Properties

Degree 48
Conductor $ 3^{24} \cdot 5^{24} \cdot 7^{24} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 32·4-s + 16·5-s + 80·8-s + 128·10-s + 64·13-s + 138·16-s + 24·17-s + 512·20-s + 8·23-s + 60·25-s + 512·26-s + 96·31-s + 232·32-s + 192·34-s + 8·37-s + 1.28e3·40-s + 320·41-s − 112·43-s + 64·46-s + 64·47-s + 480·50-s + 2.04e3·52-s − 72·53-s − 496·61-s + 768·62-s + 640·64-s + ⋯
L(s)  = 1  + 4·2-s + 8·4-s + 16/5·5-s + 10·8-s + 64/5·10-s + 4.92·13-s + 69/8·16-s + 1.41·17-s + 25.5·20-s + 8/23·23-s + 12/5·25-s + 19.6·26-s + 3.09·31-s + 29/4·32-s + 5.64·34-s + 8/37·37-s + 32·40-s + 7.80·41-s − 2.60·43-s + 1.39·46-s + 1.36·47-s + 48/5·50-s + 39.3·52-s − 1.35·53-s − 8.13·61-s + 12.3·62-s + 10·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(48\)
\( N \)  =  \(3^{24} \cdot 5^{24} \cdot 7^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((48,\ 3^{24} \cdot 5^{24} \cdot 7^{24} ,\ ( \ : [1]^{24} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(87.5595\)
\(L(\frac12)\)  \(\approx\)  \(87.5595\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 48. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 47.
$p$$F_p(T)$
bad3 \( ( 1 + p^{2} T^{4} )^{6} \)
5 \( 1 - 16 T + 196 T^{2} - 1744 T^{3} + 13602 T^{4} - 83664 T^{5} + 454484 T^{6} - 1875856 T^{7} + 5655567 T^{8} + 486816 p T^{9} - 1158456 p^{3} T^{10} + 10665888 p^{3} T^{11} - 11842276 p^{4} T^{12} + 10665888 p^{5} T^{13} - 1158456 p^{7} T^{14} + 486816 p^{7} T^{15} + 5655567 p^{8} T^{16} - 1875856 p^{10} T^{17} + 454484 p^{12} T^{18} - 83664 p^{14} T^{19} + 13602 p^{16} T^{20} - 1744 p^{18} T^{21} + 196 p^{20} T^{22} - 16 p^{22} T^{23} + p^{24} T^{24} \)
7 \( ( 1 + p^{2} T^{4} )^{6} \)
good2 \( 1 - p^{3} T + p^{5} T^{2} - 5 p^{4} T^{3} + 59 p T^{4} - 9 p^{3} T^{5} - 17 p^{4} T^{7} + 1471 T^{8} - 287 p^{4} T^{9} + 219 p^{6} T^{10} - 341 p^{7} T^{11} + 25349 p^{2} T^{12} - 8203 p^{4} T^{13} + 85 p^{7} T^{14} + 3563 p^{6} T^{15} + 28687 T^{16} - 231369 p^{3} T^{17} + 175345 p^{5} T^{18} - 781707 p^{4} T^{19} + 14818635 p T^{20} - 7868353 p^{3} T^{21} + 648453 p^{7} T^{22} - 63443 p^{4} T^{23} - 177314735 T^{24} - 63443 p^{6} T^{25} + 648453 p^{11} T^{26} - 7868353 p^{9} T^{27} + 14818635 p^{9} T^{28} - 781707 p^{14} T^{29} + 175345 p^{17} T^{30} - 231369 p^{17} T^{31} + 28687 p^{16} T^{32} + 3563 p^{24} T^{33} + 85 p^{27} T^{34} - 8203 p^{26} T^{35} + 25349 p^{26} T^{36} - 341 p^{33} T^{37} + 219 p^{34} T^{38} - 287 p^{34} T^{39} + 1471 p^{32} T^{40} - 17 p^{38} T^{41} - 9 p^{41} T^{43} + 59 p^{41} T^{44} - 5 p^{46} T^{45} + p^{49} T^{46} - p^{49} T^{47} + p^{48} T^{48} \)
11 \( ( 1 + 776 T^{2} + 1856 T^{3} + 287190 T^{4} + 1318464 T^{5} + 68903336 T^{6} + 456391168 T^{7} + 12276983919 T^{8} + 102521035776 T^{9} + 1782212123984 T^{10} + 16577727233664 T^{11} + 226777567370612 T^{12} + 16577727233664 p^{2} T^{13} + 1782212123984 p^{4} T^{14} + 102521035776 p^{6} T^{15} + 12276983919 p^{8} T^{16} + 456391168 p^{10} T^{17} + 68903336 p^{12} T^{18} + 1318464 p^{14} T^{19} + 287190 p^{16} T^{20} + 1856 p^{18} T^{21} + 776 p^{20} T^{22} + p^{24} T^{24} )^{2} \)
