# Properties

 Degree $16$ Conductor $4.348\times 10^{12}$ Sign $1$ Motivic weight $4$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 32·4-s + 18·5-s − 162·7-s + 190·9-s − 6·11-s + 640·16-s + 510·17-s − 12·19-s − 576·20-s − 396·23-s − 609·25-s + 5.18e3·28-s − 2.91e3·35-s − 6.08e3·36-s − 8.65e3·43-s + 192·44-s + 3.42e3·45-s + 3.21e3·47-s + 8.12e3·49-s − 108·55-s + 1.31e3·61-s − 3.07e4·63-s − 1.02e4·64-s − 1.63e4·68-s + 2.33e4·73-s + 384·76-s + 972·77-s + ⋯
 L(s)  = 1 − 2·4-s + 0.719·5-s − 3.30·7-s + 2.34·9-s − 0.0495·11-s + 5/2·16-s + 1.76·17-s − 0.0332·19-s − 1.43·20-s − 0.748·23-s − 0.974·25-s + 6.61·28-s − 2.38·35-s − 4.69·36-s − 4.68·43-s + 0.0991·44-s + 1.68·45-s + 1.45·47-s + 3.38·49-s − 0.0357·55-s + 0.353·61-s − 7.75·63-s − 5/2·64-s − 3.52·68-s + 4.39·73-s + 0.0664·76-s + 0.163·77-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 19^{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 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$2^{8} \cdot 19^{8}$$ Sign: $1$ Motivic weight: $$4$$ Character: induced by $\chi_{38} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 2^{8} \cdot 19^{8} ,\ ( \ : [2]^{8} ),\ 1 )$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$1.02036$$ $$L(\frac12)$$ $$\approx$$ $$1.02036$$ $$L(3)$$ 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}$$
19 $$1 + 12 T + 303560 T^{2} - 1063956 p T^{3} + 6709482 p^{3} T^{4} - 1063956 p^{5} T^{5} + 303560 p^{8} T^{6} + 12 p^{12} T^{7} + p^{16} T^{8}$$
good3 $$1 - 190 T^{2} + 9523 p T^{4} - 297718 p^{2} T^{6} + 114172 p^{7} T^{8} - 297718 p^{10} T^{10} + 9523 p^{17} T^{12} - 190 p^{24} T^{14} + p^{32} T^{16}$$
5 $$( 1 - 9 T + 426 T^{2} - 10251 T^{3} + 716146 T^{4} - 10251 p^{4} T^{5} + 426 p^{8} T^{6} - 9 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
7 $$( 1 + 81 T + 5777 T^{2} + 49050 p T^{3} + 20657670 T^{4} + 49050 p^{5} T^{5} + 5777 p^{8} T^{6} + 81 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
11 $$( 1 + 3 T + 20874 T^{2} + 203751 p T^{3} + 23528702 p T^{4} + 203751 p^{5} T^{5} + 20874 p^{8} T^{6} + 3 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
13 $$1 - 87902 T^{2} + 4031399833 T^{4} - 153863832997766 T^{6} + 5034444913352767444 T^{8} - 153863832997766 p^{8} T^{10} + 4031399833 p^{16} T^{12} - 87902 p^{24} T^{14} + p^{32} T^{16}$$
17 $$( 1 - 15 p T + 266529 T^{2} - 48908322 T^{3} + 30973446334 T^{4} - 48908322 p^{4} T^{5} + 266529 p^{8} T^{6} - 15 p^{13} T^{7} + p^{16} T^{8} )^{2}$$
23 $$( 1 + 198 T + 793305 T^{2} + 151845534 T^{3} + 305726793412 T^{4} + 151845534 p^{4} T^{5} + 793305 p^{8} T^{6} + 198 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
29 $$1 - 1884206 T^{2} + 2536130733097 T^{4} - 2411241746303389430 T^{6} +$$$$19\!\cdots\!76$$$$T^{8} - 2411241746303389430 p^{8} T^{10} + 2536130733097 p^{16} T^{12} - 1884206 p^{24} T^{14} + p^{32} T^{16}$$
31 $$1 - 5185760 T^{2} + 13147692978364 T^{4} - 21074811809060168480 T^{6} +$$$$23\!\cdots\!86$$$$T^{8} - 21074811809060168480 p^{8} T^{10} + 13147692978364 p^{16} T^{12} - 5185760 p^{24} T^{14} + p^{32} T^{16}$$
