Properties

Label 32-5e64-1.1-c1e16-0-3
Degree $32$
Conductor $5.421\times 10^{44}$
Sign $1$
Analytic cond. $1.48085\times 10^{11}$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 15·4-s − 20·7-s + 35·8-s − 3·11-s + 5·13-s − 100·14-s + 73·16-s + 25·17-s + 10·19-s − 15·22-s + 15·23-s + 25·26-s − 5·27-s − 300·28-s − 8·31-s + 140·32-s + 125·34-s + 5·37-s + 50·38-s − 8·41-s − 45·44-s + 75·46-s + 5·47-s + 140·49-s + 75·52-s + 10·53-s + ⋯
L(s)  = 1  + 3.53·2-s + 15/2·4-s − 7.55·7-s + 12.3·8-s − 0.904·11-s + 1.38·13-s − 26.7·14-s + 73/4·16-s + 6.06·17-s + 2.29·19-s − 3.19·22-s + 3.12·23-s + 4.90·26-s − 0.962·27-s − 56.6·28-s − 1.43·31-s + 24.7·32-s + 21.4·34-s + 0.821·37-s + 8.11·38-s − 1.24·41-s − 6.78·44-s + 11.0·46-s + 0.729·47-s + 20·49-s + 10.4·52-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(5^{64}\)
Sign: $1$
Analytic conductor: \(1.48085\times 10^{11}\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 5^{64} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(38.35740388\)
\(L(\frac12)\) \(\approx\) \(38.35740388\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 5 T + 5 p T^{2} - 5 p T^{3} + p T^{4} + 15 T^{5} - 5 p^{3} T^{6} + 65 T^{7} - 9 p^{3} T^{8} + 15 T^{9} + 5 p^{5} T^{10} - 25 p^{4} T^{11} + 509 T^{12} - 115 p T^{13} - 525 T^{14} + 1385 T^{15} - 2105 T^{16} + 1385 p T^{17} - 525 p^{2} T^{18} - 115 p^{4} T^{19} + 509 p^{4} T^{20} - 25 p^{9} T^{21} + 5 p^{11} T^{22} + 15 p^{7} T^{23} - 9 p^{11} T^{24} + 65 p^{9} T^{25} - 5 p^{13} T^{26} + 15 p^{11} T^{27} + p^{13} T^{28} - 5 p^{14} T^{29} + 5 p^{15} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
3 \( 1 + 5 T^{3} + 7 T^{4} - 10 T^{5} + 5 T^{6} + 65 T^{7} + 8 T^{8} - 130 T^{9} + 250 T^{10} + 395 T^{11} - 89 p^{2} T^{12} - 1190 T^{13} + 2900 T^{14} + 1195 T^{15} - 14105 T^{16} + 1195 p T^{17} + 2900 p^{2} T^{18} - 1190 p^{3} T^{19} - 89 p^{6} T^{20} + 395 p^{5} T^{21} + 250 p^{6} T^{22} - 130 p^{7} T^{23} + 8 p^{8} T^{24} + 65 p^{9} T^{25} + 5 p^{10} T^{26} - 10 p^{11} T^{27} + 7 p^{12} T^{28} + 5 p^{13} T^{29} + p^{16} T^{32} \)
7 \( ( 1 + 10 T + 80 T^{2} + 65 p T^{3} + 318 p T^{4} + 9040 T^{5} + 32660 T^{6} + 102420 T^{7} + 289171 T^{8} + 102420 p T^{9} + 32660 p^{2} T^{10} + 9040 p^{3} T^{11} + 318 p^{5} T^{12} + 65 p^{6} T^{13} + 80 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
