Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

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 \( 1 - 588T - 2.88e6T^{2} + 2.58e9T^{3} + 1.65e12T^{4} - 3.08e15T^{5} + 3.60e18T^{6} - 1.76e21T^{7} - 4.56e24T^{8} + 7.69e27T^{9} - 2.45e30T^{10} - 5.75e33T^{11} + 8.10e36T^{12} - 3.13e39T^{13} - 4.70e42T^{14} + 9.31e45T^{15} - 3.86e48T^{16} - 6.29e51T^{17} + 9.54e54T^{18} - 3.03e57T^{19}+O(T^{20}) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.60627940952550749518246007842, −2.59684262565933651842884293654, −2.56171081373911547111668642267, −2.28136591066239083552444470907, −2.21093546168783796977701253188, −2.10342002309472312389137088771, −2.02513945855890556117314590143, −1.96769807473116773700288705308, −1.84721454707034965672713924622, −1.78071700580366098120759164201, −1.76078645700658011303519445294, −1.75994662149358056112491320194, −1.73697533590655057560116691915, −1.61137019726309867837660849835, −1.59408566353163872504724116725, −1.54500569622278170409350453987, −1.25741966306243671265690169112, −0.973865684584080583019039752334, −0.958069253197356653259170408975, −0.69317691044304699924153782926, −0.56196377356538095600270773961, −0.44454764896787524085977176106, −0.37068278149258161712504420677, −0.14902179528642485069975522841, −0.01913878939440817432839098584, 0.01913878939440817432839098584, 0.14902179528642485069975522841, 0.37068278149258161712504420677, 0.44454764896787524085977176106, 0.56196377356538095600270773961, 0.69317691044304699924153782926, 0.958069253197356653259170408975, 0.973865684584080583019039752334, 1.25741966306243671265690169112, 1.54500569622278170409350453987, 1.59408566353163872504724116725, 1.61137019726309867837660849835, 1.73697533590655057560116691915, 1.75994662149358056112491320194, 1.76078645700658011303519445294, 1.78071700580366098120759164201, 1.84721454707034965672713924622, 1.96769807473116773700288705308, 2.02513945855890556117314590143, 2.10342002309472312389137088771, 2.21093546168783796977701253188, 2.28136591066239083552444470907, 2.56171081373911547111668642267, 2.59684262565933651842884293654, 2.60627940952550749518246007842

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.