# Properties

 Degree 48 Conductor $2^{48} \cdot 3^{72}$ Sign $1$ Motivic weight 3 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 6·4-s + 72·5-s − 216·13-s − 432·20-s + 1.96e3·25-s − 576·29-s − 72·32-s + 1.24e3·37-s − 1.11e3·41-s − 1.88e3·49-s + 1.29e3·52-s − 264·61-s + 232·64-s − 1.55e4·65-s − 4.77e3·73-s + 588·97-s − 1.18e4·100-s + 792·101-s + 3.31e3·109-s + 8.85e3·113-s + 3.45e3·116-s + 7.86e3·121-s + 1.72e4·125-s + 127-s + 2.01e3·128-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 3/4·4-s + 6.43·5-s − 4.60·13-s − 4.82·20-s + 15.7·25-s − 3.68·29-s − 0.397·32-s + 5.54·37-s − 4.25·41-s − 5.49·49-s + 3.45·52-s − 0.554·61-s + 0.453·64-s − 29.6·65-s − 7.65·73-s + 0.615·97-s − 11.8·100-s + 0.780·101-s + 2.91·109-s + 7.37·113-s + 2.76·116-s + 5.90·121-s + 12.3·125-s + 0.000698·127-s + 1.39·128-s + 0.000666·131-s + 0.000623·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{72}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$48$$ $$N$$ = $$2^{48} \cdot 3^{72}$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(48,\ 2^{48} \cdot 3^{72} ,\ ( \ : [3/2]^{24} ),\ 1 )$$ $$L(2)$$ $$\approx$$ $$0.120776$$ $$L(\frac12)$$ $$\approx$$ $$0.120776$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 48. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 47.
$p$$F_p(T)$
bad2 $$1 + 3 p T^{2} + 9 p^{2} T^{4} + 9 p^{3} T^{5} - p^{4} T^{6} - 9 p^{7} T^{7} + 3 p^{7} T^{8} - 27 p^{8} T^{9} + 69 p^{8} T^{10} - 153 p^{9} T^{11} + 39 p^{10} T^{12} - 153 p^{12} T^{13} + 69 p^{14} T^{14} - 27 p^{17} T^{15} + 3 p^{19} T^{16} - 9 p^{22} T^{17} - p^{22} T^{18} + 9 p^{24} T^{19} + 9 p^{26} T^{20} + 3 p^{31} T^{22} + p^{36} T^{24}$$
3 $$1$$
good5 $$( 1 - 36 T + 192 p T^{2} - 19008 T^{3} + 317052 T^{4} - 4555332 T^{5} + 54439076 T^{6} - 538556076 T^{7} + 3798167244 T^{8} - 8329850784 T^{9} - 303382886376 T^{10} + 6944132468052 T^{11} - 90876215659866 T^{12} + 6944132468052 p^{3} T^{13} - 303382886376 p^{6} T^{14} - 8329850784 p^{9} T^{15} + 3798167244 p^{12} T^{16} - 538556076 p^{15} T^{17} + 54439076 p^{18} T^{18} - 4555332 p^{21} T^{19} + 317052 p^{24} T^{20} - 19008 p^{27} T^{21} + 192 p^{31} T^{22} - 36 p^{33} T^{23} + p^{36} T^{24} )^{2}$$
7 $$1 + 1884 T^{2} + 1601280 T^{4} + 793359976 T^{6} + 266231005416 T^{8} + 92097922073916 T^{10} + 49507838923948012 T^{12} + 25659663639756717636 T^{14} +$$$$92\!\cdots\!76$$$$T^{16} +$$$$24\!\cdots\!72$$$$T^{18} +$$$$79\!\cdots\!28$$$$T^{20} +$$$$39\!\cdots\!04$$$$T^{22} +$$$$16\!\cdots\!74$$$$T^{24} +$$$$39\!\cdots\!04$$$$p^{6} T^{26} +$$$$79\!\cdots\!28$$$$p^{12} T^{28} +$$$$24\!\cdots\!72$$$$p^{18} T^{30} +$$$$92\!\cdots\!76$$$$p^{24} T^{32} + 25659663639756717636 p^{30} T^{34} + 49507838923948012 p^{36} T^{36} + 92097922073916 p^{42} T^{38} + 266231005416 p^{48} T^{40} + 793359976 p^{54} T^{42} + 1601280 p^{60} T^{44} + 1884 p^{66} T^{46} + p^{72} T^{48}$$