11 \( 1 + 3 T - 9 T^{2} - 15 T^{3} - 10 T^{4} - 51 p T^{5} + 57 T^{6} + 4659 T^{7} - 6525 T^{8} - 37050 T^{9} - 127560 T^{10} - 240180 T^{11} + 116740 T^{12} + 793125 T^{13} + 30432675 T^{14} + 58722705 T^{15} - 220058035 T^{16} + 58722705 p T^{17} + 30432675 p^{2} T^{18} + 793125 p^{3} T^{19} + 116740 p^{4} T^{20} - 240180 p^{5} T^{21} - 127560 p^{6} T^{22} - 37050 p^{7} T^{23} - 6525 p^{8} T^{24} + 4659 p^{9} T^{25} + 57 p^{10} T^{26} - 51 p^{12} T^{27} - 10 p^{12} T^{28} - 15 p^{13} T^{29} - 9 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \)
13 \( 1 - 5 T - 5 T^{2} + 45 T^{3} + 162 T^{4} - 435 T^{5} - 1380 T^{6} + 8115 T^{7} - 33327 T^{8} + 94725 T^{9} - 162555 T^{10} + 862905 T^{11} + 2306604 T^{12} - 11837340 T^{13} - 16579125 T^{14} - 231314780 T^{15} + 2328058495 T^{16} - 231314780 p T^{17} - 16579125 p^{2} T^{18} - 11837340 p^{3} T^{19} + 2306604 p^{4} T^{20} + 862905 p^{5} T^{21} - 162555 p^{6} T^{22} + 94725 p^{7} T^{23} - 33327 p^{8} T^{24} + 8115 p^{9} T^{25} - 1380 p^{10} T^{26} - 435 p^{11} T^{27} + 162 p^{12} T^{28} + 45 p^{13} T^{29} - 5 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
17 \( 1 - 25 T + 260 T^{2} - 5 p^{2} T^{3} + 4772 T^{4} - 11070 T^{5} + 21835 T^{6} - 76280 T^{7} + 1183053 T^{8} - 10728915 T^{9} + 172790 p^{2} T^{10} - 144218690 T^{11} + 375762864 T^{12} - 1313586580 T^{13} + 7767601200 T^{14} - 3439979095 p T^{15} + 309819923595 T^{16} - 3439979095 p^{2} T^{17} + 7767601200 p^{2} T^{18} - 1313586580 p^{3} T^{19} + 375762864 p^{4} T^{20} - 144218690 p^{5} T^{21} + 172790 p^{8} T^{22} - 10728915 p^{7} T^{23} + 1183053 p^{8} T^{24} - 76280 p^{9} T^{25} + 21835 p^{10} T^{26} - 11070 p^{11} T^{27} + 4772 p^{12} T^{28} - 5 p^{15} T^{29} + 260 p^{14} T^{30} - 25 p^{15} T^{31} + p^{16} T^{32} \)
19 \( 1 - 10 T + p T^{2} + 200 T^{3} - 1265 T^{4} + 250 p T^{5} - 11445 T^{6} - 70350 T^{7} + 864810 T^{8} - 2986300 T^{9} + 1946622 T^{10} + 44106680 T^{11} - 352684432 T^{12} + 1420694875 T^{13} - 409211820 T^{14} - 23205110225 T^{15} + 148171055515 T^{16} - 23205110225 p T^{17} - 409211820 p^{2} T^{18} + 1420694875 p^{3} T^{19} - 352684432 p^{4} T^{20} + 44106680 p^{5} T^{21} + 1946622 p^{6} T^{22} - 2986300 p^{7} T^{23} + 864810 p^{8} T^{24} - 70350 p^{9} T^{25} - 11445 p^{10} T^{26} + 250 p^{12} T^{27} - 1265 p^{12} T^{28} + 200 p^{13} T^{29} + p^{15} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
23 \( 1 - 15 T + 100 T^{2} - 390 T^{3} + 1147 T^{4} - 4740 T^{5} + 32480 T^{6} - 212625 T^{7} + 1048283 T^{8} - 4037745 T^{9} + 7507450 T^{10} + 71659875 T^{11} - 890209216 T^{12} + 5634687060 T^{13} - 25069109950 T^{14} + 80094669720 T^{15} - 265999125455 T^{16} + 80094669720 p T^{17} - 25069109950 p^{2} T^{18} + 5634687060 p^{3} T^{19} - 890209216 p^{4} T^{20} + 71659875 p^{5} T^{21} + 7507450 p^{6} T^{22} - 4037745 p^{7} T^{23} + 1048283 p^{8} T^{24} - 212625 p^{9} T^{25} + 32480 p^{10} T^{26} - 4740 p^{11} T^{27} + 1147 p^{12} T^{28} - 390 p^{13} T^{29} + 100 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \)
