# Properties

 Degree 32 Conductor $2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}$ Sign $1$ Motivic weight 2 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 24·3-s − 8·4-s + 4·7-s + 300·9-s − 4·11-s + 192·12-s + 24·16-s + 12·17-s − 72·19-s − 96·21-s − 12·23-s + 20·25-s − 2.59e3·27-s − 32·28-s + 72·29-s + 120·31-s + 96·33-s − 2.40e3·36-s + 44·37-s − 56·43-s + 32·44-s − 24·47-s − 576·48-s − 12·49-s − 288·51-s + 32·53-s + 1.72e3·57-s + ⋯
 L(s)  = 1 − 8·3-s − 2·4-s + 4/7·7-s + 33.3·9-s − 0.363·11-s + 16·12-s + 3/2·16-s + 0.705·17-s − 3.78·19-s − 4.57·21-s − 0.521·23-s + 4/5·25-s − 96·27-s − 8/7·28-s + 2.48·29-s + 3.87·31-s + 2.90·33-s − 66.6·36-s + 1.18·37-s − 1.30·43-s + 8/11·44-s − 0.510·47-s − 12·48-s − 0.244·49-s − 5.64·51-s + 0.603·53-s + 30.3·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$32$$ $$N$$ = $$2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}$$ $$\varepsilon$$ = $1$ motivic weight = $$2$$ character : induced by $\chi_{210} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(32,\ 2^{16} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )$$ $$L(\frac{3}{2})$$ $$\approx$$ $$0.370562$$ $$L(\frac12)$$ $$\approx$$ $$0.370562$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 32. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 31.
$p$$F_p(T)$
bad2 $$( 1 + p T^{2} + p^{2} T^{4} )^{4}$$
3 $$( 1 + p T + p T^{2} )^{8}$$
5 $$( 1 - p T^{2} + p^{2} T^{4} )^{4}$$
7 $$1 - 4 T + 4 p T^{2} - 352 T^{3} + 414 p T^{4} - 16836 T^{5} + 70272 T^{6} - 23652 p^{2} T^{7} + 159067 p^{2} T^{8} - 23652 p^{4} T^{9} + 70272 p^{4} T^{10} - 16836 p^{6} T^{11} + 414 p^{9} T^{12} - 352 p^{10} T^{13} + 4 p^{13} T^{14} - 4 p^{14} T^{15} + p^{16} T^{16}$$
good11 $$1 + 4 T - 364 T^{2} - 232 p T^{3} + 54108 T^{4} + 655724 T^{5} - 3422504 T^{6} - 97911028 T^{7} - 146055574 T^{8} + 11147160032 T^{9} + 52178728220 T^{10} - 1271915177612 T^{11} - 12776124769680 T^{12} + 140823145942564 T^{13} + 3084355376442156 T^{14} - 699823554207552 p T^{15} - 472018303966569005 T^{16} - 699823554207552 p^{3} T^{17} + 3084355376442156 p^{4} T^{18} + 140823145942564 p^{6} T^{19} - 12776124769680 p^{8} T^{20} - 1271915177612 p^{10} T^{21} + 52178728220 p^{12} T^{22} + 11147160032 p^{14} T^{23} - 146055574 p^{16} T^{24} - 97911028 p^{18} T^{25} - 3422504 p^{20} T^{26} + 655724 p^{22} T^{27} + 54108 p^{24} T^{28} - 232 p^{27} T^{29} - 364 p^{28} T^{30} + 4 p^{30} T^{31} + p^{32} T^{32}$$
13 $$1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} - 75519988778385192 p^{4} T^{18} + 1484717408181296 p^{8} T^{20} - 13349067637096 p^{12} T^{22} + 77337463770 p^{16} T^{24} - 307870064 p^{20} T^{26} + 822548 p^{24} T^{28} - 1336 p^{28} T^{30} + p^{32} T^{32}$$