13 \( 1 - 64 T + 2048 T^{2} - 48224 T^{3} + 940716 T^{4} - 14929888 T^{5} + 191703552 T^{6} - 1917474048 T^{7} + 10480201794 T^{8} + 89331585536 T^{9} - 3600227034624 T^{10} + 66197176785056 T^{11} - 863110165149572 T^{12} + 7351088820145184 T^{13} - 12770658566877184 T^{14} - 1090068393976087616 T^{15} + 29421158299700587183 T^{16} - \)\(48\!\cdots\!88\)\( T^{17} + \)\(61\!\cdots\!44\)\( T^{18} - \)\(59\!\cdots\!48\)\( T^{19} + \)\(29\!\cdots\!36\)\( T^{20} + \)\(32\!\cdots\!88\)\( T^{21} - \)\(11\!\cdots\!56\)\( T^{22} + \)\(22\!\cdots\!88\)\( T^{23} - \)\(32\!\cdots\!76\)\( T^{24} + \)\(22\!\cdots\!88\)\( p^{2} T^{25} - \)\(11\!\cdots\!56\)\( p^{4} T^{26} + \)\(32\!\cdots\!88\)\( p^{6} T^{27} + \)\(29\!\cdots\!36\)\( p^{8} T^{28} - \)\(59\!\cdots\!48\)\( p^{10} T^{29} + \)\(61\!\cdots\!44\)\( p^{12} T^{30} - \)\(48\!\cdots\!88\)\( p^{14} T^{31} + 29421158299700587183 p^{16} T^{32} - 1090068393976087616 p^{18} T^{33} - 12770658566877184 p^{20} T^{34} + 7351088820145184 p^{22} T^{35} - 863110165149572 p^{24} T^{36} + 66197176785056 p^{26} T^{37} - 3600227034624 p^{28} T^{38} + 89331585536 p^{30} T^{39} + 10480201794 p^{32} T^{40} - 1917474048 p^{34} T^{41} + 191703552 p^{36} T^{42} - 14929888 p^{38} T^{43} + 940716 p^{40} T^{44} - 48224 p^{42} T^{45} + 2048 p^{44} T^{46} - 64 p^{46} T^{47} + p^{48} T^{48} \)
17 \( 1 - 24 T + 288 T^{2} + 1656 T^{3} - 295820 T^{4} + 4179496 T^{5} - 13740576 T^{6} - 1397619464 T^{7} + 30944616386 T^{8} + 221336399832 T^{9} - 11869224732832 T^{10} + 289208649415112 T^{11} - 501960249695836 T^{12} - 91514917379169704 T^{13} + 1635639566048283552 T^{14} - 18003476549084560504 T^{15} - \)\(24\!\cdots\!49\)\( T^{16} + \)\(68\!\cdots\!32\)\( T^{17} - \)\(42\!\cdots\!64\)\( T^{18} - \)\(68\!\cdots\!28\)\( T^{19} + \)\(38\!\cdots\!28\)\( T^{20} - \)\(32\!\cdots\!80\)\( T^{21} - \)\(14\!\cdots\!68\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{23} - \)\(37\!\cdots\!00\)\( T^{24} + \)\(12\!\cdots\!00\)\( p^{2} T^{25} - \)\(14\!\cdots\!68\)\( p^{4} T^{26} - \)\(32\!\cdots\!80\)\( p^{6} T^{27} + \)\(38\!\cdots\!28\)\( p^{8} T^{28} - \)\(68\!\cdots\!28\)\( p^{10} T^{29} - \)\(42\!\cdots\!64\)\( p^{12} T^{30} + \)\(68\!\cdots\!32\)\( p^{14} T^{31} - \)\(24\!\cdots\!49\)\( p^{16} T^{32} - 18003476549084560504 p^{18} T^{33} + 1635639566048283552 p^{20} T^{34} - 91514917379169704 p^{22} T^{35} - 501960249695836 p^{24} T^{36} + 289208649415112 p^{26} T^{37} - 11869224732832 p^{28} T^{38} + 221336399832 p^{30} T^{39} + 30944616386 p^{32} T^{40} - 1397619464 p^{34} T^{41} - 13740576 p^{36} T^{42} + 4179496 p^{38} T^{43} - 295820 p^{40} T^{44} + 1656 p^{42} T^{45} + 288 p^{44} T^{46} - 24 p^{46} T^{47} + p^{48} T^{48} \)
19 \( 1 - 4128 T^{2} + 8376332 T^{4} - 588221408 p T^{6} + 11086148160002 T^{8} - 462042040956000 p T^{10} + 5827157656857086556 T^{12} - \)\(33\!\cdots\!84\)\( T^{14} + \)\(17\!\cdots\!95\)\( T^{16} - \)\(81\!\cdots\!32\)\( T^{18} + \)\(35\!\cdots\!60\)\( T^{20} - \)\(14\!\cdots\!44\)\( T^{22} + \)\(53\!\cdots\!28\)\( T^{24} - \)\(14\!\cdots\!44\)\( p^{4} T^{26} + \)\(35\!\cdots\!60\)\( p^{8} T^{28} - \)\(81\!\cdots\!32\)\( p^{12} T^{30} + \)\(17\!\cdots\!95\)\( p^{16} T^{32} - \)\(33\!\cdots\!84\)\( p^{20} T^{34} + 5827157656857086556 p^{24} T^{36} - 462042040956000 p^{29} T^{38} + 11086148160002 p^{32} T^{40} - 588221408 p^{37} T^{42} + 8376332 p^{40} T^{44} - 4128 p^{44} T^{46} + p^{48} T^{48} \)
