# Properties

 Degree 48 Conductor $2^{24} \cdot 23^{24} \cdot 131^{24}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 24

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 24·2-s − 3-s + 300·4-s − 5-s + 24·6-s − 7·7-s − 2.60e3·8-s − 22·9-s + 24·10-s − 4·11-s − 300·12-s − 5·13-s + 168·14-s + 15-s + 1.75e4·16-s + 5·17-s + 528·18-s − 20·19-s − 300·20-s + 7·21-s + 96·22-s − 24·23-s + 2.60e3·24-s − 59·25-s + 120·26-s + 23·27-s − 2.10e3·28-s + ⋯
 L(s)  = 1 − 16.9·2-s − 0.577·3-s + 150·4-s − 0.447·5-s + 9.79·6-s − 2.64·7-s − 919.·8-s − 7.33·9-s + 7.58·10-s − 1.20·11-s − 86.6·12-s − 1.38·13-s + 44.8·14-s + 0.258·15-s + 4.38e3·16-s + 1.21·17-s + 124.·18-s − 4.58·19-s − 67.0·20-s + 1.52·21-s + 20.4·22-s − 5.00·23-s + 530.·24-s − 11.7·25-s + 23.5·26-s + 4.42·27-s − 396.·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{24} \cdot 131^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{24} \cdot 131^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{24} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$48$$ $$N$$ = $$2^{24} \cdot 23^{24} \cdot 131^{24}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{6026} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 24 Selberg data = $(48,\ 2^{24} \cdot 23^{24} \cdot 131^{24} ,\ ( \ : [1/2]^{24} ),\ 1 )$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;23,\;131\}$, $$F_p$$ is a polynomial of degree 48. If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 47.
$p$$F_p$
bad2 $$( 1 + T )^{24}$$
23 $$( 1 + T )^{24}$$
131 $$( 1 + T )^{24}$$
good3 $$1 + T + 23 T^{2} + 22 T^{3} + 278 T^{4} + 89 p T^{5} + 2351 T^{6} + 2291 T^{7} + 15635 T^{8} + 15415 T^{9} + 87164 T^{10} + 85943 T^{11} + 423568 T^{12} + 137258 p T^{13} + 1841495 T^{14} + 1743587 T^{15} + 2432230 p T^{16} + 6682796 T^{17} + 26731135 T^{18} + 23657704 T^{19} + 91595612 T^{20} + 78575828 T^{21} + 10974824 p^{3} T^{22} + 247406810 T^{23} + 33736304 p^{3} T^{24} + 247406810 p T^{25} + 10974824 p^{5} T^{26} + 78575828 p^{3} T^{27} + 91595612 p^{4} T^{28} + 23657704 p^{5} T^{29} + 26731135 p^{6} T^{30} + 6682796 p^{7} T^{31} + 2432230 p^{9} T^{32} + 1743587 p^{9} T^{33} + 1841495 p^{10} T^{34} + 137258 p^{12} T^{35} + 423568 p^{12} T^{36} + 85943 p^{13} T^{37} + 87164 p^{14} T^{38} + 15415 p^{15} T^{39} + 15635 p^{16} T^{40} + 2291 p^{17} T^{41} + 2351 p^{18} T^{42} + 89 p^{20} T^{43} + 278 p^{20} T^{44} + 22 p^{21} T^{45} + 23 p^{22} T^{46} + p^{23} T^{47} + p^{24} T^{48}$$
5 $$1 + T + 12 p T^{2} + 56 T^{3} + 1811 T^{4} + 1574 T^{5} + 36653 T^{6} + 5951 p T^{7} + 22393 p^{2} T^{8} + 426849 T^{9} + 6887103 T^{10} + 4963083 T^{11} + 71118502 T^{12} + 48715591 T^{13} + 634054521 T^{14} + 414644006 T^{15} + 995766044 p T^{16} + 3116904283 T^{17} + 6985050489 p T^{18} + 20955309394 T^{19} + 221017903249 T^{20} + 127083725544 T^{21} + 1269873149626 T^{22} + 698903851616 T^{23} + 6647921580832 T^{24} + 698903851616 p T^{25} + 1269873149626 p^{2} T^{26} + 127083725544 p^{3} T^{27} + 221017903249 p^{4} T^{28} + 20955309394 p^{5} T^{29} + 6985050489 p^{7} T^{30} + 3116904283 p^{7} T^{31} + 995766044 p^{9} T^{32} + 414644006 p^{9} T^{33} + 634054521 p^{10} T^{34} + 48715591 p^{11} T^{35} + 71118502 p^{12} T^{36} + 4963083 p^{13} T^{37} + 6887103 p^{14} T^{38} + 426849 p^{15} T^{39} + 22393 p^{18} T^{40} + 5951 p^{18} T^{41} + 36653 p^{18} T^{42} + 1574 p^{19} T^{43} + 1811 p^{20} T^{44} + 56 p^{21} T^{45} + 12 p^{23} T^{46} + p^{23} T^{47} + p^{24} T^{48}$$