17 $$1 - 12 T + 420 T^{2} - 4464 T^{3} + 92044 T^{4} - 785292 T^{5} + 23989560 T^{6} + 227912772 T^{7} - 3236850998 T^{8} + 177940653384 T^{9} - 2004860299572 T^{10} + 43692613137348 T^{11} - 466147839356624 T^{12} + 8985915182672820 T^{13} - 29797959923559588 T^{14} - 45291766011984888 p T^{15} + 119434782862777075 p^{2} T^{16} - 45291766011984888 p^{3} T^{17} - 29797959923559588 p^{4} T^{18} + 8985915182672820 p^{6} T^{19} - 466147839356624 p^{8} T^{20} + 43692613137348 p^{10} T^{21} - 2004860299572 p^{12} T^{22} + 177940653384 p^{14} T^{23} - 3236850998 p^{16} T^{24} + 227912772 p^{18} T^{25} + 23989560 p^{20} T^{26} - 785292 p^{22} T^{27} + 92044 p^{24} T^{28} - 4464 p^{26} T^{29} + 420 p^{28} T^{30} - 12 p^{30} T^{31} + p^{32} T^{32}$$
19 $$1 + 72 T + 3812 T^{2} + 150048 T^{3} + 257018 p T^{4} + 136887912 T^{5} + 178133848 p T^{6} + 75691203624 T^{7} + 82482274371 p T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 11988069986573136 p^{2} T^{13} + 88720456587214840044 T^{14} +$$$$17\!\cdots\!80$$$$T^{15} +$$$$34\!\cdots\!48$$$$T^{16} +$$$$17\!\cdots\!80$$$$p^{2} T^{17} + 88720456587214840044 p^{4} T^{18} + 11988069986573136 p^{8} T^{19} + 213751725988675694 p^{8} T^{20} + 10979211701479464 p^{10} T^{21} + 578634766673960 p^{12} T^{22} + 30524546336184 p^{14} T^{23} + 82482274371 p^{17} T^{24} + 75691203624 p^{18} T^{25} + 178133848 p^{21} T^{26} + 136887912 p^{22} T^{27} + 257018 p^{25} T^{28} + 150048 p^{26} T^{29} + 3812 p^{28} T^{30} + 72 p^{30} T^{31} + p^{32} T^{32}$$
23 $$1 + 12 T - 1804 T^{2} - 41256 T^{3} + 1078172 T^{4} + 48109188 T^{5} - 226519400 T^{6} - 29579538300 T^{7} + 68149425194 T^{8} + 16885870838304 T^{9} + 1241401844668 T^{10} - 10296857200481412 T^{11} - 100210062266615184 T^{12} + 3750005340988049388 T^{13} + 80231845600155444876 T^{14} -$$$$51\!\cdots\!40$$$$T^{15} -$$$$38\!\cdots\!85$$$$T^{16} -$$$$51\!\cdots\!40$$$$p^{2} T^{17} + 80231845600155444876 p^{4} T^{18} + 3750005340988049388 p^{6} T^{19} - 100210062266615184 p^{8} T^{20} - 10296857200481412 p^{10} T^{21} + 1241401844668 p^{12} T^{22} + 16885870838304 p^{14} T^{23} + 68149425194 p^{16} T^{24} - 29579538300 p^{18} T^{25} - 226519400 p^{20} T^{26} + 48109188 p^{22} T^{27} + 1078172 p^{24} T^{28} - 41256 p^{26} T^{29} - 1804 p^{28} T^{30} + 12 p^{30} T^{31} + p^{32} T^{32}$$
29 $$( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 30035473452 p^{2} T^{9} + 84102156 p^{4} T^{10} + 68800452 p^{6} T^{11} - 1274860 p^{8} T^{12} + 19116 p^{10} T^{13} + 1444 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
31 $$1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} +$$$$21\!\cdots\!82$$$$T^{12} -$$$$69\!\cdots\!28$$$$T^{13} +$$$$23\!\cdots\!88$$$$T^{14} -$$$$74\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!16$$$$T^{16} -$$$$74\!\cdots\!40$$$$p^{2} T^{17} +$$$$23\!\cdots\!88$$$$p^{4} T^{18} -$$$$69\!\cdots\!28$$$$p^{6} T^{19} +$$$$21\!\cdots\!82$$$$p^{8} T^{20} - 6377368910762118360 p^{10} T^{21} + 196357797692622056 p^{12} T^{22} - 5869377562140936 p^{14} T^{23} + 170270490089625 p^{16} T^{24} - 4558808688792 p^{18} T^{25} + 113293093576 p^{20} T^{26} - 2488742424 p^{22} T^{27} + 49123694 p^{24} T^{28} - 814560 p^{26} T^{29} + 11588 p^{28} T^{30} - 120 p^{30} T^{31} + p^{32} T^{32}$$