29 \( 1 - 66 T^{2} - 270 T^{3} + 2615 T^{4} + 9060 T^{5} - 31590 T^{6} - 228885 T^{7} + 574440 T^{8} - 2953905 T^{9} + 16797702 T^{10} + 7431375 p T^{11} + 529638328 T^{12} - 11984933085 T^{13} - 42592397205 T^{14} + 106619386380 T^{15} + 2300604709055 T^{16} + 106619386380 p T^{17} - 42592397205 p^{2} T^{18} - 11984933085 p^{3} T^{19} + 529638328 p^{4} T^{20} + 7431375 p^{6} T^{21} + 16797702 p^{6} T^{22} - 2953905 p^{7} T^{23} + 574440 p^{8} T^{24} - 228885 p^{9} T^{25} - 31590 p^{10} T^{26} + 9060 p^{11} T^{27} + 2615 p^{12} T^{28} - 270 p^{13} T^{29} - 66 p^{14} T^{30} + p^{16} T^{32} \)
31 \( 1 + 8 T - 24 T^{2} + 15 T^{3} + 3435 T^{4} + 1624 T^{5} - 111973 T^{6} + 170814 T^{7} + 2790225 T^{8} - 660500 p T^{9} - 72204210 T^{10} + 687544295 T^{11} - 990552660 T^{12} - 16376534175 T^{13} + 129865157650 T^{14} + 370971337455 T^{15} - 3153884931985 T^{16} + 370971337455 p T^{17} + 129865157650 p^{2} T^{18} - 16376534175 p^{3} T^{19} - 990552660 p^{4} T^{20} + 687544295 p^{5} T^{21} - 72204210 p^{6} T^{22} - 660500 p^{8} T^{23} + 2790225 p^{8} T^{24} + 170814 p^{9} T^{25} - 111973 p^{10} T^{26} + 1624 p^{11} T^{27} + 3435 p^{12} T^{28} + 15 p^{13} T^{29} - 24 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \)
37 \( 1 - 5 T + 35 T^{2} - 225 T^{3} + 3717 T^{4} - 16620 T^{5} + 114285 T^{6} - 579430 T^{7} + 244639 p T^{8} - 35154080 T^{9} + 283630610 T^{10} - 1507702750 T^{11} + 16364292029 T^{12} - 64949275205 T^{13} + 497018291375 T^{14} - 2230099731565 T^{15} + 24059676863795 T^{16} - 2230099731565 p T^{17} + 497018291375 p^{2} T^{18} - 64949275205 p^{3} T^{19} + 16364292029 p^{4} T^{20} - 1507702750 p^{5} T^{21} + 283630610 p^{6} T^{22} - 35154080 p^{7} T^{23} + 244639 p^{9} T^{24} - 579430 p^{9} T^{25} + 114285 p^{10} T^{26} - 16620 p^{11} T^{27} + 3717 p^{12} T^{28} - 225 p^{13} T^{29} + 35 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