11 $$1 - 7866 T^{2} + 29330823 T^{4} - 66879731030 T^{6} + 103347856157043 T^{8} - 11171858335691040 p T^{10} +$$$$16\!\cdots\!24$$$$T^{12} -$$$$34\!\cdots\!76$$$$T^{14} +$$$$73\!\cdots\!89$$$$T^{16} -$$$$12\!\cdots\!50$$$$T^{18} +$$$$13\!\cdots\!65$$$$p^{2} T^{20} -$$$$16\!\cdots\!18$$$$T^{22} +$$$$18\!\cdots\!50$$$$T^{24} -$$$$16\!\cdots\!18$$$$p^{6} T^{26} +$$$$13\!\cdots\!65$$$$p^{14} T^{28} -$$$$12\!\cdots\!50$$$$p^{18} T^{30} +$$$$73\!\cdots\!89$$$$p^{24} T^{32} -$$$$34\!\cdots\!76$$$$p^{30} T^{34} +$$$$16\!\cdots\!24$$$$p^{36} T^{36} - 11171858335691040 p^{43} T^{38} + 103347856157043 p^{48} T^{40} - 66879731030 p^{54} T^{42} + 29330823 p^{60} T^{44} - 7866 p^{66} T^{46} + p^{72} T^{48}$$
13 $$( 1 + 108 T - 1824 T^{2} - 366800 T^{3} + 16126356 T^{4} + 1167827796 T^{5} - 68920556876 T^{6} - 2858292413316 T^{7} + 176214474146628 T^{8} + 260085101275408 p T^{9} - 455486127290340984 T^{10} - 2747096533695618492 T^{11} +$$$$99\!\cdots\!82$$$$T^{12} - 2747096533695618492 p^{3} T^{13} - 455486127290340984 p^{6} T^{14} + 260085101275408 p^{10} T^{15} + 176214474146628 p^{12} T^{16} - 2858292413316 p^{15} T^{17} - 68920556876 p^{18} T^{18} + 1167827796 p^{21} T^{19} + 16126356 p^{24} T^{20} - 366800 p^{27} T^{21} - 1824 p^{30} T^{22} + 108 p^{33} T^{23} + p^{36} T^{24} )^{2}$$
17 $$( 1 - 28014 T^{2} + 413781087 T^{4} - 4166069827462 T^{6} + 32083747543301019 T^{8} -$$$$20\!\cdots\!44$$$$T^{10} +$$$$10\!\cdots\!26$$$$T^{12} -$$$$20\!\cdots\!44$$$$p^{6} T^{14} + 32083747543301019 p^{12} T^{16} - 4166069827462 p^{18} T^{18} + 413781087 p^{24} T^{20} - 28014 p^{30} T^{22} + p^{36} T^{24} )^{2}$$
19 $$( 1 - 56334 T^{2} + 1518719367 T^{4} - 26149020628126 T^{6} + 17067129428854377 p T^{8} -$$$$30\!\cdots\!56$$$$T^{10} +$$$$23\!\cdots\!90$$$$T^{12} -$$$$30\!\cdots\!56$$$$p^{6} T^{14} + 17067129428854377 p^{13} T^{16} - 26149020628126 p^{18} T^{18} + 1518719367 p^{24} T^{20} - 56334 p^{30} T^{22} + p^{36} T^{24} )^{2}$$
23 $$1 - 61716 T^{2} + 1863008016 T^{4} - 34234578277592 T^{6} + 390753097842006168 T^{8} -$$$$22\!\cdots\!20$$$$T^{10} -$$$$15\!\cdots\!68$$$$T^{12} +$$$$76\!\cdots\!88$$$$T^{14} +$$$$80\!\cdots\!60$$$$T^{16} -$$$$13\!\cdots\!96$$$$T^{18} -$$$$81\!\cdots\!24$$$$T^{20} +$$$$23\!\cdots\!12$$$$T^{22} -$$$$36\!\cdots\!06$$$$T^{24} +$$$$23\!\cdots\!12$$$$p^{6} T^{26} -$$$$81\!\cdots\!24$$$$p^{12} T^{28} -$$$$13\!\cdots\!96$$$$p^{18} T^{30} +$$$$80\!\cdots\!60$$$$p^{24} T^{32} +$$$$76\!\cdots\!88$$$$p^{30} T^{34} -$$$$15\!\cdots\!68$$$$p^{36} T^{36} -$$$$22\!\cdots\!20$$$$p^{42} T^{38} + 390753097842006168 p^{48} T^{40} - 34234578277592 p^{54} T^{42} + 1863008016 p^{60} T^{44} - 61716 p^{66} T^{46} + p^{72} T^{48}$$