37 $$1 - 44 T - 1956 T^{2} - 11728 T^{3} + 3652806 T^{4} + 242885524 T^{5} - 1534355576 T^{6} - 317275895964 T^{7} - 9605253544351 T^{8} + 2052828037948 p T^{9} + 6072970847630504 T^{10} + 248299820626414612 T^{11} + 4168440621293518230 T^{12} +$$$$29\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!28$$$$T^{14} -$$$$69\!\cdots\!04$$$$T^{15} -$$$$27\!\cdots\!16$$$$T^{16} -$$$$69\!\cdots\!04$$$$p^{2} T^{17} +$$$$10\!\cdots\!28$$$$p^{4} T^{18} +$$$$29\!\cdots\!60$$$$p^{6} T^{19} + 4168440621293518230 p^{8} T^{20} + 248299820626414612 p^{10} T^{21} + 6072970847630504 p^{12} T^{22} + 2052828037948 p^{15} T^{23} - 9605253544351 p^{16} T^{24} - 317275895964 p^{18} T^{25} - 1534355576 p^{20} T^{26} + 242885524 p^{22} T^{27} + 3652806 p^{24} T^{28} - 11728 p^{26} T^{29} - 1956 p^{28} T^{30} - 44 p^{30} T^{31} + p^{32} T^{32}$$
41 $$1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} +$$$$50\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!32$$$$T^{14} +$$$$18\!\cdots\!14$$$$T^{16} -$$$$10\!\cdots\!32$$$$p^{4} T^{18} +$$$$50\!\cdots\!56$$$$p^{8} T^{20} - 2208514761298495288 p^{12} T^{22} + 835818107018940 p^{16} T^{24} - 265318066664 p^{20} T^{26} + 66218264 p^{24} T^{28} - 11416 p^{28} T^{30} + p^{32} T^{32}$$
43 $$( 1 + 28 T + 5732 T^{2} + 194288 T^{3} + 21755610 T^{4} + 646615948 T^{5} + 59916131504 T^{6} + 37466423748 p T^{7} + 122468758030675 T^{8} + 37466423748 p^{3} T^{9} + 59916131504 p^{4} T^{10} + 646615948 p^{6} T^{11} + 21755610 p^{8} T^{12} + 194288 p^{10} T^{13} + 5732 p^{12} T^{14} + 28 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
47 $$1 + 24 T + 10040 T^{2} + 236352 T^{3} + 48191588 T^{4} + 529473000 T^{5} + 142018937488 T^{6} - 1533668366952 T^{7} + 289429105887690 T^{8} - 11114511768111792 T^{9} + 616929826339364168 T^{10} - 28449623988679606104 T^{11} +$$$$19\!\cdots\!48$$$$T^{12} -$$$$39\!\cdots\!52$$$$T^{13} +$$$$60\!\cdots\!84$$$$T^{14} -$$$$28\!\cdots\!36$$$$T^{15} +$$$$14\!\cdots\!27$$$$T^{16} -$$$$28\!\cdots\!36$$$$p^{2} T^{17} +$$$$60\!\cdots\!84$$$$p^{4} T^{18} -$$$$39\!\cdots\!52$$$$p^{6} T^{19} +$$$$19\!\cdots\!48$$$$p^{8} T^{20} - 28449623988679606104 p^{10} T^{21} + 616929826339364168 p^{12} T^{22} - 11114511768111792 p^{14} T^{23} + 289429105887690 p^{16} T^{24} - 1533668366952 p^{18} T^{25} + 142018937488 p^{20} T^{26} + 529473000 p^{22} T^{27} + 48191588 p^{24} T^{28} + 236352 p^{26} T^{29} + 10040 p^{28} T^{30} + 24 p^{30} T^{31} + p^{32} T^{32}$$