23 \( 1 - 8 T + 32 T^{2} - 13480 T^{3} - 851732 T^{4} + 10736008 T^{5} + 32222560 T^{6} + 9692099688 T^{7} + 322838683650 T^{8} - 5183608917624 T^{9} - 14435550968096 T^{10} - 2732637207182808 T^{11} - 121450906418053828 T^{12} + 1116011670504380472 T^{13} + 413507015357687968 T^{14} + \)\(59\!\cdots\!64\)\( T^{15} + \)\(55\!\cdots\!23\)\( T^{16} - \)\(80\!\cdots\!16\)\( p T^{17} - \)\(19\!\cdots\!52\)\( p T^{18} - \)\(16\!\cdots\!12\)\( T^{19} - \)\(18\!\cdots\!80\)\( T^{20} + \)\(35\!\cdots\!40\)\( T^{21} + \)\(19\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!48\)\( T^{23} + \)\(50\!\cdots\!92\)\( T^{24} + \)\(16\!\cdots\!48\)\( p^{2} T^{25} + \)\(19\!\cdots\!52\)\( p^{4} T^{26} + \)\(35\!\cdots\!40\)\( p^{6} T^{27} - \)\(18\!\cdots\!80\)\( p^{8} T^{28} - \)\(16\!\cdots\!12\)\( p^{10} T^{29} - \)\(19\!\cdots\!52\)\( p^{13} T^{30} - \)\(80\!\cdots\!16\)\( p^{15} T^{31} + \)\(55\!\cdots\!23\)\( p^{16} T^{32} + \)\(59\!\cdots\!64\)\( p^{18} T^{33} + 413507015357687968 p^{20} T^{34} + 1116011670504380472 p^{22} T^{35} - 121450906418053828 p^{24} T^{36} - 2732637207182808 p^{26} T^{37} - 14435550968096 p^{28} T^{38} - 5183608917624 p^{30} T^{39} + 322838683650 p^{32} T^{40} + 9692099688 p^{34} T^{41} + 32222560 p^{36} T^{42} + 10736008 p^{38} T^{43} - 851732 p^{40} T^{44} - 13480 p^{42} T^{45} + 32 p^{44} T^{46} - 8 p^{46} T^{47} + p^{48} T^{48} \)
29 \( 1 - 8 p^{2} T^{2} + 23616980 T^{4} - 58432585144 T^{6} + 115062373742274 T^{8} - 191572252081321368 T^{10} + \)\(27\!\cdots\!44\)\( T^{12} - \)\(36\!\cdots\!84\)\( T^{14} + \)\(43\!\cdots\!15\)\( T^{16} - \)\(47\!\cdots\!56\)\( T^{18} + \)\(47\!\cdots\!12\)\( T^{20} - \)\(44\!\cdots\!88\)\( T^{22} + \)\(38\!\cdots\!04\)\( T^{24} - \)\(44\!\cdots\!88\)\( p^{4} T^{26} + \)\(47\!\cdots\!12\)\( p^{8} T^{28} - \)\(47\!\cdots\!56\)\( p^{12} T^{30} + \)\(43\!\cdots\!15\)\( p^{16} T^{32} - \)\(36\!\cdots\!84\)\( p^{20} T^{34} + \)\(27\!\cdots\!44\)\( p^{24} T^{36} - 191572252081321368 p^{28} T^{38} + 115062373742274 p^{32} T^{40} - 58432585144 p^{36} T^{42} + 23616980 p^{40} T^{44} - 8 p^{46} T^{46} + p^{48} T^{48} \)
31 \( ( 1 - 48 T + 7464 T^{2} - 324528 T^{3} + 27629894 T^{4} - 1074522544 T^{5} + 66925589416 T^{6} - 2323898253616 T^{7} + 118165131421263 T^{8} - 3677957302447264 T^{9} + 160567489451425328 T^{10} - 4489098229417174176 T^{11} + \)\(17\!\cdots\!60\)\( T^{12} - 4489098229417174176 p^{2} T^{13} + 160567489451425328 p^{4} T^{14} - 3677957302447264 p^{6} T^{15} + 118165131421263 p^{8} T^{16} - 2323898253616 p^{10} T^{17} + 66925589416 p^{12} T^{18} - 1074522544 p^{14} T^{19} + 27629894 p^{16} T^{20} - 324528 p^{18} T^{21} + 7464 p^{20} T^{22} - 48 p^{22} T^{23} + p^{24} T^{24} )^{2} \)
37 \( 1 - 8 T + 32 T^{2} - 146328 T^{3} - 4837580 T^{4} + 263099768 T^{5} + 8755946208 T^{6} + 884318092200 T^{7} - 23721048837886 T^{8} - 1619246068205560 T^{9} - 38719121469261216 T^{10} + 4573040801911064 T^{11} + \)\(20\!\cdots\!32\)\( T^{12} + \)\(18\!\cdots\!76\)\( T^{13} - \)\(36\!\cdots\!96\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} - \)\(34\!\cdots\!17\)\( T^{16} + \)\(12\!\cdots\!88\)\( T^{17} + \)\(46\!\cdots\!96\)\( T^{18} + \)\(24\!\cdots\!20\)\( T^{19} - \)\(69\!\cdots\!04\)\( T^{20} - \)\(41\!\cdots\!48\)\( T^{21} - \)\(58\!\cdots\!04\)\( T^{22} + \)\(11\!\cdots\!40\)\( T^{23} + \)\(30\!\cdots\!08\)\( T^{24} + \)\(11\!\cdots\!40\)\( p^{2} T^{25} - \)\(58\!\cdots\!04\)\( p^{4} T^{26} - \)\(41\!\cdots\!48\)\( p^{6} T^{27} - \)\(69\!\cdots\!04\)\( p^{8} T^{28} + \)\(24\!\cdots\!20\)\( p^{10} T^{29} + \)\(46\!\cdots\!96\)\( p^{12} T^{30} + \)\(12\!