7 $$1 + p T + 115 T^{2} + 654 T^{3} + 6179 T^{4} + 30143 T^{5} + 212675 T^{6} + 131525 p T^{7} + 765469 p T^{8} + 21048313 T^{9} + 106303368 T^{10} + 384661490 T^{11} + 1736800133 T^{12} + 5849275522 T^{13} + 24064197488 T^{14} + 75964562171 T^{15} + 288428824769 T^{16} + 122494979655 p T^{17} + 433087319449 p T^{18} + 1216126718172 p T^{19} + 28203518640940 T^{20} + 74923189819937 T^{21} + 233636460397121 T^{22} + 587400813938335 T^{23} + 1729256645857206 T^{24} + 587400813938335 p T^{25} + 233636460397121 p^{2} T^{26} + 74923189819937 p^{3} T^{27} + 28203518640940 p^{4} T^{28} + 1216126718172 p^{6} T^{29} + 433087319449 p^{7} T^{30} + 122494979655 p^{8} T^{31} + 288428824769 p^{8} T^{32} + 75964562171 p^{9} T^{33} + 24064197488 p^{10} T^{34} + 5849275522 p^{11} T^{35} + 1736800133 p^{12} T^{36} + 384661490 p^{13} T^{37} + 106303368 p^{14} T^{38} + 21048313 p^{15} T^{39} + 765469 p^{17} T^{40} + 131525 p^{18} T^{41} + 212675 p^{18} T^{42} + 30143 p^{19} T^{43} + 6179 p^{20} T^{44} + 654 p^{21} T^{45} + 115 p^{22} T^{46} + p^{24} T^{47} + p^{24} T^{48}$$
11 $$1 + 4 T + 155 T^{2} + 591 T^{3} + 1088 p T^{4} + 43364 T^{5} + 613794 T^{6} + 2112323 T^{7} + 23525977 T^{8} + 76947665 T^{9} + 718650079 T^{10} + 2236057411 T^{11} + 18208068991 T^{12} + 53936906065 T^{13} + 392922046262 T^{14} + 1108574645770 T^{15} + 7355045342754 T^{16} + 19763273825156 T^{17} + 120962141157883 T^{18} + 309375542454218 T^{19} + 1763324830118919 T^{20} + 4287603519008267 T^{21} + 22916557830341915 T^{22} + 52874780989173916 T^{23} + 266396842144263196 T^{24} + 52874780989173916 p T^{25} + 22916557830341915 p^{2} T^{26} + 4287603519008267 p^{3} T^{27} + 1763324830118919 p^{4} T^{28} + 309375542454218 p^{5} T^{29} + 120962141157883 p^{6} T^{30} + 19763273825156 p^{7} T^{31} + 7355045342754 p^{8} T^{32} + 1108574645770 p^{9} T^{33} + 392922046262 p^{10} T^{34} + 53936906065 p^{11} T^{35} + 18208068991 p^{12} T^{36} + 2236057411 p^{13} T^{37} + 718650079 p^{14} T^{38} + 76947665 p^{15} T^{39} + 23525977 p^{16} T^{40} + 2112323 p^{17} T^{41} + 613794 p^{18} T^{42} + 43364 p^{19} T^{43} + 1088 p^{21} T^{44} + 591 p^{21} T^{45} + 155 p^{22} T^{46} + 4 p^{23} T^{47} + p^{24} T^{48}$$
13 $$1 + 5 T + 186 T^{2} + 799 T^{3} + 16663 T^{4} + 61391 T^{5} + 961982 T^{6} + 3017519 T^{7} + 40465361 T^{8} + 106683203 T^{9} + 1331972946 T^{10} + 2898930780 T^{11} + 36038543925 T^{12} + 63395273347 T^{13} + 831672676154 T^{14} + 1159177911823 T^{15} + 16833375537119 T^{16} + 18389278546836 T^{17} + 304729154161452 T^{18} + 20240817153828 p T^{19} + 384044744882372 p T^{20} + 271920258124303 p T^{21} + 74462666225223376 T^{22} + 3554504409883366 p T^{23} + 1013216158003164222 T^{24} + 3554504409883366 p^{2} T^{25} + 74462666225223376 p^{2} T^{26} + 271920258124303 p^{4} T^{27} + 384044744882372 p^{5} T^{28} + 20240817153828 p^{6} T^{29} + 304729154161452 p^{6} T^{30} + 18389278546836 p^{7} T^{31} + 16833375537119 p^{8} T^{32} + 1159177911823 p^{9} T^{33} + 831672676154 p^{10} T^{34} + 63395273347 p^{11} T^{35} + 36038543925 p^{12} T^{36} + 2898930780 p^{13} T^{37} + 1331972946 p^{14} T^{38} + 106683203 p^{15} T^{39} + 40465361 p^{16} T^{40} + 3017519 p^{17} T^{41} + 961982 p^{18} T^{42} + 61391 p^{19} T^{43} + 16663 p^{20} T^{44} + 799 p^{21} T^{45} + 186 p^{22} T^{46} + 5 p^{23} T^{47} + p^{24} T^{48}$$
17 $$1 - 5 T + 258 T^{2} - 1136 T^{3} + 32672 T^{4} - 128556 T^{5} + 2721244 T^{6} - 9677905 T^{7} + 168168853 T^{8} - 545576109 T^{9} + 8234022053 T^{10} - 24555427480 T^{11} + 332654650478 T^{12} - 53990208822 p T^{13} + 11390807457988 T^{14} - 29236885014128 T^{15} + 336738737930143 T^{16} - 807717050900818 T^{17} + 8704905539787903 T^{18} - 19585967146681248 T^{19} + 198509440398190350 T^{20} - 420199961637866862 T^{21} + 4016373502858132250 T^{22} - 8015424487889574153 T^{23} + 72332282528348506366 T^{24} - 8015424487889574153 p T^{25} + 4016373502858132250 p^{2} T^{26} - 420199961637866862 p^{3} T^{27} + 198509440398190350 p^{4} T^{28} - 19585967146681248 p^{5} T^{29} + 8704905539787903 p^{6} T^{30} - 807717050900818 p^{7} T^{31} + 336738737930143 p^{8} T^{32} - 29236885014128 p^{9} T^{33} + 11390807457988 p^{10} T^{34} - 53990208822 p^{12} T^{35} + 332654650478 p^{12} T^{36} - 24555427480 p^{13} T^{37} + 8234022053 p^{14} T^{38} - 545576109 p^{15} T^{39} + 168168853 p^{16} T^{40} - 9677905 p^{17} T^{41} + 2721244 p^{18} T^{42} - 128556 p^{19} T^{43} + 32672 p^{20} T^{44} - 1136 p^{21} T^{45} + 258 p^{22} T^{46} - 5 p^{23} T^{47} + p^{24} T^{48}$$
