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

Origins of factors

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Normalization:  

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.