\cdots\!88\)\( p^{14} T^{31} - \)\(34\!\cdots\!17\)\( p^{16} T^{32} - \)\(11\!\cdots\!00\)\( p^{18} T^{33} - \)\(36\!\cdots\!96\)\( p^{20} T^{34} + \)\(18\!\cdots\!76\)\( p^{22} T^{35} + \)\(20\!\cdots\!32\)\( p^{24} T^{36} + 4573040801911064 p^{26} T^{37} - 38719121469261216 p^{28} T^{38} - 1619246068205560 p^{30} T^{39} - 23721048837886 p^{32} T^{40} + 884318092200 p^{34} T^{41} + 8755946208 p^{36} T^{42} + 263099768 p^{38} T^{43} - 4837580 p^{40} T^{44} - 146328 p^{42} T^{45} + 32 p^{44} T^{46} - 8 p^{46} T^{47} + p^{48} T^{48} \)
41 \( ( 1 - 160 T + 22420 T^{2} - 2185856 T^{3} + 190290306 T^{4} - 13883072256 T^{5} + 22598370180 p T^{6} - 55049672493216 T^{7} + 3047735763652527 T^{8} - 3770358655935936 p T^{9} + 7405275084219033800 T^{10} - \)\(32\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(32\!\cdots\!36\)\( p^{2} T^{13} + 7405275084219033800 p^{4} T^{14} - 3770358655935936 p^{7} T^{15} + 3047735763652527 p^{8} T^{16} - 55049672493216 p^{10} T^{17} + 22598370180 p^{13} T^{18} - 13883072256 p^{14} T^{19} + 190290306 p^{16} T^{20} - 2185856 p^{18} T^{21} + 22420 p^{20} T^{22} - 160 p^{22} T^{23} + p^{24} T^{24} )^{2} \)
43 \( 1 + 112 T + 6272 T^{2} + 297264 T^{3} + 19664076 T^{4} + 1226165520 T^{5} + 58180396416 T^{6} + 2516187808976 T^{7} + 107322597083394 T^{8} + 4302037155270032 T^{9} + 164350089714721664 T^{10} + 6210174340943468496 T^{11} + \)\(12\!\cdots\!48\)\( T^{12} - \)\(24\!\cdots\!28\)\( T^{13} - \)\(23\!\cdots\!88\)\( T^{14} - \)\(11\!\cdots\!56\)\( T^{15} - \)\(86\!\cdots\!17\)\( T^{16} - \)\(54\!\cdots\!96\)\( T^{17} - \)\(23\!\cdots\!60\)\( T^{18} - \)\(88\!\cdots\!36\)\( T^{19} - \)\(37\!\cdots\!04\)\( T^{20} - \)\(15\!\cdots\!88\)\( T^{21} - \)\(54\!\cdots\!04\)\( T^{22} - \)\(22\!\cdots\!92\)\( T^{23} - \)\(96\!\cdots\!96\)\( T^{24} - \)\(22\!\cdots\!92\)\( p^{2} T^{25} - \)\(54\!\cdots\!04\)\( p^{4} T^{26} - \)\(15\!\cdots\!88\)\( p^{6} T^{27} - \)\(37\!\cdots\!04\)\( p^{8} T^{28} - \)\(88\!\cdots\!36\)\( p^{10} T^{29} - \)\(23\!\cdots\!60\)\( p^{12} T^{30} - \)\(54\!\cdots\!96\)\( p^{14} T^{31} - \)\(86\!\cdots\!17\)\( p^{16} T^{32} - \)\(11\!\cdots\!56\)\( p^{18} T^{33} - \)\(23\!\cdots\!88\)\( p^{20} T^{34} - \)\(24\!\cdots\!28\)\( p^{22} T^{35} + \)\(12\!\cdots\!48\)\( p^{24} T^{36} + 6210174340943468496 p^{26} T^{37} + 164350089714721664 p^{28} T^{38} + 4302037155270032 p^{30} T^{39} + 107322597083394 p^{32} T^{40} + 2516187808976 p^{34} T^{41} + 58180396416 p^{36} T^{42} + 1226165520 p^{38} T^{43} + 19664076 p^{40} T^{44} + 297264 p^{42} T^{45} + 6272 p^{44} T^{46} + 112 p^{46} T^{47} + p^{48} T^{48} \)
47 \( 1 - 64 T + 2048 T^{2} - 295744 T^{3} + 4106892 T^{4} + 432123968 T^{5} + 7665408000 T^{6} + 1261053337408 T^{7} - 103402334780094 T^{8} - 1493848667448256 T^{9} - 3690102803974144 T^{10} + 3615108459475872064 T^{11} + \)\(59\!\cdots\!64\)\( T^{12} - \)\(19\!\cdots\!72\)\( T^{13} - \)\(56\!\cdots\!08\)\( T^{14} - \)\(51\!\cdots\!32\)\( T^{15} + \)\(10\!\cdots\!87\)\( T^{16} + \)\(86\!\cdots\!80\)\( T^{17} - \)\(13\!\cdots\!04\)\( T^{18} + \)\(45\!\cdots\!80\)\( T^{19} - \)\(96\!\cdots\!36\)\( T^{20} - \)\(97\!\cdots\!76\)\( T^{21} + \)\(97\!\cdots\!68\)\( T^{22} - \)\(43\!\cdots\!96\)\( T^{23} + \)\(54\!\cdots\!72\)\( T^{24} - \)\(43\!\cdots\!96\)\( p^{2} T^{25} + \)\(97\!\cdots\!68\)\( p^{4} T^{26} - \)\(97\!\cdots\!76\)\( p^{6} T^{27} - \)\(96\!