53 $$1 - 32 T - 12072 T^{2} + 345280 T^{3} + 71529764 T^{4} - 1574568864 T^{5} - 290206976304 T^{6} + 2770742930720 T^{7} + 1006589576083018 T^{8} + 3889510628997504 T^{9} - 3360408915215254232 T^{10} - 25194864185334307040 T^{11} +$$$$10\!\cdots\!32$$$$T^{12} +$$$$34\!\cdots\!88$$$$T^{13} -$$$$28\!\cdots\!00$$$$T^{14} -$$$$63\!\cdots\!04$$$$T^{15} +$$$$75\!\cdots\!55$$$$T^{16} -$$$$63\!\cdots\!04$$$$p^{2} T^{17} -$$$$28\!\cdots\!00$$$$p^{4} T^{18} +$$$$34\!\cdots\!88$$$$p^{6} T^{19} +$$$$10\!\cdots\!32$$$$p^{8} T^{20} - 25194864185334307040 p^{10} T^{21} - 3360408915215254232 p^{12} T^{22} + 3889510628997504 p^{14} T^{23} + 1006589576083018 p^{16} T^{24} + 2770742930720 p^{18} T^{25} - 290206976304 p^{20} T^{26} - 1574568864 p^{22} T^{27} + 71529764 p^{24} T^{28} + 345280 p^{26} T^{29} - 12072 p^{28} T^{30} - 32 p^{30} T^{31} + p^{32} T^{32}$$
59 $$1 - 132 T + 9660 T^{2} - 508464 T^{3} + 17070268 T^{4} - 1368116580 T^{5} + 85370423112 T^{6} - 5735724829716 T^{7} + 274189008275050 T^{8} - 2776853544451464 T^{9} + 245103035882958612 T^{10} + 79108787586088932 p T^{11} +$$$$23\!\cdots\!68$$$$T^{12} -$$$$15\!\cdots\!44$$$$T^{13} +$$$$45\!\cdots\!16$$$$T^{14} -$$$$18\!\cdots\!68$$$$T^{15} -$$$$23\!\cdots\!33$$$$T^{16} -$$$$18\!\cdots\!68$$$$p^{2} T^{17} +$$$$45\!\cdots\!16$$$$p^{4} T^{18} -$$$$15\!\cdots\!44$$$$p^{6} T^{19} +$$$$23\!\cdots\!68$$$$p^{8} T^{20} + 79108787586088932 p^{11} T^{21} + 245103035882958612 p^{12} T^{22} - 2776853544451464 p^{14} T^{23} + 274189008275050 p^{16} T^{24} - 5735724829716 p^{18} T^{25} + 85370423112 p^{20} T^{26} - 1368116580 p^{22} T^{27} + 17070268 p^{24} T^{28} - 508464 p^{26} T^{29} + 9660 p^{28} T^{30} - 132 p^{30} T^{31} + p^{32} T^{32}$$
61 $$1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} -$$$$25\!\cdots\!96$$$$T^{11} +$$$$16\!\cdots\!12$$$$T^{12} -$$$$10\!\cdots\!72$$$$T^{13} +$$$$72\!\cdots\!12$$$$T^{14} -$$$$43\!\cdots\!24$$$$T^{15} +$$$$28\!\cdots\!43$$$$T^{16} -$$$$43\!\cdots\!24$$$$p^{2} T^{17} +$$$$72\!\cdots\!12$$$$p^{4} T^{18} -$$$$10\!\cdots\!72$$$$p^{6} T^{19} +$$$$16\!\cdots\!12$$$$p^{8} T^{20} -$$$$25\!\cdots\!96$$$$p^{10} T^{21} + 33635567216910203432 p^{12} T^{22} - 516522392725304640 p^{14} T^{23} + 6336184490820426 p^{16} T^{24} - 90213896412000 p^{18} T^{25} + 1094820209872 p^{20} T^{26} - 12864972960 p^{22} T^{27} + 159354980 p^{24} T^{28} - 1396992 p^{26} T^{29} + 17624 p^{28} T^{30} - 96 p^{30} T^{31} + p^{32} T^{32}$$
67 $$1 + 164 T - 6340 T^{2} - 2597968 T^{3} - 31915482 T^{4} + 22697073892 T^{5} + 967634972296 T^{6} - 123392891205548 T^{7} - 9455087268413407 T^{8} + 333882071018564764 T^{9} + 51639925158822741896 T^{10} +$$$$26\!\cdots\!32$$$$T^{11} -$$$$16\!\cdots\!86$$$$T^{12} -$$$$90\!\cdots\!08$$$$p T^{13} +$$$$18\!\cdots\!60$$$$T^{14} +$$$$15\!\cdots\!28$$$$T^{15} +$$$$54\!\cdots\!88$$$$T^{16} +$$$$15\!\cdots\!28$$$$p^{2} T^{17} +$$$$18\!\cdots\!60$$$$p^{4} T^{18} -$$$$90\!\cdots\!08$$$$p^{7} T^{19} -$$$$16\!\cdots\!86$$$$p^{8} T^{20} +$$$$26\!\cdots\!32$$$$p^{10} T^{21} + 51639925158822741896 p^{12} T^{22} + 333882071018564764 p^{14} T^{23} - 9455087268413407 p^{16} T^{24} - 123392891205548 p^{18} T^{25} + 967634972296 p^{20} T^{26} + 22697073892 p^{22} T^{27} - 31915482 p^{24} T^{28} - 2597968 p^{26} T^{29} - 6340 p^{28} T^{30} + 164 p^{30} T^{31} + p^{32} T^{32}$$