37 $$1 - 8989760 T^{2} + 32058963480700 T^{4} - 61405971768293818304 T^{6} +$$$$98\!\cdots\!62$$$$T^{8} - 61405971768293818304 p^{8} T^{10} + 32058963480700 p^{16} T^{12} - 8989760 p^{24} T^{14} + p^{32} T^{16}$$
41 $$1 - 15418400 T^{2} + 116131174644796 T^{4} -$$$$56\!\cdots\!20$$$$T^{6} +$$$$18\!\cdots\!30$$$$T^{8} -$$$$56\!\cdots\!20$$$$p^{8} T^{10} + 116131174644796 p^{16} T^{12} - 15418400 p^{24} T^{14} + p^{32} T^{16}$$
43 $$( 1 + 4327 T + 17040418 T^{2} + 41562268261 T^{3} + 93546947175394 T^{4} + 41562268261 p^{4} T^{5} + 17040418 p^{8} T^{6} + 4327 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
47 $$( 1 - 1605 T + 15846978 T^{2} - 24977693547 T^{3} + 106935047396698 T^{4} - 24977693547 p^{4} T^{5} + 15846978 p^{8} T^{6} - 1605 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
53 $$1 - 26276750 T^{2} + 354841909979401 T^{4} -$$$$39\!\cdots\!70$$$$T^{6} +$$$$37\!\cdots\!60$$$$T^{8} -$$$$39\!\cdots\!70$$$$p^{8} T^{10} + 354841909979401 p^{16} T^{12} - 26276750 p^{24} T^{14} + p^{32} T^{16}$$
59 $$1 - 70986542 T^{2} + 2266647860250313 T^{4} -$$$$44\!\cdots\!70$$$$T^{6} +$$$$62\!\cdots\!16$$$$T^{8} -$$$$44\!\cdots\!70$$$$p^{8} T^{10} + 2266647860250313 p^{16} T^{12} - 70986542 p^{24} T^{14} + p^{32} T^{16}$$
61 $$( 1 - 657 T + 13651298 T^{2} + 36749564541 T^{3} + 190575679355970 T^{4} + 36749564541 p^{4} T^{5} + 13651298 p^{8} T^{6} - 657 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
67 $$1 - 11587598 T^{2} + 1243984422721705 T^{4} -$$$$11\!\cdots\!74$$$$T^{6} +$$$$71\!\cdots\!88$$$$T^{8} -$$$$11\!\cdots\!74$$$$p^{8} T^{10} + 1243984422721705 p^{16} T^{12} - 11587598 p^{24} T^{14} + p^{32} T^{16}$$
71 $$1 - 27489248 T^{2} + 2078050971545020 T^{4} -$$$$45\!\cdots\!44$$$$T^{6} +$$$$19\!\cdots\!38$$$$T^{8} -$$$$45\!\cdots\!44$$$$p^{8} T^{10} + 2078050971545020 p^{16} T^{12} - 27489248 p^{24} T^{14} + p^{32} T^{16}$$
73 $$( 1 - 11699 T + 88585093 T^{2} - 366782768102 T^{3} + 1847388837013414 T^{4} - 366782768102 p^{4} T^{5} + 88585093 p^{8} T^{6} - 11699 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
79 $$1 - 198341024 T^{2} + 17865606744931900 T^{4} -$$$$10\!\cdots\!64$$$$T^{6} +$$$$43\!\cdots\!70$$$$T^{8} -$$$$10\!\cdots\!64$$$$p^{8} T^{10} + 17865606744931900 p^{16} T^{12} - 198341024 p^{24} T^{14} + p^{32} T^{16}$$
83 $$( 1 + 5220 T + 132074424 T^{2} + 739113578028 T^{3} + 8089121462686894 T^{4} + 739113578028 p^{4} T^{5} + 132074424 p^{8} T^{6} + 5220 p^{12} T^{7} + p^{16} T^{8} )^{2}$$
89 $$1 - 219510728 T^{2} + 29593428331358620 T^{4} -$$$$27\!\cdots\!44$$$$T^{6} +$$$$20\!\cdots\!18$$$$T^{8} -$$$$27\!\cdots\!44$$$$p^{8} T^{10} + 29593428331358620 p^{16} T^{12} - 219510728 p^{24} T^{14} + p^{32} T^{16}$$
97 $$1 - 272222816 T^{2} + 39430845297013180 T^{4} -$$$$47\!\cdots\!04$$$$T^{6} +$$$$48\!\cdots\!14$$$$T^{8} -$$$$47\!\cdots\!04$$$$p^{8} T^{10} + 39430845297013180 p^{16} T^{12} - 272222816 p^{24} T^{14} + p^{32} T^{16}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$