\cdots\!36\)\( p^{8} T^{28} + \)\(45\!\cdots\!80\)\( p^{10} T^{29} - \)\(13\!\cdots\!04\)\( p^{12} T^{30} + \)\(86\!\cdots\!80\)\( p^{14} T^{31} + \)\(10\!\cdots\!87\)\( p^{16} T^{32} - \)\(51\!\cdots\!32\)\( p^{18} T^{33} - \)\(56\!\cdots\!08\)\( p^{20} T^{34} - \)\(19\!\cdots\!72\)\( p^{22} T^{35} + \)\(59\!\cdots\!64\)\( p^{24} T^{36} + 3615108459475872064 p^{26} T^{37} - 3690102803974144 p^{28} T^{38} - 1493848667448256 p^{30} T^{39} - 103402334780094 p^{32} T^{40} + 1261053337408 p^{34} T^{41} + 7665408000 p^{36} T^{42} + 432123968 p^{38} T^{43} + 4106892 p^{40} T^{44} - 295744 p^{42} T^{45} + 2048 p^{44} T^{46} - 64 p^{46} T^{47} + p^{48} T^{48} \)
53 \( 1 + 72 T + 2592 T^{2} + 727272 T^{3} + 35368684 T^{4} - 124356168 T^{5} + 163833007968 T^{6} + 6216604735320 T^{7} - 512857854227518 T^{8} + 16221026763504888 T^{9} + 1231578384865873632 T^{10} - \)\(15\!\cdots\!52\)\( T^{11} + \)\(25\!\cdots\!36\)\( T^{12} + \)\(41\!\cdots\!88\)\( T^{13} - \)\(23\!\cdots\!68\)\( T^{14} - \)\(45\!\cdots\!36\)\( T^{15} + \)\(10\!\cdots\!03\)\( T^{16} - \)\(21\!\cdots\!60\)\( T^{17} - \)\(20\!\cdots\!00\)\( T^{18} + \)\(18\!\cdots\!00\)\( T^{19} - \)\(60\!\cdots\!00\)\( T^{20} - \)\(59\!\cdots\!80\)\( T^{21} + \)\(18\!\cdots\!16\)\( T^{22} + \)\(31\!\cdots\!16\)\( T^{23} - \)\(11\!\cdots\!92\)\( T^{24} + \)\(31\!\cdots\!16\)\( p^{2} T^{25} + \)\(18\!\cdots\!16\)\( p^{4} T^{26} - \)\(59\!\cdots\!80\)\( p^{6} T^{27} - \)\(60\!\cdots\!00\)\( p^{8} T^{28} + \)\(18\!\cdots\!00\)\( p^{10} T^{29} - \)\(20\!\cdots\!00\)\( p^{12} T^{30} - \)\(21\!\cdots\!60\)\( p^{14} T^{31} + \)\(10\!\cdots\!03\)\( p^{16} T^{32} - \)\(45\!\cdots\!36\)\( p^{18} T^{33} - \)\(23\!\cdots\!68\)\( p^{20} T^{34} + \)\(41\!\cdots\!88\)\( p^{22} T^{35} + \)\(25\!\cdots\!36\)\( p^{24} T^{36} - \)\(15\!\cdots\!52\)\( p^{26} T^{37} + 1231578384865873632 p^{28} T^{38} + 16221026763504888 p^{30} T^{39} - 512857854227518 p^{32} T^{40} + 6216604735320 p^{34} T^{41} + 163833007968 p^{36} T^{42} - 124356168 p^{38} T^{43} + 35368684 p^{40} T^{44} + 727272 p^{42} T^{45} + 2592 p^{44} T^{46} + 72 p^{46} T^{47} + p^{48} T^{48} \)
59 \( 1 - 54024 T^{2} + 1430801812 T^{4} - 24759183049336 T^{6} + 314902406644546498 T^{8} - \)\(31\!\cdots\!12\)\( T^{10} + \)\(43\!\cdots\!76\)\( p T^{12} - \)\(17\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!83\)\( T^{16} - \)\(53\!\cdots\!72\)\( T^{18} + \)\(24\!\cdots\!80\)\( T^{20} - \)\(99\!\cdots\!32\)\( T^{22} + \)\(36\!\cdots\!84\)\( T^{24} - \)\(99\!\cdots\!32\)\( p^{4} T^{26} + \)\(24\!\cdots\!80\)\( p^{8} T^{28} - \)\(53\!\cdots\!72\)\( p^{12} T^{30} + \)\(10\!\cdots\!83\)\( p^{16} T^{32} - \)\(17\!\cdots\!84\)\( p^{20} T^{34} + \)\(43\!\cdots\!76\)\( p^{25} T^{36} - \)\(31\!\cdots\!12\)\( p^{28} T^{38} + 314902406644546498 p^{32} T^{40} - 24759183049336 p^{36} T^{42} + 1430801812 p^{40} T^{44} - 54024 p^{44} T^{46} + p^{48} T^{48} \)
61 \( ( 1 + 248 T + 44012 T^{2} + 5742984 T^{3} + 660244514 T^{4} + 65489503976 T^{5} + 6014949611324 T^{6} + 502810792608792 T^{7} + 652730467352571 p T^{8} + 2931247048649320112 T^{9} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{11} + \)\(85\!\cdots\!60\)\( T^{12} + \)\(13\!\cdots\!08\)\( p^{2} T^{13} + \)\(20\!\cdots\!28\)\( p^{4} T^{14} + 2931247048649320112 p^{6} T^{15} + 652730467352571 p^{9} T^{16} + 502810792608792 p^{10} T^{17} + 6014949611324 p^{12} T^{18} + 65489503976 p^{14} T^{19} + 660244514 p^{16} T^{20} + 5742984 p^{18} T^{21} + 44012 p^{20} T^{22} + 248 p^{22} T^{23} + p^{24} T^{24} )^{2} \)