29 $$( 1 + 288 T + 104472 T^{2} + 22125312 T^{3} + 4433744652 T^{4} + 23652994320 p T^{5} + 94177853862644 T^{6} + 7876469041637760 T^{7} + 363718818603559788 T^{8} -$$$$15\!\cdots\!28$$$$T^{9} -$$$$40\!\cdots\!92$$$$T^{10} -$$$$91\!\cdots\!72$$$$T^{11} -$$$$14\!\cdots\!30$$$$T^{12} -$$$$91\!\cdots\!72$$$$p^{3} T^{13} -$$$$40\!\cdots\!92$$$$p^{6} T^{14} -$$$$15\!\cdots\!28$$$$p^{9} T^{15} + 363718818603559788 p^{12} T^{16} + 7876469041637760 p^{15} T^{17} + 94177853862644 p^{18} T^{18} + 23652994320 p^{22} T^{19} + 4433744652 p^{24} T^{20} + 22125312 p^{27} T^{21} + 104472 p^{30} T^{22} + 288 p^{33} T^{23} + p^{36} T^{24} )^{2}$$
31 $$1 + 181188 T^{2} + 17878846512 T^{4} + 1277349503435608 T^{6} + 72953942761200147960 T^{8} +$$$$35\!\cdots\!60$$$$T^{10} +$$$$14\!\cdots\!96$$$$T^{12} +$$$$53\!\cdots\!80$$$$T^{14} +$$$$18\!\cdots\!24$$$$T^{16} +$$$$55\!\cdots\!20$$$$T^{18} +$$$$16\!\cdots\!08$$$$T^{20} +$$$$46\!\cdots\!48$$$$T^{22} +$$$$13\!\cdots\!50$$$$T^{24} +$$$$46\!\cdots\!48$$$$p^{6} T^{26} +$$$$16\!\cdots\!08$$$$p^{12} T^{28} +$$$$55\!\cdots\!20$$$$p^{18} T^{30} +$$$$18\!\cdots\!24$$$$p^{24} T^{32} +$$$$53\!\cdots\!80$$$$p^{30} T^{34} +$$$$14\!\cdots\!96$$$$p^{36} T^{36} +$$$$35\!\cdots\!60$$$$p^{42} T^{38} + 72953942761200147960 p^{48} T^{40} + 1277349503435608 p^{54} T^{42} + 17878846512 p^{60} T^{44} + 181188 p^{66} T^{46} + p^{72} T^{48}$$
37 $$( 1 - 312 T + 207606 T^{2} - 44489464 T^{3} + 19423334967 T^{4} - 89960027952 p T^{5} + 1172110607756724 T^{6} - 89960027952 p^{4} T^{7} + 19423334967 p^{6} T^{8} - 44489464 p^{9} T^{9} + 207606 p^{12} T^{10} - 312 p^{15} T^{11} + p^{18} T^{12} )^{4}$$
41 $$( 1 + 558 T + 388071 T^{2} + 158629914 T^{3} + 67288237971 T^{4} + 25108476714216 T^{5} + 8739573904551224 T^{6} + 2996686992495855852 T^{7} +$$$$92\!\cdots\!29$$$$T^{8} +$$$$28\!\cdots\!26$$$$T^{9} +$$$$82\!\cdots\!61$$$$T^{10} +$$$$22\!\cdots\!30$$$$T^{11} +$$$$62\!\cdots\!18$$$$T^{12} +$$$$22\!\cdots\!30$$$$p^{3} T^{13} +$$$$82\!\cdots\!61$$$$p^{6} T^{14} +$$$$28\!\cdots\!26$$$$p^{9} T^{15} +$$$$92\!\cdots\!29$$$$p^{12} T^{16} + 2996686992495855852 p^{15} T^{17} + 8739573904551224 p^{18} T^{18} + 25108476714216 p^{21} T^{19} + 67288237971 p^{24} T^{20} + 158629914 p^{27} T^{21} + 388071 p^{30} T^{22} + 558 p^{33} T^{23} + p^{36} T^{24} )^{2}$$