19 $$1 + 20 T + 490 T^{2} + 6982 T^{3} + 103225 T^{4} + 1174047 T^{5} + 13247794 T^{6} + 127359739 T^{7} + 1195396998 T^{8} + 10045181382 T^{9} + 81885257805 T^{10} + 614491708914 T^{11} + 4461565790469 T^{12} + 30334336328633 T^{13} + 199347040353749 T^{14} + 65282865554601 p T^{15} + 7457682616472469 T^{16} + 42766679864052973 T^{17} + 236988134516328265 T^{18} + 1258671642322768232 T^{19} + 6460592866148915406 T^{20} + 31881961281004432566 T^{21} +$$$$15\!\cdots\!77$$$$T^{22} +$$$$69\!\cdots\!35$$$$T^{23} +$$$$31\!\cdots\!12$$$$T^{24} +$$$$69\!\cdots\!35$$$$p T^{25} +$$$$15\!\cdots\!77$$$$p^{2} T^{26} + 31881961281004432566 p^{3} T^{27} + 6460592866148915406 p^{4} T^{28} + 1258671642322768232 p^{5} T^{29} + 236988134516328265 p^{6} T^{30} + 42766679864052973 p^{7} T^{31} + 7457682616472469 p^{8} T^{32} + 65282865554601 p^{10} T^{33} + 199347040353749 p^{10} T^{34} + 30334336328633 p^{11} T^{35} + 4461565790469 p^{12} T^{36} + 614491708914 p^{13} T^{37} + 81885257805 p^{14} T^{38} + 10045181382 p^{15} T^{39} + 1195396998 p^{16} T^{40} + 127359739 p^{17} T^{41} + 13247794 p^{18} T^{42} + 1174047 p^{19} T^{43} + 103225 p^{20} T^{44} + 6982 p^{21} T^{45} + 490 p^{22} T^{46} + 20 p^{23} T^{47} + p^{24} T^{48}$$
29 $$1 + 6 T + 394 T^{2} + 2401 T^{3} + 78255 T^{4} + 473484 T^{5} + 10370834 T^{6} + 61349902 T^{7} + 1025822189 T^{8} + 5870081808 T^{9} + 80444264837 T^{10} + 441994657243 T^{11} + 5195459243599 T^{12} + 27272499674146 T^{13} + 9789171831058 p T^{14} + 1419444142532171 T^{15} + 13399823642971622 T^{16} + 63740787680143664 T^{17} + 555877546349625711 T^{18} + 2516535791027049955 T^{19} + 20565355651828337046 T^{20} + 88739381401340927489 T^{21} +$$$$68\!\cdots\!42$$$$T^{22} +$$$$28\!\cdots\!03$$$$T^{23} +$$$$20\!\cdots\!04$$$$T^{24} +$$$$28\!\cdots\!03$$$$p T^{25} +$$$$68\!\cdots\!42$$$$p^{2} T^{26} + 88739381401340927489 p^{3} T^{27} + 20565355651828337046 p^{4} T^{28} + 2516535791027049955 p^{5} T^{29} + 555877546349625711 p^{6} T^{30} + 63740787680143664 p^{7} T^{31} + 13399823642971622 p^{8} T^{32} + 1419444142532171 p^{9} T^{33} + 9789171831058 p^{11} T^{34} + 27272499674146 p^{11} T^{35} + 5195459243599 p^{12} T^{36} + 441994657243 p^{13} T^{37} + 80444264837 p^{14} T^{38} + 5870081808 p^{15} T^{39} + 1025822189 p^{16} T^{40} + 61349902 p^{17} T^{41} + 10370834 p^{18} T^{42} + 473484 p^{19} T^{43} + 78255 p^{20} T^{44} + 2401 p^{21} T^{45} + 394 p^{22} T^{46} + 6 p^{23} T^{47} + p^{24} T^{48}$$
31 $$1 + 23 T + 643 T^{2} + 10401 T^{3} + 176918 T^{4} + 2278420 T^{5} + 29765468 T^{6} + 324564780 T^{7} + 3549238943 T^{8} + 33972212976 T^{9} + 324949466670 T^{10} + 2794982476024 T^{11} + 24013017898963 T^{12} + 188661495463650 T^{13} + 1481901837097116 T^{14} + 10763442621718523 T^{15} + 78279612137243727 T^{16} + 530455276568398642 T^{17} + 3605561464705006268 T^{18} + 22955262308180130849 T^{19} +$$$$14\!\cdots\!57$$$$T^{20} +$$$$88\!\cdots\!48$$$$T^{21} +$$$$53\!\cdots\!71$$$$T^{22} +$$$$30\!\cdots\!06$$$$T^{23} +$$$$17\!\cdots\!66$$$$T^{24} +$$$$30\!\cdots\!06$$$$p T^{25} +$$$$53\!\cdots\!71$$$$p^{2} T^{26} +$$$$88\!\cdots\!48$$$$p^{3} T^{27} +$$$$14\!\cdots\!57$$$$p^{4} T^{28} + 22955262308180130849 p^{5} T^{29} + 3605561464705006268 p^{6} T^{30} + 530455276568398642 p^{7} T^{31} + 78279612137243727 p^{8} T^{32} + 10763442621718523 p^{9} T^{33} + 1481901837097116 p^{10} T^{34} + 188661495463650 p^{11} T^{35} + 24013017898963 p^{12} T^{36} + 2794982476024 p^{13} T^{37} + 324949466670 p^{14} T^{38} + 33972212976 p^{15} T^{39} + 3549238943 p^{16} T^{40} + 324564780 p^{17} T^{41} + 29765468 p^{18} T^{42} + 2278420 p^{19} T^{43} + 176918 p^{20} T^{44} + 10401 p^{21} T^{45} + 643 p^{22} T^{46} + 23 p^{23} T^{47} + p^{24} T^{48}$$