67 \( 1 + 192 T + 18432 T^{2} + 1919680 T^{3} + 219659468 T^{4} + 18999916608 T^{5} + 1441806325760 T^{6} + 125613809882688 T^{7} + 10365549743634562 T^{8} + 720446338855147840 T^{9} + 52198207821115455488 T^{10} + \)\(41\!\cdots\!12\)\( T^{11} + \)\(29\!\cdots\!96\)\( T^{12} + \)\(19\!\cdots\!44\)\( T^{13} + \)\(14\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!20\)\( T^{15} + \)\(71\!\cdots\!55\)\( T^{16} + \)\(48\!\cdots\!04\)\( T^{17} + \)\(36\!\cdots\!68\)\( T^{18} + \)\(25\!\cdots\!56\)\( T^{19} + \)\(15\!\cdots\!08\)\( T^{20} + \)\(11\!\cdots\!96\)\( T^{21} + \)\(79\!\cdots\!20\)\( T^{22} + \)\(51\!\cdots\!56\)\( T^{23} + \)\(32\!\cdots\!36\)\( T^{24} + \)\(51\!\cdots\!56\)\( p^{2} T^{25} + \)\(79\!\cdots\!20\)\( p^{4} T^{26} + \)\(11\!\cdots\!96\)\( p^{6} T^{27} + \)\(15\!\cdots\!08\)\( p^{8} T^{28} + \)\(25\!\cdots\!56\)\( p^{10} T^{29} + \)\(36\!\cdots\!68\)\( p^{12} T^{30} + \)\(48\!\cdots\!04\)\( p^{14} T^{31} + \)\(71\!\cdots\!55\)\( p^{16} T^{32} + \)\(10\!\cdots\!20\)\( p^{18} T^{33} + \)\(14\!\cdots\!24\)\( p^{20} T^{34} + \)\(19\!\cdots\!44\)\( p^{22} T^{35} + \)\(29\!\cdots\!96\)\( p^{24} T^{36} + \)\(41\!\cdots\!12\)\( p^{26} T^{37} + 52198207821115455488 p^{28} T^{38} + 720446338855147840 p^{30} T^{39} + 10365549743634562 p^{32} T^{40} + 125613809882688 p^{34} T^{41} + 1441806325760 p^{36} T^{42} + 18999916608 p^{38} T^{43} + 219659468 p^{40} T^{44} + 1919680 p^{42} T^{45} + 18432 p^{44} T^{46} + 192 p^{46} T^{47} + p^{48} T^{48} \)
71 \( ( 1 + 72 T + 37912 T^{2} + 1474712 T^{3} + 596434966 T^{4} + 4576000248 T^{5} + 5495661646456 T^{6} - 135505260222744 T^{7} + 37574935106262671 T^{8} - 1945517857999653552 T^{9} + \)\(22\!\cdots\!32\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!16\)\( p^{2} T^{13} + \)\(22\!\cdots\!32\)\( p^{4} T^{14} - 1945517857999653552 p^{6} T^{15} + 37574935106262671 p^{8} T^{16} - 135505260222744 p^{10} T^{17} + 5495661646456 p^{12} T^{18} + 4576000248 p^{14} T^{19} + 596434966 p^{16} T^{20} + 1474712 p^{18} T^{21} + 37912 p^{20} T^{22} + 72 p^{22} T^{23} + p^{24} T^{24} )^{2} \)
73 \( 1 - 224 T + 25088 T^{2} - 1716352 T^{3} + 96873580 T^{4} - 9201024320 T^{5} + 1103597166592 T^{6} - 101085446622432 T^{7} + 6681027042627714 T^{8} - 356491237387067232 T^{9} + 21158655073555671040 T^{10} - \)\(15\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!44\)\( T^{12} - \)\(61\!\cdots\!44\)\( T^{13} + \)\(33\!\cdots\!72\)\( T^{14} - \)\(26\!\cdots\!76\)\( T^{15} + \)\(25\!\cdots\!07\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{17} + \)\(13\!\cdots\!64\)\( T^{18} - \)\(11\!\cdots\!04\)\( T^{19} + \)\(15\!\cdots\!36\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} + \)\(13\!\cdots\!04\)\( T^{22} - \)\(80\!\cdots\!20\)\( T^{23} + \)\(47\!\cdots\!16\)\( T^{24} - \)\(80\!\cdots\!20\)\( p^{2} T^{25} + \)\(13\!\cdots\!04\)\( p^{4} T^{26} - \)\(17\!\cdots\!16\)\( p^{6} T^{27} + \)\(15\!\cdots\!36\)\( p^{8} T^{28} - \)\(11\!\cdots\!04\)\( p^{10} T^{29} + \)\(13\!\cdots\!64\)\( p^{12} T^{30} - \)\(20\!\cdots\!80\)\( p^{14} T^{31} + \)\(25\!\cdots\!07\)\( p^{16} T^{32} - \)\(26\!\cdots\!76\)\( p^{18} T^{33} + \)\(33\!\cdots\!72\)\( p^{20} T^{34} - \)\(61\!\cdots\!44\)\( p^{22} T^{35} + \)\(10\!\cdots\!44\)\( p^{24} T^{36} - \)\(15\!\cdots\!12\)\( p^{26} T^{37} + 21158655073555671040 p^{28} T^{38} - 356491237387067232 p^{30} T^{39} + 6681027042627714 p^{32} T^{40} - 101085446622432 p^{34} T^{41} + 1103597166592 p^{36} T^{42} - 9201024320 p^{38} T^{43} + 96873580 p^{40} T^{44} - 1716352 p^{42} T^{45} + 25088 p^{44} T^{46} - 224 p^{46} T^{47} + p^{48} T^{48} \)
