-s + 0.811·41-s + 0.844·43-s − 0.411·44-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(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{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: $$6.12157\times 10^{10}$$ Root analytic conductor: $$4.72257$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$24.14324022$$ $$L(\frac12)$$ $$\approx$$ $$24.14324022$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$( 1 - p T + p^{2} T^{2} )^{4}$$
3 $$1$$
7 $$( 1 - p T + p^{2} T^{2} )^{4}$$
good5 $$1 + T - 229 T^{2} - 774 p T^{3} + 26462 T^{4} + 678974 T^{5} + 1104902 p T^{6} - 2493309 p^{2} T^{7} - 997334831 T^{8} - 2493309 p^{5} T^{9} + 1104902 p^{7} T^{10} + 678974 p^{9} T^{11} + 26462 p^{12} T^{12} - 774 p^{16} T^{13} - 229 p^{18} T^{14} + p^{21} T^{15} + p^{24} T^{16}$$
11 $$1 + 5 T - 2182 T^{2} - 158451 T^{3} + 2427827 T^{4} + 279627664 T^{5} + 9980471089 T^{6} - 295406959071 T^{7} - 17130347878430 T^{8} - 295406959071 p^{3} T^{9} + 9980471089 p^{6} T^{10} + 279627664 p^{9} T^{11} + 2427827 p^{12} T^{12} - 158451 p^{15} T^{13} - 2182 p^{18} T^{14} + 5 p^{21} T^{15} + p^{24} T^{16}$$
13 $$1 - 21 T - 508 p T^{2} + 143907 T^{3} + 25185493 T^{4} - 450600480 T^{5} - 68065247446 T^{6} + 460246830006 T^{7} + 156900588822904 T^{8} + 460246830006 p^{3} T^{9} - 68065247446 p^{6} T^{10} - 450600480 p^{9} T^{11} + 25185493 p^{12} T^{12} + 143907 p^{15} T^{13} - 508 p^{19} T^{14} - 21 p^{21} T^{15} + p^{24} T^{16}$$
17 $$( 1 - 23 T + 9893 T^{2} - 281573 T^{3} + 69040201 T^{4} - 281573 p^{3} T^{5} + 9893 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
19 $$( 1 + 94 T + 13819 T^{2} + 1292485 T^{3} + 131846581 T^{4} + 1292485 p^{3} T^{5} + 13819 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
23 $$1 + 374 T + 45833 T^{2} + 3344556 T^{3} + 859764083 T^{4} + 169835114038 T^{5} + 17527026209797 T^{6} + 1808539479332106 T^{7} + 222811406401501087 T^{8} + 1808539479332106 p^{3} T^{9} + 17527026209797 p^{6} T^{10} + 169835114038 p^{9} T^{11} + 859764083 p^{12} T^{12} + 3344556 p^{15} T^{13} + 45833 p^{18} T^{14} + 374 p^{21} T^{15} + p^{24} T^{16}$$
29 $$1 + 271 T - 38803 T^{2} - 259344 p T^{3} + 3235076084 T^{4} + 303059385266 T^{5} - 105997401830390 T^{6} - 1727808215476197 T^{7} + 3339594708008059465 T^{8} - 1727808215476197 p^{3} T^{9} - 105997401830390 p^{6} T^{10} + 303059385266 p^{9} T^{11} + 3235076084 p^{12} T^{12} - 259344 p^{16} T^{13} - 38803 p^{18} T^{14} + 271 p^{21} T^{15} + p^{24} T^{16}$$
31 $$1 - 243 T - 6151 T^{2} - 8914860 T^{3} + 2490318622 T^{4} + 117491450778 T^{5} + 42742085844890 T^{6} - 10211221082121813 T^{7} - 951217853496050879 T^{8} - 10211221082121813 p^{3} T^{9} + 42742085844890 p^{6} T^{10} + 117491450778 p^{9} T^{11} + 2490318622 p^{12} T^{12} - 8914860 p^{15} T^{13} - 6151 p^{18} T^{14} - 243 p^{21} T^{15} + p^{24} T^{16}$$
37 $$( 1 + 181 T + 42844 T^{2} - 12321968 T^{3} - 3161946908 T^{4} - 12321968 p^{3} T^{5} + 42844 p^{6} T^{6} + 181 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
41 $$1 - 213 T - 84590 T^{2} + 65999067 T^{3} - 3185697689 T^{4} - 5013788714586 T^{5} + 1428680211356995 T^{6} + 164868593391514353 T^{7} -$$$$12\!\cdots\!52$$$$T^{8} + 164868593391514353 p^{3} T^{9} + 1428680211356995 p^{6} T^{10} - 5013788714586 p^{9} T^{11} - 3185697689 p^{12} T^{12} + 65999067 p^{15} T^{13} - 84590 p^{18} T^{14} - 213 p^{21} T^{15} + p^{24} T^{16}$$
43 $$1 - 238 T - 60675 T^{2} + 5271292 T^{3} - 2134721566 T^{4} + 2420831216454 T^{5} + 42843939153790 T^{6} - 184810086452079577 T^{7} + 50838540246961940772 T^{8} - 184810086452079577 p^{3} T^{9} + 42843939153790 p^{6} T^{10} + 2420831216454 p^{9} T^{11} - 2134721566 p^{12} T^{12} + 5271292 p^{15} T^{13} - 60675 p^{18} T^{14} - 238 p^{21} T^{15} + p^{24} T^{16}$$