37 $$1 + 6 T + 398 T^{2} + 1603 T^{3} + 77548 T^{4} + 198910 T^{5} + 10195071 T^{6} + 14082672 T^{7} + 1029154063 T^{8} + 404413724 T^{9} + 2306206057 p T^{10} - 35804042897 T^{11} + 6054141308636 T^{12} - 6387470112543 T^{13} + 377506420033807 T^{14} - 571882004031089 T^{15} + 21054445739802616 T^{16} - 38166717975631419 T^{17} + 1062402096903571225 T^{18} - 2097102853144716460 T^{19} + 48855976862424087086 T^{20} - 99128051432581728589 T^{21} +$$$$20\!\cdots\!18$$$$T^{22} -$$$$41\!\cdots\!22$$$$T^{23} +$$$$79\!\cdots\!12$$$$T^{24} -$$$$41\!\cdots\!22$$$$p T^{25} +$$$$20\!\cdots\!18$$$$p^{2} T^{26} - 99128051432581728589 p^{3} T^{27} + 48855976862424087086 p^{4} T^{28} - 2097102853144716460 p^{5} T^{29} + 1062402096903571225 p^{6} T^{30} - 38166717975631419 p^{7} T^{31} + 21054445739802616 p^{8} T^{32} - 571882004031089 p^{9} T^{33} + 377506420033807 p^{10} T^{34} - 6387470112543 p^{11} T^{35} + 6054141308636 p^{12} T^{36} - 35804042897 p^{13} T^{37} + 2306206057 p^{15} T^{38} + 404413724 p^{15} T^{39} + 1029154063 p^{16} T^{40} + 14082672 p^{17} T^{41} + 10195071 p^{18} T^{42} + 198910 p^{19} T^{43} + 77548 p^{20} T^{44} + 1603 p^{21} T^{45} + 398 p^{22} T^{46} + 6 p^{23} T^{47} + p^{24} T^{48}$$
41 $$1 + T + 626 T^{2} + 422 T^{3} + 192868 T^{4} + 63647 T^{5} + 38964947 T^{6} - 1167337 T^{7} + 5803460113 T^{8} - 2329898440 T^{9} + 679414167599 T^{10} - 528041584155 T^{11} + 65115057270407 T^{12} - 74594350596711 T^{13} + 5255907270151603 T^{14} - 7829884722946315 T^{15} + 364925950060619192 T^{16} - 652255549325020286 T^{17} + 22153916198313819438 T^{18} - 44583047864980750360 T^{19} +$$$$11\!\cdots\!74$$$$T^{20} -$$$$25\!\cdots\!37$$$$T^{21} +$$$$57\!\cdots\!67$$$$T^{22} -$$$$12\!\cdots\!21$$$$T^{23} +$$$$24\!\cdots\!50$$$$T^{24} -$$$$12\!\cdots\!21$$$$p T^{25} +$$$$57\!\cdots\!67$$$$p^{2} T^{26} -$$$$25\!\cdots\!37$$$$p^{3} T^{27} +$$$$11\!\cdots\!74$$$$p^{4} T^{28} - 44583047864980750360 p^{5} T^{29} + 22153916198313819438 p^{6} T^{30} - 652255549325020286 p^{7} T^{31} + 364925950060619192 p^{8} T^{32} - 7829884722946315 p^{9} T^{33} + 5255907270151603 p^{10} T^{34} - 74594350596711 p^{11} T^{35} + 65115057270407 p^{12} T^{36} - 528041584155 p^{13} T^{37} + 679414167599 p^{14} T^{38} - 2329898440 p^{15} T^{39} + 5803460113 p^{16} T^{40} - 1167337 p^{17} T^{41} + 38964947 p^{18} T^{42} + 63647 p^{19} T^{43} + 192868 p^{20} T^{44} + 422 p^{21} T^{45} + 626 p^{22} T^{46} + p^{23} T^{47} + p^{24} T^{48}$$
43 $$1 + 44 T + 1414 T^{2} + 32800 T^{3} + 643815 T^{4} + 10725717 T^{5} + 159834834 T^{6} + 2140249339 T^{7} + 26417364246 T^{8} + 301825125230 T^{9} + 3240142174920 T^{10} + 32798335642501 T^{11} + 316085203650534 T^{12} + 2908619655955011 T^{13} + 25720639538437955 T^{14} + 219079090781822373 T^{15} + 1804898984308750392 T^{16} + 14408097774870126655 T^{17} +$$$$11\!\cdots\!87$$$$T^{18} +$$$$84\!\cdots\!71$$$$T^{19} +$$$$61\!\cdots\!97$$$$T^{20} +$$$$44\!\cdots\!08$$$$T^{21} +$$$$30\!\cdots\!34$$$$T^{22} +$$$$21\!\cdots\!77$$$$T^{23} +$$$$13\!\cdots\!94$$$$T^{24} +$$$$21\!\cdots\!77$$$$p T^{25} +$$$$30\!\cdots\!34$$$$p^{2} T^{26} +$$$$44\!\cdots\!08$$$$p^{3} T^{27} +$$$$61\!\cdots\!97$$$$p^{4} T^{28} +$$$$84\!\cdots\!71$$$$p^{5} T^{29} +$$$$11\!\cdots\!87$$$$p^{6} T^{30} + 14408097774870126655 p^{7} T^{31} + 1804898984308750392 p^{8} T^{32} + 219079090781822373 p^{9} T^{33} + 25720639538437955 p^{10} T^{34} + 2908619655955011 p^{11} T^{35} + 316085203650534 p^{12} T^{36} + 32798335642501 p^{13} T^{37} + 3240142174920 p^{14} T^{38} + 301825125230 p^{15} T^{39} + 26417364246 p^{16} T^{40} + 2140249339 p^{17} T^{41} + 159834834 p^{18} T^{42} + 10725717 p^{19} T^{43} + 643815 p^{20} T^{44} + 32800 p^{21} T^{45} + 1414 p^{22} T^{46} + 44 p^{23} T^{47} + p^{24} T^{48}$$