79 \( 1 - 100296 T^{2} + 4954216532 T^{4} - 160669672067896 T^{6} + 3847193392339896834 T^{8} - \)\(72\!\cdots\!56\)\( T^{10} + \)\(11\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!56\)\( T^{14} + \)\(16\!\cdots\!31\)\( T^{16} - \)\(15\!\cdots\!60\)\( T^{18} + \)\(13\!\cdots\!64\)\( T^{20} - \)\(99\!\cdots\!16\)\( T^{22} + \)\(65\!\cdots\!24\)\( T^{24} - \)\(99\!\cdots\!16\)\( p^{4} T^{26} + \)\(13\!\cdots\!64\)\( p^{8} T^{28} - \)\(15\!\cdots\!60\)\( p^{12} T^{30} + \)\(16\!\cdots\!31\)\( p^{16} T^{32} - \)\(14\!\cdots\!56\)\( p^{20} T^{34} + \)\(11\!\cdots\!96\)\( p^{24} T^{36} - \)\(72\!\cdots\!56\)\( p^{28} T^{38} + 3847193392339896834 p^{32} T^{40} - 160669672067896 p^{36} T^{42} + 4954216532 p^{40} T^{44} - 100296 p^{44} T^{46} + p^{48} T^{48} \)
83 \( 1 + 32 T + 512 T^{2} + 936864 T^{3} - 331572 T^{4} - 5370090016 T^{5} + 267183961600 T^{6} - 14930846145696 T^{7} - 7294634369315198 T^{8} + 96086188510242784 T^{9} + 7328962696462888448 T^{10} - \)\(27\!\cdots\!08\)\( T^{11} + \)\(17\!\cdots\!36\)\( T^{12} + \)\(19\!\cdots\!24\)\( T^{13} - \)\(47\!\cdots\!56\)\( T^{14} + \)\(69\!\cdots\!88\)\( T^{15} + \)\(85\!\cdots\!35\)\( T^{16} - \)\(52\!\cdots\!44\)\( T^{17} - \)\(98\!\cdots\!12\)\( T^{18} - \)\(14\!\cdots\!96\)\( T^{19} - \)\(30\!\cdots\!52\)\( T^{20} + \)\(25\!\cdots\!88\)\( T^{21} - \)\(58\!\cdots\!80\)\( T^{22} - \)\(13\!\cdots\!12\)\( T^{23} + \)\(71\!\cdots\!36\)\( T^{24} - \)\(13\!\cdots\!12\)\( p^{2} T^{25} - \)\(58\!\cdots\!80\)\( p^{4} T^{26} + \)\(25\!\cdots\!88\)\( p^{6} T^{27} - \)\(30\!\cdots\!52\)\( p^{8} T^{28} - \)\(14\!\cdots\!96\)\( p^{10} T^{29} - \)\(98\!\cdots\!12\)\( p^{12} T^{30} - \)\(52\!\cdots\!44\)\( p^{14} T^{31} + \)\(85\!\cdots\!35\)\( p^{16} T^{32} + \)\(69\!\cdots\!88\)\( p^{18} T^{33} - \)\(47\!\cdots\!56\)\( p^{20} T^{34} + \)\(19\!\cdots\!24\)\( p^{22} T^{35} + \)\(17\!\cdots\!36\)\( p^{24} T^{36} - \)\(27\!\cdots\!08\)\( p^{26} T^{37} + 7328962696462888448 p^{28} T^{38} + 96086188510242784 p^{30} T^{39} - 7294634369315198 p^{32} T^{40} - 14930846145696 p^{34} T^{41} + 267183961600 p^{36} T^{42} - 5370090016 p^{38} T^{43} - 331572 p^{40} T^{44} + 936864 p^{42} T^{45} + 512 p^{44} T^{46} + 32 p^{46} T^{47} + p^{48} T^{48} \)
89 \( 1 - 109496 T^{2} + 5853525460 T^{4} - 204130549495432 T^{6} + 5241644938432946050 T^{8} - \)\(10\!\cdots\!48\)\( T^{10} + \)\(17\!\cdots\!48\)\( T^{12} - \)\(25\!\cdots\!76\)\( T^{14} + \)\(31\!\cdots\!51\)\( T^{16} - \)\(35\!\cdots\!20\)\( T^{18} + \)\(35\!\cdots\!56\)\( T^{20} - \)\(32\!\cdots\!68\)\( T^{22} + \)\(26\!\cdots\!28\)\( T^{24} - \)\(32\!\cdots\!68\)\( p^{4} T^{26} + \)\(35\!\cdots\!56\)\( p^{8} T^{28} - \)\(35\!\cdots\!20\)\( p^{12} T^{30} + \)\(31\!\cdots\!51\)\( p^{16} T^{32} - \)\(25\!\cdots\!76\)\( p^{20} T^{34} + \)\(17\!\cdots\!48\)\( p^{24} T^{36} - \)\(10\!\cdots\!48\)\( p^{28} T^{38} + 5241644938432946050 p^{32} T^{40} - 204130549495432 p^{36} T^{42} + 5853525460 p^{40} T^{44} - 109496 p^{44} T^{46} + p^{48} T^{48} \)