71 $$( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 217589962697708 p^{2} T^{9} + 3188424927972 p^{4} T^{10} + 28773766364 p^{6} T^{11} + 351820772 p^{8} T^{12} + 2195028 p^{10} T^{13} + 26092 p^{12} T^{14} + 68 p^{14} T^{15} + p^{16} T^{16} )^{2}$$
73 $$1 + 348 T + 82708 T^{2} + 201840 p T^{3} + 2214323678 T^{4} + 290251205388 T^{5} + 34312832193656 T^{6} + 3733330194438684 T^{7} + 379285162573239545 T^{8} + 36467064302261366820 T^{9} +$$$$33\!\cdots\!56$$$$T^{10} +$$$$40\!\cdots\!92$$$$p T^{11} +$$$$24\!\cdots\!70$$$$T^{12} +$$$$20\!\cdots\!16$$$$T^{13} +$$$$16\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!12$$$$T^{15} +$$$$91\!\cdots\!12$$$$T^{16} +$$$$12\!\cdots\!12$$$$p^{2} T^{17} +$$$$16\!\cdots\!72$$$$p^{4} T^{18} +$$$$20\!\cdots\!16$$$$p^{6} T^{19} +$$$$24\!\cdots\!70$$$$p^{8} T^{20} +$$$$40\!\cdots\!92$$$$p^{11} T^{21} +$$$$33\!\cdots\!56$$$$p^{12} T^{22} + 36467064302261366820 p^{14} T^{23} + 379285162573239545 p^{16} T^{24} + 3733330194438684 p^{18} T^{25} + 34312832193656 p^{20} T^{26} + 290251205388 p^{22} T^{27} + 2214323678 p^{24} T^{28} + 201840 p^{27} T^{29} + 82708 p^{28} T^{30} + 348 p^{30} T^{31} + p^{32} T^{32}$$
79 $$1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 14222476938584728 p T^{9} - 61265347082641448728 T^{10} +$$$$14\!\cdots\!40$$$$T^{11} +$$$$44\!\cdots\!62$$$$T^{12} -$$$$73\!\cdots\!60$$$$T^{13} +$$$$53\!\cdots\!92$$$$T^{14} -$$$$10\!\cdots\!56$$$$T^{15} +$$$$35\!\cdots\!48$$$$T^{16} -$$$$10\!\cdots\!56$$$$p^{2} T^{17} +$$$$53\!\cdots\!92$$$$p^{4} T^{18} -$$$$73\!\cdots\!60$$$$p^{6} T^{19} +$$$$44\!\cdots\!62$$$$p^{8} T^{20} +$$$$14\!\cdots\!40$$$$p^{10} T^{21} - 61265347082641448728 p^{12} T^{22} - 14222476938584728 p^{15} T^{23} + 10355496367899353 p^{16} T^{24} + 161497193433480 p^{18} T^{25} + 55399514824 p^{20} T^{26} - 30655524856 p^{22} T^{27} + 35326638 p^{24} T^{28} + 1622944 p^{26} T^{29} + 14532 p^{28} T^{30} - 280 p^{30} T^{31} + p^{32} T^{32}$$
83 $$1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} -$$$$83\!\cdots\!72$$$$T^{10} +$$$$84\!\cdots\!60$$$$T^{12} -$$$$72\!\cdots\!04$$$$T^{14} +$$$$53\!\cdots\!06$$$$T^{16} -$$$$72\!\cdots\!04$$$$p^{4} T^{18} +$$$$84\!\cdots\!60$$$$p^{8} T^{20} -$$$$83\!\cdots\!72$$$$p^{12} T^{22} + 678322851950602236 p^{16} T^{24} - 43986302935160 p^{20} T^{26} + 2113671608 p^{24} T^{28} - 66184 p^{28} T^{30} + p^{32} T^{32}$$