47 $$1 - 32 T + 1107 T^{2} - 24151 T^{3} + 510014 T^{4} - 8673018 T^{5} + 140095921 T^{6} - 1980414138 T^{7} + 26552391390 T^{8} - 323766718885 T^{9} + 3754564997383 T^{10} - 40439898463042 T^{11} + 415813264171188 T^{12} - 4023390944012457 T^{13} + 37319628137102427 T^{14} - 328803417785187141 T^{15} + 2790415700998407596 T^{16} - 22672434068390510703 T^{17} +$$$$17\!\cdots\!41$$$$T^{18} -$$$$13\!\cdots\!88$$$$T^{19} +$$$$10\!\cdots\!36$$$$T^{20} -$$$$72\!\cdots\!86$$$$T^{21} +$$$$51\!\cdots\!73$$$$T^{22} -$$$$36\!\cdots\!77$$$$T^{23} +$$$$24\!\cdots\!22$$$$T^{24} -$$$$36\!\cdots\!77$$$$p T^{25} +$$$$51\!\cdots\!73$$$$p^{2} T^{26} -$$$$72\!\cdots\!86$$$$p^{3} T^{27} +$$$$10\!\cdots\!36$$$$p^{4} T^{28} -$$$$13\!\cdots\!88$$$$p^{5} T^{29} +$$$$17\!\cdots\!41$$$$p^{6} T^{30} - 22672434068390510703 p^{7} T^{31} + 2790415700998407596 p^{8} T^{32} - 328803417785187141 p^{9} T^{33} + 37319628137102427 p^{10} T^{34} - 4023390944012457 p^{11} T^{35} + 415813264171188 p^{12} T^{36} - 40439898463042 p^{13} T^{37} + 3754564997383 p^{14} T^{38} - 323766718885 p^{15} T^{39} + 26552391390 p^{16} T^{40} - 1980414138 p^{17} T^{41} + 140095921 p^{18} T^{42} - 8673018 p^{19} T^{43} + 510014 p^{20} T^{44} - 24151 p^{21} T^{45} + 1107 p^{22} T^{46} - 32 p^{23} T^{47} + p^{24} T^{48}$$
53 $$1 - 21 T + 931 T^{2} - 15182 T^{3} + 382407 T^{4} - 5106816 T^{5} + 94476651 T^{6} - 1064058826 T^{7} + 15962662267 T^{8} - 154151339780 T^{9} + 1981010275084 T^{10} - 16555853665676 T^{11} + 189908257260168 T^{12} - 1383611538723138 T^{13} + 14773019976007700 T^{14} - 95207503290255607 T^{15} + 994146770284282487 T^{16} - 5868704756721408097 T^{17} + 62598491044048352438 T^{18} -$$$$35\!\cdots\!33$$$$T^{19} +$$$$38\!\cdots\!91$$$$T^{20} -$$$$21\!\cdots\!91$$$$T^{21} +$$$$23\!\cdots\!64$$$$T^{22} -$$$$12\!\cdots\!77$$$$T^{23} +$$$$13\!\cdots\!06$$$$T^{24} -$$$$12\!\cdots\!77$$$$p T^{25} +$$$$23\!\cdots\!64$$$$p^{2} T^{26} -$$$$21\!\cdots\!91$$$$p^{3} T^{27} +$$$$38\!\cdots\!91$$$$p^{4} T^{28} -$$$$35\!\cdots\!33$$$$p^{5} T^{29} + 62598491044048352438 p^{6} T^{30} - 5868704756721408097 p^{7} T^{31} + 994146770284282487 p^{8} T^{32} - 95207503290255607 p^{9} T^{33} + 14773019976007700 p^{10} T^{34} - 1383611538723138 p^{11} T^{35} + 189908257260168 p^{12} T^{36} - 16555853665676 p^{13} T^{37} + 1981010275084 p^{14} T^{38} - 154151339780 p^{15} T^{39} + 15962662267 p^{16} T^{40} - 1064058826 p^{17} T^{41} + 94476651 p^{18} T^{42} - 5106816 p^{19} T^{43} + 382407 p^{20} T^{44} - 15182 p^{21} T^{45} + 931 p^{22} T^{46} - 21 p^{23} T^{47} + p^{24} T^{48}$$
59 $$1 + 24 T + 1299 T^{2} + 26025 T^{3} + 795922 T^{4} + 13771713 T^{5} + 309622095 T^{6} + 4732676634 T^{7} + 86427371764 T^{8} + 1185624291309 T^{9} + 18501345559418 T^{10} + 230391535400403 T^{11} + 3164773074654159 T^{12} + 36074154630369558 T^{13} + 444558509480287587 T^{14} + 4666919237543256123 T^{15} + 52258153313706930641 T^{16} +$$$$50\!\cdots\!23$$$$T^{17} +$$$$52\!\cdots\!23$$$$T^{18} +$$$$46\!\cdots\!53$$$$T^{19} +$$$$44\!\cdots\!21$$$$T^{20} +$$$$37\!\cdots\!66$$$$T^{21} +$$$$32\!\cdots\!66$$$$T^{22} +$$$$25\!\cdots\!47$$$$T^{23} +$$$$20\!\cdots\!32$$$$T^{24} +$$$$25\!\cdots\!47$$$$p T^{25} +$$$$32\!