41 \( 1 + 8 T - 14 T^{2} - 35 T^{3} + 1610 T^{4} + 21624 T^{5} + 138527 T^{6} + 304429 T^{7} + 4569300 T^{8} + 41329200 T^{9} + 113769490 T^{10} + 794298370 T^{11} + 355004590 p T^{12} + 95849230025 T^{13} + 523856338350 T^{14} + 2452556495155 T^{15} + 4927540217965 T^{16} + 2452556495155 p T^{17} + 523856338350 p^{2} T^{18} + 95849230025 p^{3} T^{19} + 355004590 p^{5} T^{20} + 794298370 p^{5} T^{21} + 113769490 p^{6} T^{22} + 41329200 p^{7} T^{23} + 4569300 p^{8} T^{24} + 304429 p^{9} T^{25} + 138527 p^{10} T^{26} + 21624 p^{11} T^{27} + 1610 p^{12} T^{28} - 35 p^{13} T^{29} - 14 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32} \)
43 \( ( 1 + 245 T^{2} - 180 T^{3} + 28176 T^{4} - 33015 T^{5} + 2039890 T^{6} - 2606415 T^{7} + 103354571 T^{8} - 2606415 p T^{9} + 2039890 p^{2} T^{10} - 33015 p^{3} T^{11} + 28176 p^{4} T^{12} - 180 p^{5} T^{13} + 245 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
47 \( 1 - 5 T - 100 T^{2} + 1310 T^{3} + 1202 T^{4} - 93975 T^{5} + 527350 T^{6} + 1630010 T^{7} - 34802037 T^{8} + 193245675 T^{9} + 488276750 T^{10} - 12181794550 T^{11} + 52177254249 T^{12} + 296460190195 T^{13} - 2835757066725 T^{14} - 1157173604890 T^{15} + 116382820426395 T^{16} - 1157173604890 p T^{17} - 2835757066725 p^{2} T^{18} + 296460190195 p^{3} T^{19} + 52177254249 p^{4} T^{20} - 12181794550 p^{5} T^{21} + 488276750 p^{6} T^{22} + 193245675 p^{7} T^{23} - 34802037 p^{8} T^{24} + 1630010 p^{9} T^{25} + 527350 p^{10} T^{26} - 93975 p^{11} T^{27} + 1202 p^{12} T^{28} + 1310 p^{13} T^{29} - 100 p^{14} T^{30} - 5 p^{15} T^{31} + p^{16} T^{32} \)
53 \( 1 - 10 T + 10 T^{2} - 440 T^{3} + 8192 T^{4} - 65820 T^{5} + 390995 T^{6} - 1574615 T^{7} + 27076773 T^{8} - 338568120 T^{9} + 2153851360 T^{10} - 9684081260 T^{11} + 66499418399 T^{12} - 1130858776690 T^{13} + 9852730492275 T^{14} - 44126780340755 T^{15} + 228685354261795 T^{16} - 44126780340755 p T^{17} + 9852730492275 p^{2} T^{18} - 1130858776690 p^{3} T^{19} + 66499418399 p^{4} T^{20} - 9684081260 p^{5} T^{21} + 2153851360 p^{6} T^{22} - 338568120 p^{7} T^{23} + 27076773 p^{8} T^{24} - 1574615 p^{9} T^{25} + 390995 p^{10} T^{26} - 65820 p^{11} T^{27} + 8192 p^{12} T^{28} - 440 p^{13} T^{29} + 10 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
59 \( 1 + 15 T - 36 T^{2} - 1320 T^{3} + 4115 T^{4} + 81510 T^{5} - 539115 T^{6} - 3799710 T^{7} + 58259415 T^{8} + 234817920 T^{9} - 3909789198 T^{10} - 11861716095 T^{11} + 255871984288 T^{12} + 596986886715 T^{13} - 15365412197205 T^{14} - 14379881823195 T^{15} + 922851690650855 T^{16} - 14379881823195 p T^{17} - 15365412197205 p^{2} T^{18} + 596986886715 p^{3} T^{19} + 255871984288 p^{4} T^{20} - 11861716095 p^{5} T^{21} - 3909789198 p^{6} T^{22} + 234817920 p^{7} T^{23} + 58259415 p^{8} T^{24} - 3799710 p^{9} T^{25} - 539115 p^{10} T^{26} + 81510 p^{11} T^{27} + 4115 p^{12} T^{28} - 1320 p^{13} T^{29} - 36 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \)
