# Properties

 Degree $32$ Conductor $3.418\times 10^{34}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·16-s + 16·19-s − 32·43-s − 48·49-s − 32·61-s − 8·64-s − 16·67-s + 96·103-s − 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
 L(s)  = 1 − 16-s + 3.67·19-s − 4.87·43-s − 6.85·49-s − 4.09·61-s − 64-s − 1.95·67-s + 9.45·103-s − 3.06·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$32$$ Conductor: $$2^{64} \cdot 3^{32}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{144} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(32,\ 2^{64} \cdot 3^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.778923$$ $$L(\frac12)$$ $$\approx$$ $$0.778923$$ $$L(\frac{3}{2})$$ 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^{2} T^{4} + p^{3} T^{6} + p^{2} T^{8} + p^{5} T^{10} + p^{6} T^{12} + p^{8} T^{16}$$
3 $$1$$
good5 $$1 - 8 T^{4} - 804 T^{8} + 2376 T^{12} + 502406 T^{16} + 2376 p^{4} T^{20} - 804 p^{8} T^{24} - 8 p^{12} T^{28} + p^{16} T^{32}$$
7 $$( 1 + 12 T^{2} + 16 T^{3} + 74 T^{4} + 16 p T^{5} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
11 $$1 - 184 T^{4} + 8796 T^{8} + 2616696 T^{12} - 490804602 T^{16} + 2616696 p^{4} T^{20} + 8796 p^{8} T^{24} - 184 p^{12} T^{28} + p^{16} T^{32}$$
13 $$( 1 + 64 T^{3} - 36 T^{4} - 704 T^{5} + 2048 T^{6} + 384 T^{7} - 43930 T^{8} + 384 p T^{9} + 2048 p^{2} T^{10} - 704 p^{3} T^{11} - 36 p^{4} T^{12} + 64 p^{5} T^{13} + p^{8} T^{16} )^{2}$$
17 $$( 1 - 64 T^{2} + 1956 T^{4} - 40128 T^{6} + 697542 T^{8} - 40128 p^{2} T^{10} + 1956 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
19 $$( 1 - 8 T + 32 T^{2} - 184 T^{3} + 388 T^{4} + 1992 T^{5} - 11424 T^{6} + 82616 T^{7} - 538074 T^{8} + 82616 p T^{9} - 11424 p^{2} T^{10} + 1992 p^{3} T^{11} + 388 p^{4} T^{12} - 184 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2}$$
23 $$( 1 - 88 T^{2} + 3004 T^{4} - 46440 T^{6} + 546118 T^{8} - 46440 p^{2} T^{10} + 3004 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
29 $$1 - 1672 T^{4} + 1133916 T^{8} - 595679544 T^{12} + 410245939974 T^{16} - 595679544 p^{4} T^{20} + 1133916 p^{8} T^{24} - 1672 p^{12} T^{28} + p^{16} T^{32}$$
31 $$( 1 - 56 T^{2} + 3492 T^{4} - 137064 T^{6} + 4924550 T^{8} - 137064 p^{2} T^{10} + 3492 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
37 $$( 1 - 512 T^{3} + 700 T^{4} + 9728 T^{5} + 131072 T^{6} - 437248 T^{7} - 3354522 T^{8} - 437248 p T^{9} + 131072 p^{2} T^{10} + 9728 p^{3} T^{11} + 700 p^{4} T^{12} - 512 p^{5} T^{13} + p^{8} T^{16} )^{2}$$
41 $$( 1 + 160 T^{2} + 12772 T^{4} + 745952 T^{6} + 34631942 T^{8} + 745952 p^{2} T^{10} + 12772 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
43 $$( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 6980 T^{4} - 80 T^{5} - 174720 T^{6} - 2568048 T^{7} - 25827098 T^{8} - 2568048 p T^{9} - 174720 p^{2} T^{10} - 80 p^{3} T^{11} + 6980 p^{4} T^{12} + 1200 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2}$$
47 $$( 1 + 216 T^{2} + 24572 T^{4} + 1856872 T^{6} + 101744902 T^{8} + 1856872 p^{2} T^{10} + 24572 p^{4} T^{12} + 216 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
53 $$1 + 2936 T^{4} + 12654300 T^{8} + 22427423688 T^{12} + 91065624180102 T^{16} + 22427423688 p^{4} T^{20} + 12654300 p^{8} T^{24} + 2936 p^{12} T^{28} + p^{16} T^{32}$$
59 $$1 - 40 p T^{4} + 15662172 T^{8} - 90024948744 T^{12} + 175698504275846 T^{16} - 90024948744 p^{4} T^{20} + 15662172 p^{8} T^{24} - 40 p^{13} T^{28} + p^{16} T^{32}$$
61 $$( 1 + 16 T + 128 T^{2} + 1392 T^{3} + 16892 T^{4} + 124816 T^{5} + 803712 T^{6} + 7357296 T^{7} + 67536550 T^{8} + 7357296 p T^{9} + 803712 p^{2} T^{10} + 124816 p^{3} T^{11} + 16892 p^{4} T^{12} + 1392 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2}$$
67 $$( 1 + 8 T + 32 T^{2} + 1240 T^{3} + 5540 T^{4} - 65000 T^{5} + 71520 T^{6} - 2339576 T^{7} - 86245658 T^{8} - 2339576 p T^{9} + 71520 p^{2} T^{10} - 65000 p^{3} T^{11} + 5540 p^{4} T^{12} + 1240 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2}$$
71 $$( 1 - 376 T^{2} + 67068 T^{4} - 7636296 T^{6} + 626574150 T^{8} - 7636296 p^{2} T^{10} + 67068 p^{4} T^{12} - 376 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
73 $$( 1 - 408 T^{2} + 80156 T^{4} - 9970856 T^{6} + 862104454 T^{8} - 9970856 p^{2} T^{10} + 80156 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
79 $$( 1 - 408 T^{2} + 84004 T^{4} - 11101576 T^{6} + 1033756678 T^{8} - 11101576 p^{2} T^{10} + 84004 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
83 $$1 - 5432 T^{4} + 103273180 T^{8} - 984945914120 T^{12} + 5200099763862790 T^{16} - 984945914120 p^{4} T^{20} + 103273180 p^{8} T^{24} - 5432 p^{12} T^{28} + p^{16} T^{32}$$
89 $$( 1 + 512 T^{2} + 123588 T^{4} + 18716160 T^{6} + 1970104134 T^{8} + 18716160 p^{2} T^{10} + 123588 p^{4} T^{12} + 512 p^{6} T^{14} + p^{8} T^{16} )^{2}$$
97 $$( 1 + 316 T^{2} - 256 T^{3} + 42310 T^{4} - 256 p T^{5} + 316 p^{2} T^{6} + p^{4} T^{8} )^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$