\cdots\!66$$$$p^{2} T^{26} +$$$$37\!\cdots\!66$$$$p^{3} T^{27} +$$$$44\!\cdots\!21$$$$p^{4} T^{28} +$$$$46\!\cdots\!53$$$$p^{5} T^{29} +$$$$52\!\cdots\!23$$$$p^{6} T^{30} +$$$$50\!\cdots\!23$$$$p^{7} T^{31} + 52258153313706930641 p^{8} T^{32} + 4666919237543256123 p^{9} T^{33} + 444558509480287587 p^{10} T^{34} + 36074154630369558 p^{11} T^{35} + 3164773074654159 p^{12} T^{36} + 230391535400403 p^{13} T^{37} + 18501345559418 p^{14} T^{38} + 1185624291309 p^{15} T^{39} + 86427371764 p^{16} T^{40} + 4732676634 p^{17} T^{41} + 309622095 p^{18} T^{42} + 13771713 p^{19} T^{43} + 795922 p^{20} T^{44} + 26025 p^{21} T^{45} + 1299 p^{22} T^{46} + 24 p^{23} T^{47} + p^{24} T^{48}$$
61 $$1 + 40 T + 1796 T^{2} + 49085 T^{3} + 1326714 T^{4} + 28170634 T^{5} + 575906351 T^{6} + 10113259892 T^{7} + 169938300860 T^{8} + 2559147229703 T^{9} + 36866671036205 T^{10} + 487443418859263 T^{11} + 6175533024701033 T^{12} + 72905579821691272 T^{13} + 826818983401784191 T^{14} + 8833196306734108785 T^{15} + 90949434183688257384 T^{16} +$$$$88\!\cdots\!05$$$$T^{17} +$$$$84\!\cdots\!83$$$$T^{18} +$$$$76\!\cdots\!82$$$$T^{19} +$$$$67\!\cdots\!21$$$$T^{20} +$$$$57\!\cdots\!90$$$$T^{21} +$$$$47\!\cdots\!62$$$$T^{22} +$$$$38\!\cdots\!53$$$$T^{23} +$$$$30\!\cdots\!26$$$$T^{24} +$$$$38\!\cdots\!53$$$$p T^{25} +$$$$47\!\cdots\!62$$$$p^{2} T^{26} +$$$$57\!\cdots\!90$$$$p^{3} T^{27} +$$$$67\!\cdots\!21$$$$p^{4} T^{28} +$$$$76\!\cdots\!82$$$$p^{5} T^{29} +$$$$84\!\cdots\!83$$$$p^{6} T^{30} +$$$$88\!\cdots\!05$$$$p^{7} T^{31} + 90949434183688257384 p^{8} T^{32} + 8833196306734108785 p^{9} T^{33} + 826818983401784191 p^{10} T^{34} + 72905579821691272 p^{11} T^{35} + 6175533024701033 p^{12} T^{36} + 487443418859263 p^{13} T^{37} + 36866671036205 p^{14} T^{38} + 2559147229703 p^{15} T^{39} + 169938300860 p^{16} T^{40} + 10113259892 p^{17} T^{41} + 575906351 p^{18} T^{42} + 28170634 p^{19} T^{43} + 1326714 p^{20} T^{44} + 49085 p^{21} T^{45} + 1796 p^{22} T^{46} + 40 p^{23} T^{47} + p^{24} T^{48}$$
67 $$1 + 17 T + 861 T^{2} + 11194 T^{3} + 339554 T^{4} + 3585376 T^{5} + 85823862 T^{6} + 761485444 T^{7} + 16107809713 T^{8} + 122380805294 T^{9} + 2434435940334 T^{10} + 16003783583915 T^{11} + 311296616010438 T^{12} + 1783287086233978 T^{13} + 34778449172136607 T^{14} + 174832314150273629 T^{15} + 3467026590618723866 T^{16} + 15433737729441642896 T^{17} +$$$$31\!\cdots\!04$$$$T^{18} +$$$$12\!\cdots\!95$$$$T^{19} +$$$$25\!\cdots\!80$$$$T^{20} +$$$$93\!\cdots\!04$$$$T^{21} +$$$$19\!\cdots\!52$$$$T^{22} +$$$$66\!\cdots\!76$$$$T^{23} +$$$$13\!\cdots\!84$$$$T^{24} +$$$$66\!\cdots\!76$$$$p T^{25} +$$$$19\!\cdots\!52$$$$p^{2} T^{26} +$$$$93\!\cdots\!04$$$$p^{3} T^{27} +$$$$25\!\cdots\!80$$$$p^{4} T^{28} +$$$$12\!\cdots\!95$$$$p^{5} T^{29} +$$$$31\!\cdots\!04$$$$p^{6} T^{30} + 15433737729441642896 p^{7} T^{31} + 3467026590618723866 p^{8} T^{32} + 174832314150273629 p^{9} T^{33} + 34778449172136607 p^{10} T^{34} + 1783287086233978 p^{11} T^{35} + 311296616010438 p^{12} T^{36} + 16003783583915 p^{13} T^{37} + 2434435940334 p^{14} T^{38} + 122380805294 p^{15} T^{39} + 16107809713 p^{16} T^{40} + 761485444 p^{17} T^{41} + 85823862 p^{18} T^{42} + 3585376 p^{19} T^{43} + 339554 p^{20} T^{44} + 11194 p^{21} T^{45} + 861 p^{22} T^{46} + 17 p^{23} T^{47} + p^{24} T^{48}$$