89 $$1 + 300 T + 54668 T^{2} + 7400400 T^{3} + 757620572 T^{4} + 53632718700 T^{5} + 870152589352 T^{6} - 480156239482500 T^{7} - 98736006237054870 T^{8} - 12632355432189921000 T^{9} -$$$$12\!\cdots\!92$$$$T^{10} -$$$$89\!\cdots\!00$$$$T^{11} -$$$$37\!\cdots\!44$$$$T^{12} +$$$$17\!\cdots\!40$$$$T^{13} +$$$$60\!\cdots\!12$$$$T^{14} +$$$$83\!\cdots\!20$$$$T^{15} +$$$$83\!\cdots\!23$$$$T^{16} +$$$$83\!\cdots\!20$$$$p^{2} T^{17} +$$$$60\!\cdots\!12$$$$p^{4} T^{18} +$$$$17\!\cdots\!40$$$$p^{6} T^{19} -$$$$37\!\cdots\!44$$$$p^{8} T^{20} -$$$$89\!\cdots\!00$$$$p^{10} T^{21} -$$$$12\!\cdots\!92$$$$p^{12} T^{22} - 12632355432189921000 p^{14} T^{23} - 98736006237054870 p^{16} T^{24} - 480156239482500 p^{18} T^{25} + 870152589352 p^{20} T^{26} + 53632718700 p^{22} T^{27} + 757620572 p^{24} T^{28} + 7400400 p^{26} T^{29} + 54668 p^{28} T^{30} + 300 p^{30} T^{31} + p^{32} T^{32}$$
97 $$1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} -$$$$21\!\cdots\!72$$$$T^{10} +$$$$29\!\cdots\!68$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$34\!\cdots\!38$$$$T^{16} -$$$$34\!\cdots\!84$$$$p^{4} T^{18} +$$$$29\!\cdots\!68$$$$p^{8} T^{20} -$$$$21\!\cdots\!72$$$$p^{12} T^{22} + 1323480529449253148 p^{16} T^{24} - 66719407643536 p^{20} T^{26} + 2604307832 p^{24} T^{28} - 70448 p^{28} T^{30} + p^{32} T^{32}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−3.31164782553778583425388238616, −3.19363976635606995269604861580, −3.16037557574744135352548187731, −3.03384971086687030987615879955, −2.96349713982430887246875798612, −2.78921908763615392791861669542, −2.73475786541821102652990224151, −2.63672782757444040459692455821, −2.26795925201286165099887601527, −2.16786917271214546274879760060, −2.08495543081594910551898217251, −2.04009372462701913533350307535, −2.02682800163684990068561994024, −1.58976726596958184663327864551, −1.52505091249007077451594289916, −1.46185547647178607828312924954, −1.40788048862622439143579048612, −1.15902809124987229929026598409, −0.824193187289460405166805549776, −0.75788115018570857711974294500, −0.63310162333397786645667231835, −0.53948927312179983093619001115, −0.44930254686225314879534086806, −0.38804118052352755605831340381, −0.35182848913325975009181206190, 0.35182848913325975009181206190, 0.38804118052352755605831340381, 0.44930254686225314879534086806, 0.53948927312179983093619001115, 0.63310162333397786645667231835, 0.75788115018570857711974294500, 0.824193187289460405166805549776, 1.15902809124987229929026598409, 1.40788048862622439143579048612, 1.46185547647178607828312924954, 1.52505091249007077451594289916, 1.58976726596958184663327864551, 2.02682800163684990068561994024, 2.04009372462701913533350307535, 2.08495543081594910551898217251, 2.16786917271214546274879760060, 2.26795925201286165099887601527, 2.63672782757444040459692455821, 2.73475786541821102652990224151, 2.78921908763615392791861669542, 2.96349713982430887246875798612, 3.03384971086687030987615879955, 3.16037557574744135352548187731, 3.19363976635606995269604861580, 3.31164782553778583425388238616

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

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