47 $$1 + 675 T + 37387 T^{2} - 77590386 T^{3} - 14073185960 T^{4} + 2780172589698 T^{5} + 544267648367926 T^{6} + 322605518680562283 T^{7} +$$$$21\!\cdots\!97$$$$T^{8} + 322605518680562283 p^{3} T^{9} + 544267648367926 p^{6} T^{10} + 2780172589698 p^{9} T^{11} - 14073185960 p^{12} T^{12} - 77590386 p^{15} T^{13} + 37387 p^{18} T^{14} + 675 p^{21} T^{15} + p^{24} T^{16}$$
53 $$( 1 + 54 T + 236591 T^{2} + 19653129 T^{3} + 33755250168 T^{4} + 19653129 p^{3} T^{5} + 236591 p^{6} T^{6} + 54 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
59 $$1 + 202 T - 531049 T^{2} - 16684212 T^{3} + 159300075830 T^{4} - 10988486623156 T^{5} - 38404349933253128 T^{6} + 1299442121783896557 T^{7} +$$$$80\!\cdots\!84$$$$T^{8} + 1299442121783896557 p^{3} T^{9} - 38404349933253128 p^{6} T^{10} - 10988486623156 p^{9} T^{11} + 159300075830 p^{12} T^{12} - 16684212 p^{15} T^{13} - 531049 p^{18} T^{14} + 202 p^{21} T^{15} + p^{24} T^{16}$$
61 $$1 - 1212 T + 239267 T^{2} - 3162822 T^{3} + 187949681281 T^{4} - 58043170902294 T^{5} - 44955529435193749 T^{6} + 9664786778799768882 T^{7} +$$$$71\!\cdots\!33$$$$T^{8} + 9664786778799768882 p^{3} T^{9} - 44955529435193749 p^{6} T^{10} - 58043170902294 p^{9} T^{11} + 187949681281 p^{12} T^{12} - 3162822 p^{15} T^{13} + 239267 p^{18} T^{14} - 1212 p^{21} T^{15} + p^{24} T^{16}$$
67 $$1 + 139 T - 800538 T^{2} - 280034389 T^{3} + 334951438031 T^{4} + 134625567082980 T^{5} - 75122243735690903 T^{6} - 22025505408916196753 T^{7} +$$$$16\!\cdots\!74$$$$T^{8} - 22025505408916196753 p^{3} T^{9} - 75122243735690903 p^{6} T^{10} + 134625567082980 p^{9} T^{11} + 334951438031 p^{12} T^{12} - 280034389 p^{15} T^{13} - 800538 p^{18} T^{14} + 139 p^{21} T^{15} + p^{24} T^{16}$$
71 $$( 1 - 1295 T + 1795244 T^{2} - 1349389658 T^{3} + 1011484858822 T^{4} - 1349389658 p^{3} T^{5} + 1795244 p^{6} T^{6} - 1295 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
73 $$( 1 + 2000 T + 2663509 T^{2} + 2429329973 T^{3} + 1727799097657 T^{4} + 2429329973 p^{3} T^{5} + 2663509 p^{6} T^{6} + 2000 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
79 $$1 - 1545 T + 135641 T^{2} + 576786432 T^{3} + 262453195432 T^{4} - 392965726353228 T^{5} - 106866556646468452 T^{6} + 77122026263059272753 T^{7} +$$$$55\!\cdots\!45$$$$T^{8} + 77122026263059272753 p^{3} T^{9} - 106866556646468452 p^{6} T^{10} - 392965726353228 p^{9} T^{11} + 262453195432 p^{12} T^{12} + 576786432 p^{15} T^{13} + 135641 p^{18} T^{14} - 1545 p^{21} T^{15} + p^{24} T^{16}$$
83 $$1 - 142 T - 981853 T^{2} - 1435324800 T^{3} + 772358229251 T^{4} + 1109364392322268 T^{5} + 9657132248274443 p T^{6} -$$$$60\!\cdots\!94$$$$T^{7} -$$$$56\!\cdots\!45$$$$T^{8} -$$$$60\!\cdots\!94$$$$p^{3} T^{9} + 9657132248274443 p^{7} T^{10} + 1109364392322268 p^{9} T^{11} + 772358229251 p^{12} T^{12} - 1435324800 p^{15} T^{13} - 981853 p^{18} T^{14} - 142 p^{21} T^{15} + p^{24} T^{16}$$
89 $$( 1 - 132 T + 738863 T^{2} + 1004820993 T^{3} - 79516549278 T^{4} + 1004820993 p^{3} T^{5} + 738863 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} )^{2}$$
97 $$1 - 638 T - 197967 T^{2} + 1627259216 T^{3} - 1785731534086 T^{4} + 398431063574808 T^{5} + 521780813880337198 T^{6} -$$$$12\!\cdots\!03$$$$T^{7} +$$$$97\!\cdots\!70$$$$T^{8} -$$$$12\!\cdots\!03$$$$p^{3} T^{9} + 521780813880337198 p^{6} T^{10} + 398431063574808 p^{9} T^{11} - 1785731534086 p^{12} T^{12} + 1627259216 p^{15} T^{13} - 197967 p^{18} T^{14} - 638 p^{21} T^{15} + p^{24} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$