71 $$1 - 4 T + 800 T^{2} - 2121 T^{3} + 321094 T^{4} - 487689 T^{5} + 87023871 T^{6} - 47176083 T^{7} + 17979582718 T^{8} + 5576383528 T^{9} + 3022157546045 T^{10} + 3170203127326 T^{11} + 430160015949438 T^{12} + 720140893796705 T^{13} + 53244149593758192 T^{14} + 116081115099411155 T^{15} + 5838044856515116691 T^{16} + 14949109846199877904 T^{17} +$$$$57\!\cdots\!03$$$$T^{18} +$$$$16\!\cdots\!45$$$$T^{19} +$$$$51\!\cdots\!82$$$$T^{20} +$$$$15\!\cdots\!26$$$$T^{21} +$$$$41\!\cdots\!45$$$$T^{22} +$$$$12\!\cdots\!00$$$$T^{23} +$$$$30\!\cdots\!16$$$$T^{24} +$$$$12\!\cdots\!00$$$$p T^{25} +$$$$41\!\cdots\!45$$$$p^{2} T^{26} +$$$$15\!\cdots\!26$$$$p^{3} T^{27} +$$$$51\!\cdots\!82$$$$p^{4} T^{28} +$$$$16\!\cdots\!45$$$$p^{5} T^{29} +$$$$57\!\cdots\!03$$$$p^{6} T^{30} + 14949109846199877904 p^{7} T^{31} + 5838044856515116691 p^{8} T^{32} + 116081115099411155 p^{9} T^{33} + 53244149593758192 p^{10} T^{34} + 720140893796705 p^{11} T^{35} + 430160015949438 p^{12} T^{36} + 3170203127326 p^{13} T^{37} + 3022157546045 p^{14} T^{38} + 5576383528 p^{15} T^{39} + 17979582718 p^{16} T^{40} - 47176083 p^{17} T^{41} + 87023871 p^{18} T^{42} - 487689 p^{19} T^{43} + 321094 p^{20} T^{44} - 2121 p^{21} T^{45} + 800 p^{22} T^{46} - 4 p^{23} T^{47} + p^{24} T^{48}$$
73 $$1 + 16 T + 1043 T^{2} + 14904 T^{3} + 540238 T^{4} + 7010495 T^{5} + 185648730 T^{6} + 2215064733 T^{7} + 47653368222 T^{8} + 527698617114 T^{9} + 9744171210741 T^{10} + 100855511297202 T^{11} + 1651696745094894 T^{12} + 16062621941313261 T^{13} + 238285330941324690 T^{14} + 2185421143505393392 T^{15} + 29792097070105859963 T^{16} +$$$$25\!\cdots\!26$$$$T^{17} +$$$$32\!\cdots\!19$$$$T^{18} +$$$$26\!\cdots\!43$$$$T^{19} +$$$$31\!\cdots\!08$$$$T^{20} +$$$$24\!\cdots\!75$$$$T^{21} +$$$$27\!\cdots\!29$$$$T^{22} +$$$$20\!\cdots\!99$$$$T^{23} +$$$$21\!\cdots\!04$$$$T^{24} +$$$$20\!\cdots\!99$$$$p T^{25} +$$$$27\!\cdots\!29$$$$p^{2} T^{26} +$$$$24\!\cdots\!75$$$$p^{3} T^{27} +$$$$31\!\cdots\!08$$$$p^{4} T^{28} +$$$$26\!\cdots\!43$$$$p^{5} T^{29} +$$$$32\!\cdots\!19$$$$p^{6} T^{30} +$$$$25\!\cdots\!26$$$$p^{7} T^{31} + 29792097070105859963 p^{8} T^{32} + 2185421143505393392 p^{9} T^{33} + 238285330941324690 p^{10} T^{34} + 16062621941313261 p^{11} T^{35} + 1651696745094894 p^{12} T^{36} + 100855511297202 p^{13} T^{37} + 9744171210741 p^{14} T^{38} + 527698617114 p^{15} T^{39} + 47653368222 p^{16} T^{40} + 2215064733 p^{17} T^{41} + 185648730 p^{18} T^{42} + 7010495 p^{19} T^{43} + 540238 p^{20} T^{44} + 14904 p^{21} T^{45} + 1043 p^{22} T^{46} + 16 p^{23} T^{47} + p^{24} T^{48}$$
79 $$1 + 53 T + 2386 T^{2} + 74996 T^{3} + 2089715 T^{4} + 48943365 T^{5} + 1046993479 T^{6} + 20035162200 T^{7} + 357038154507 T^{8} + 5865979014546 T^{9} + 90952990076296 T^{10} + 1323284162298701 T^{11} + 18343307966952679 T^{12} + 241340766777496862 T^{13} + 3046566979549042693 T^{14} + 36790260104747777674 T^{15} +$$$$42\!\cdots\!27$$$$T^{16} +$$$$48\!\cdots\!83$$$$T^{17} +$$$$52\!\cdots\!33$$$$T^{18} +$$$$54\!\cdots\!06$$$$T^{19} +$$$$55\!\cdots\!94$$$$T^{20} +$$$$54\!\cdots\!40$$$$T^{21} +$$$$52\!\cdots\!45$$$$T^{22} +$$$$48\!\cdots\!48$$$$T^{23} +$$$$43\!\cdots\!62$$$$T^{24} +$$$$48\!\cdots\!48$$$$p T^{25} +$$$$52\!\cdots\!45$$$$p^{2} T^{26} +$$$$54\!\cdots\!40$$$$p^{3} T^{27} +$$$$55\!\cdots\!94$$$$p^{4} T^{28} +$$$$54\!\cdots\!06$$$$p^{5} T^{29} +$$$$52\!\cdots\!33$$$$p^{6} T^{30} +$$$$48\!\cdots\!83$$$$p^{7} T^{31} +$$$$42\!\cdots\!27$$$$p^{8} T^{32} + 36790260104747777674 p^{9} T^{33} + 3046566979549042693 p^{10} T^{34} + 241340766777496862 p^{11} T^{35} + 18343307966952679 p^{12} T^{36} + 1323284162298701 p^{13} T^{37} + 90952990076296 p^{14} T^{38} + 5865979014546 p^{15} T^{39} + 357038154507 p^{16} T^{40} + 20035162200 p^{17} T^{41} + 1046993479 p^{18} T^{42} + 48943365 p^{19} T^{43} + 2089715 p^{20} T^{44} + 74996 p^{21} T^{45} + 2386 p^{22} T^{46} + 53 p^{23} T^{47} + p^{24} T^{48}$$