43 $$1 + 5.29e5T^{2} + 1.36e11T^{4} + 2.32e16T^{6} + 3.06e21T^{8} + 3.44e26T^{10} + 3.49e31T^{12} + 3.29e36T^{14} + 2.91e41T^{16} + 2.44e46T^{18} + 1.97e51T^{20} + 1.55e56T^{22}+O(T^{24})$$
47 $$1 - 9.57e5T^{2} + 4.74e11T^{4} - 1.63e17T^{6} + 4.40e22T^{8} - 9.92e27T^{10} + 1.93e33T^{12} - 3.32e38T^{14} + 5.14e43T^{16} - 7.22e48T^{18} + 9.25e53T^{20} - 1.08e59T^{22}+O(T^{24})$$
53 $$1 - 2.35e6T^{2} + 2.79e12T^{4} - 2.19e18T^{6} + 1.28e24T^{8} - 5.95e29T^{10} + 2.26e35T^{12} - 7.22e40T^{14} + 1.97e46T^{16} - 4.65e51T^{18} + 9.59e56T^{20} - 1.73e62T^{22}+O(T^{23})$$
59 $$1 - 1.70e6T^{2} + 1.50e12T^{4} - 9.10e17T^{6} + 4.19e23T^{8} - 1.57e29T^{10} + 4.99e34T^{12} - 1.39e40T^{14} + 3.54e45T^{16} - 8.40e50T^{18} + 1.89e56T^{20} - 4.12e61T^{22}+O(T^{23})$$
61 $$1 + 264T - 1.81e6T^{2} - 8.21e8T^{3} + 1.66e12T^{4} + 1.04e15T^{5} - 9.42e17T^{6} - 8.10e20T^{7} + 3.42e23T^{8} + 4.37e26T^{9} - 6.43e28T^{10} - 1.71e32T^{11} - 7.09e33T^{12} + 4.79e37T^{13} + 9.30e39T^{14} - 8.56e42T^{15} - 2.93e45T^{16} + 3.71e47T^{17} + 2.35e50T^{18} + 3.07e53T^{19} + 1.94e56T^{20} - 1.00e59T^{21} - 1.06e62T^{22}+O(T^{23})$$
67 $$1 + 1.78e6T^{2} + 1.72e12T^{4} + 1.20e18T^{6} + 6.67e23T^{8} + 3.09e29T^{10} + 1.24e35T^{12} + 4.46e40T^{14} + 1.46e46T^{16} + 4.54e51T^{18} + 1.36e57T^{20}+O(T^{22})$$
71 $$1 + 6.53e6T^{2} + 2.07e13T^{4} + 4.24e19T^{6} + 6.31e25T^{8} + 7.30e31T^{10} + 6.82e37T^{12} + 5.30e43T^{14} + 3.50e49T^{16} + 1.99e55T^{18} + 9.86e60T^{20}+O(T^{22})$$
73 $$1 + 4.77e3T + 1.67e7T^{2} + 4.42e10T^{3} + 9.89e13T^{4} + 1.91e17T^{5} + 3.31e20T^{6} + 5.19e23T^{7} + 7.48e26T^{8} + 9.99e29T^{9} + 1.24e33T^{10} + 1.45e36T^{11} + 1.60e39T^{12} + 1.67e42T^{13} + 1.65e45T^{14} + 1.55e48T^{15} + 1.39e51T^{16} + 1.19e54T^{17} + 9.79e56T^{18} + 7.67e59T^{19} + 5.76e62T^{20} + 4.14e65T^{21}+O(T^{22})$$
79 $$1 + 3.51e6T^{2} + 6.02e12T^{4} + 7.03e18T^{6} + 6.54e24T^{8} + 5.25e30T^{10} + 3.74e36T^{12} + 2.41e42T^{14} + 1.43e48T^{16} + 8.04e53T^{18} + 4.29e59T^{20}+O(T^{21})$$
83 $$1 - 4.81e6T^{2} + 1.17e13T^{4} - 1.99e19T^{6} + 2.66e25T^{8} - 2.99e31T^{10} + 2.92e37T^{12} - 2.55e43T^{14} + 2.02e49T^{16} - 1.47e55T^{18} + 9.97e60T^{20}+O(T^{21})$$
89 $$1 - 7.60e6T^{2} + 2.88e13T^{4} - 7.28e19T^{6} + 1.39e26T^{8} - 2.15e32T^{10} + 2.81e38T^{12} - 3.22e44T^{14} + 3.29e50T^{16} - 3.07e56T^{18} + 2.64e62T^{20}+O(T^{21})$$
97 $$1 - 588T - 2.88e6T^{2} + 2.58e9T^{3} + 1.65e12T^{4} - 3.08e15T^{5} + 3.60e18T^{6} - 1.76e21T^{7} - 4.56e24T^{8} + 7.69e27T^{9} - 2.45e30T^{10} - 5.75e33T^{11} + 8.10e36T^{12} - 3.13e39T^{13} - 4.70e42T^{14} + 9.31e45T^{15} - 3.86e48T^{16} - 6.29e51T^{17} + 9.54e54T^{18} - 3.03e57T^{19}+O(T^{20})$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}