61 \( 1 - 17 T - 89 T^{2} + 2475 T^{3} + 3525 T^{4} - 163506 T^{5} + 124152 T^{6} + 2592984 T^{7} - 26583375 T^{8} + 282882000 T^{9} + 2857045515 T^{10} - 23749916505 T^{11} - 87129786210 T^{12} + 531768922500 T^{13} - 6796087395750 T^{14} - 3843311957645 T^{15} + 889691988734215 T^{16} - 3843311957645 p T^{17} - 6796087395750 p^{2} T^{18} + 531768922500 p^{3} T^{19} - 87129786210 p^{4} T^{20} - 23749916505 p^{5} T^{21} + 2857045515 p^{6} T^{22} + 282882000 p^{7} T^{23} - 26583375 p^{8} T^{24} + 2592984 p^{9} T^{25} + 124152 p^{10} T^{26} - 163506 p^{11} T^{27} + 3525 p^{12} T^{28} + 2475 p^{13} T^{29} - 89 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32} \)
67 \( 1 - 10 T - 195 T^{2} + 1830 T^{3} + 18207 T^{4} - 91160 T^{5} - 1518130 T^{6} - 6555735 T^{7} + 135255243 T^{8} + 1323182560 T^{9} - 139220235 p T^{10} - 95368518535 T^{11} + 235972720089 T^{12} + 3411565527645 T^{13} + 32820294659200 T^{14} - 49083954618615 T^{15} - 3984940968636805 T^{16} - 49083954618615 p T^{17} + 32820294659200 p^{2} T^{18} + 3411565527645 p^{3} T^{19} + 235972720089 p^{4} T^{20} - 95368518535 p^{5} T^{21} - 139220235 p^{7} T^{22} + 1323182560 p^{7} T^{23} + 135255243 p^{8} T^{24} - 6555735 p^{9} T^{25} - 1518130 p^{10} T^{26} - 91160 p^{11} T^{27} + 18207 p^{12} T^{28} + 1830 p^{13} T^{29} - 195 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \)
71 \( 1 + 13 T - 154 T^{2} - 1975 T^{3} + 18725 T^{4} + 219669 T^{5} - 1067228 T^{6} - 18773851 T^{7} - 4794375 T^{8} + 1588936500 T^{9} + 8864854265 T^{10} - 115243152805 T^{11} - 15882419985 p T^{12} + 6472364466625 T^{13} + 104451477100500 T^{14} - 173436705728770 T^{15} - 8017708259357385 T^{16} - 173436705728770 p T^{17} + 104451477100500 p^{2} T^{18} + 6472364466625 p^{3} T^{19} - 15882419985 p^{5} T^{20} - 115243152805 p^{5} T^{21} + 8864854265 p^{6} T^{22} + 1588936500 p^{7} T^{23} - 4794375 p^{8} T^{24} - 18773851 p^{9} T^{25} - 1067228 p^{10} T^{26} + 219669 p^{11} T^{27} + 18725 p^{12} T^{28} - 1975 p^{13} T^{29} - 154 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \)