83 $$1 + 9 T + 974 T^{2} + 9544 T^{3} + 473205 T^{4} + 5007759 T^{5} + 154741502 T^{6} + 1732777854 T^{7} + 38684998323 T^{8} + 445829846120 T^{9} + 7918914385277 T^{10} + 91358828292349 T^{11} + 1379730431843416 T^{12} + 15598274609173114 T^{13} + 209190197814749639 T^{14} + 2288024019519042892 T^{15} + 27983722915764809902 T^{16} +$$$$29\!\cdots\!79$$$$T^{17} +$$$$33\!\cdots\!88$$$$T^{18} +$$$$33\!\cdots\!07$$$$T^{19} +$$$$35\!\cdots\!99$$$$T^{20} +$$$$34\!\cdots\!93$$$$T^{21} +$$$$34\!\cdots\!40$$$$T^{22} +$$$$38\!\cdots\!64$$$$p T^{23} +$$$$30\!\cdots\!28$$$$T^{24} +$$$$38\!\cdots\!64$$$$p^{2} T^{25} +$$$$34\!\cdots\!40$$$$p^{2} T^{26} +$$$$34\!\cdots\!93$$$$p^{3} T^{27} +$$$$35\!\cdots\!99$$$$p^{4} T^{28} +$$$$33\!\cdots\!07$$$$p^{5} T^{29} +$$$$33\!\cdots\!88$$$$p^{6} T^{30} +$$$$29\!\cdots\!79$$$$p^{7} T^{31} + 27983722915764809902 p^{8} T^{32} + 2288024019519042892 p^{9} T^{33} + 209190197814749639 p^{10} T^{34} + 15598274609173114 p^{11} T^{35} + 1379730431843416 p^{12} T^{36} + 91358828292349 p^{13} T^{37} + 7918914385277 p^{14} T^{38} + 445829846120 p^{15} T^{39} + 38684998323 p^{16} T^{40} + 1732777854 p^{17} T^{41} + 154741502 p^{18} T^{42} + 5007759 p^{19} T^{43} + 473205 p^{20} T^{44} + 9544 p^{21} T^{45} + 974 p^{22} T^{46} + 9 p^{23} T^{47} + p^{24} T^{48}$$
89 $$1 + 46 T + 2281 T^{2} + 71105 T^{3} + 2166044 T^{4} + 52871232 T^{5} + 1240522696 T^{6} + 25371617020 T^{7} + 498046935032 T^{8} + 8868644992905 T^{9} + 151874547153072 T^{10} + 2412088470816229 T^{11} + 36939132494452800 T^{12} + 531801775441244606 T^{13} + 7400732107934477224 T^{14} + 97678649691227821534 T^{15} +$$$$12\!\cdots\!82$$$$T^{16} +$$$$15\!\cdots\!22$$$$T^{17} +$$$$18\!\cdots\!89$$$$T^{18} +$$$$20\!\cdots\!21$$$$T^{19} +$$$$22\!\cdots\!96$$$$T^{20} +$$$$23\!\cdots\!87$$$$T^{21} +$$$$24\!\cdots\!26$$$$T^{22} +$$$$24\!\cdots\!41$$$$T^{23} +$$$$23\!\cdots\!98$$$$T^{24} +$$$$24\!\cdots\!41$$$$p T^{25} +$$$$24\!\cdots\!26$$$$p^{2} T^{26} +$$$$23\!\cdots\!87$$$$p^{3} T^{27} +$$$$22\!\cdots\!96$$$$p^{4} T^{28} +$$$$20\!\cdots\!21$$$$p^{5} T^{29} +$$$$18\!\cdots\!89$$$$p^{6} T^{30} +$$$$15\!\cdots\!22$$$$p^{7} T^{31} +$$$$12\!\cdots\!82$$$$p^{8} T^{32} + 97678649691227821534 p^{9} T^{33} + 7400732107934477224 p^{10} T^{34} + 531801775441244606 p^{11} T^{35} + 36939132494452800 p^{12} T^{36} + 2412088470816229 p^{13} T^{37} + 151874547153072 p^{14} T^{38} + 8868644992905 p^{15} T^{39} + 498046935032 p^{16} T^{40} + 25371617020 p^{17} T^{41} + 1240522696 p^{18} T^{42} + 52871232 p^{19} T^{43} + 2166044 p^{20} T^{44} + 71105 p^{21} T^{45} + 2281 p^{22} T^{46} + 46 p^{23} T^{47} + p^{24} T^{48}$$
97 $$1 + 20 T + 1269 T^{2} + 18445 T^{3} + 716479 T^{4} + 7816566 T^{5} + 251418712 T^{6} + 2015845024 T^{7} + 63791582271 T^{8} + 346412420959 T^{9} + 12899480967910 T^{10} + 39126635918526 T^{11} + 2237426767369748 T^{12} + 2073830556368296 T^{13} + 350010287938899431 T^{14} - 249146403556837652 T^{15} + 50432341026696499682 T^{16} - 92214924424717672953 T^{17} +$$$$66\!\cdots\!69$$$$T^{18} -$$$$16\!\cdots\!68$$$$T^{19} +$$$$80\!\cdots\!01$$$$T^{20} -$$$$23\!\cdots\!97$$$$T^{21} +$$$$89\!\cdots\!75$$$$T^{22} -$$$$27\!\cdots\!30$$$$T^{23} +$$$$90\!\cdots\!20$$$$T^{24} -$$$$27\!\cdots\!30$$$$p T^{25} +$$$$89\!\cdots\!75$$$$p^{2} T^{26} -$$$$23\!\cdots\!97$$$$p^{3} T^{27} +$$$$80\!\cdots\!01$$$$p^{4} T^{28} -$$$$16\!\cdots\!68$$$$p^{5} T^{29} +$$$$66\!\cdots\!69$$$$p^{6} T^{30} - 92214924424717672953 p^{7} T^{31} + 50432341026696499682 p^{8} T^{32} - 249146403556837652 p^{9} T^{33} + 350010287938899431 p^{10} T^{34} + 2073830556368296 p^{11} T^{35} + 2237426767369748 p^{12} T^{36} + 39126635918526 p^{13} T^{37} + 12899480967910 p^{14} T^{38} + 346412420959 p^{15} T^{39} + 63791582271 p^{16} T^{40} + 2015845024 p^{17} T^{41} + 251418712 p^{18} T^{42} + 7816566 p^{19} T^{43} + 716479 p^{20} T^{44} + 18445 p^{21} T^{45} + 1269 p^{22} T^{46} + 20 p^{23} T^{47} + p^{24} T^{48}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}