97 \( 1 + 816 T + 332928 T^{2} + 94278416 T^{3} + 21264490476 T^{4} + 4056292711792 T^{5} + 674600429363072 T^{6} + 100193972996462672 T^{7} + 13551879143313614850 T^{8} + \)\(16\!\cdots\!12\)\( T^{9} + \)\(19\!\cdots\!04\)\( T^{10} + \)\(21\!\cdots\!72\)\( T^{11} + \)\(22\!\cdots\!88\)\( T^{12} + \)\(22\!\cdots\!60\)\( T^{13} + \)\(21\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!27\)\( T^{16} + \)\(68\!\cdots\!52\)\( T^{17} - \)\(38\!\cdots\!24\)\( T^{18} - \)\(77\!\cdots\!68\)\( T^{19} - \)\(14\!\cdots\!96\)\( T^{20} - \)\(19\!\cdots\!64\)\( T^{21} - \)\(23\!\cdots\!00\)\( T^{22} - \)\(25\!\cdots\!04\)\( T^{23} - \)\(25\!\cdots\!72\)\( T^{24} - \)\(25\!\cdots\!04\)\( p^{2} T^{25} - \)\(23\!\cdots\!00\)\( p^{4} T^{26} - \)\(19\!\cdots\!64\)\( p^{6} T^{27} - \)\(14\!\cdots\!96\)\( p^{8} T^{28} - \)\(77\!\cdots\!68\)\( p^{10} T^{29} - \)\(38\!\cdots\!24\)\( p^{12} T^{30} + \)\(68\!\cdots\!52\)\( p^{14} T^{31} + \)\(13\!\cdots\!27\)\( p^{16} T^{32} + \)\(18\!\cdots\!00\)\( p^{18} T^{33} + \)\(21\!\cdots\!40\)\( p^{20} T^{34} + \)\(22\!\cdots\!60\)\( p^{22} T^{35} + \)\(22\!\cdots\!88\)\( p^{24} T^{36} + \)\(21\!\cdots\!72\)\( p^{26} T^{37} + \)\(19\!\cdots\!04\)\( p^{28} T^{38} + \)\(16\!\cdots\!12\)\( p^{30} T^{39} + 13551879143313614850 p^{32} T^{40} + 100193972996462672 p^{34} T^{41} + 674600429363072 p^{36} T^{42} + 4056292711792 p^{38} T^{43} + 21264490476 p^{40} T^{44} + 94278416 p^{42} T^{45} + 332928 p^{44} T^{46} + 816 p^{46} T^{47} + p^{48} T^{48} \)
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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

−3.02502138203634278805040117166, −2.98487043885350033638709752303, −2.76831756049325157203915062548, −2.74107191830797683522933022906, −2.72933658145651962435059370241, −2.71830777360441439920981105213, −2.68791798712816473359462782124, −2.60175289541509538553588069582, −2.47851443057465957393736170049, −2.40725196670167229761268255753, −2.29913221563469416565161512649, −2.22717373917717638106180586373, −1.82596503179745369521312910646, −1.77388266408271202743049270835, −1.62701005541909201789357564195, −1.59476577704947311700677452057, −1.51017704379425771164674717762, −1.49957020755019577682609859025, −1.39771994173501321168004238808, −1.28539181077891087823073581093, −1.20092475946463513272744461365, −1.01484168047419156876720035858, −0.841155734951819678245516687037, −0.64703589347616853402397174321, −0.10249653671813863887519817901, 0.10249653671813863887519817901, 0.64703589347616853402397174321, 0.841155734951819678245516687037, 1.01484168047419156876720035858, 1.20092475946463513272744461365, 1.28539181077891087823073581093, 1.39771994173501321168004238808, 1.49957020755019577682609859025, 1.51017704379425771164674717762, 1.59476577704947311700677452057, 1.62701005541909201789357564195, 1.77388266408271202743049270835, 1.82596503179745369521312910646, 2.22717373917717638106180586373, 2.29913221563469416565161512649, 2.40725196670167229761268255753, 2.47851443057465957393736170049, 2.60175289541509538553588069582, 2.68791798712816473359462782124, 2.71830777360441439920981105213, 2.72933658145651962435059370241, 2.74107191830797683522933022906, 2.76831756049325157203915062548, 2.98487043885350033638709752303, 3.02502138203634278805040117166

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

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