73 \( 1 + 40 T + 535 T^{2} + 990 T^{3} - 42213 T^{4} - 442710 T^{5} - 1970010 T^{6} + 11479815 T^{7} + 500721543 T^{8} + 5688805740 T^{9} + 18909249810 T^{10} - 183341486985 T^{11} - 3105469787016 T^{12} - 25869712795740 T^{13} - 654337720650 p T^{14} + 2042924370786370 T^{15} + 28191003308880745 T^{16} + 2042924370786370 p T^{17} - 654337720650 p^{3} T^{18} - 25869712795740 p^{3} T^{19} - 3105469787016 p^{4} T^{20} - 183341486985 p^{5} T^{21} + 18909249810 p^{6} T^{22} + 5688805740 p^{7} T^{23} + 500721543 p^{8} T^{24} + 11479815 p^{9} T^{25} - 1970010 p^{10} T^{26} - 442710 p^{11} T^{27} - 42213 p^{12} T^{28} + 990 p^{13} T^{29} + 535 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \)
79 \( 1 + 55 T + 1529 T^{2} + 30795 T^{3} + 532410 T^{4} + 8346765 T^{5} + 119133630 T^{6} + 1566872135 T^{7} + 19353553285 T^{8} + 226206942955 T^{9} + 2508998830097 T^{10} + 26625850605785 T^{11} + 272123218477838 T^{12} + 2681352537356860 T^{13} + 25555470848405705 T^{14} + 236878335501934670 T^{15} + 2136290254386622115 T^{16} + 236878335501934670 p T^{17} + 25555470848405705 p^{2} T^{18} + 2681352537356860 p^{3} T^{19} + 272123218477838 p^{4} T^{20} + 26625850605785 p^{5} T^{21} + 2508998830097 p^{6} T^{22} + 226206942955 p^{7} T^{23} + 19353553285 p^{8} T^{24} + 1566872135 p^{9} T^{25} + 119133630 p^{10} T^{26} + 8346765 p^{11} T^{27} + 532410 p^{12} T^{28} + 30795 p^{13} T^{29} + 1529 p^{14} T^{30} + 55 p^{15} T^{31} + p^{16} T^{32} \)
83 \( 1 - 15 T + 45 T^{2} + 1125 T^{3} - 8488 T^{4} - 91635 T^{5} + 1295610 T^{6} + 1777665 T^{7} - 128944137 T^{8} + 363546585 T^{9} + 4847775345 T^{10} - 2170361625 T^{11} - 547928809136 T^{12} + 1267804113660 T^{13} + 33208564470975 T^{14} - 2073213208980 T^{15} - 2389863999741805 T^{16} - 2073213208980 p T^{17} + 33208564470975 p^{2} T^{18} + 1267804113660 p^{3} T^{19} - 547928809136 p^{4} T^{20} - 2170361625 p^{5} T^{21} + 4847775345 p^{6} T^{22} + 363546585 p^{7} T^{23} - 128944137 p^{8} T^{24} + 1777665 p^{9} T^{25} + 1295610 p^{10} T^{26} - 91635 p^{11} T^{27} - 8488 p^{12} T^{28} + 1125 p^{13} T^{29} + 45 p^{14} T^{30} - 15 p^{15} T^{31} + p^{16} T^{32} \)
89 \( 1 - 396 T^{2} + 1035 T^{3} + 73325 T^{4} - 412380 T^{5} - 7913205 T^{6} + 75560430 T^{7} + 429045675 T^{8} - 8473999710 T^{9} + 9586471662 T^{10} + 637211884125 T^{11} - 4060050701972 T^{12} - 31203954775695 T^{13} + 448788105606750 T^{14} + 755178159411885 T^{15} - 40194365934387715 T^{16} + 755178159411885 p T^{17} + 448788105606750 p^{2} T^{18} - 31203954775695 p^{3} T^{19} - 4060050701972 p^{4} T^{20} + 637211884125 p^{5} T^{21} + 9586471662 p^{6} T^{22} - 8473999710 p^{7} T^{23} + 429045675 p^{8} T^{24} + 75560430 p^{9} T^{25} - 7913205 p^{10} T^{26} - 412380 p^{11} T^{27} + 73325 p^{12} T^{28} + 1035 p^{13} T^{29} - 396 p^{14} T^{30} + p^{16} T^{32} \)
97 \( 1 + 40 T + 340 T^{2} - 5615 T^{3} - 85598 T^{4} + 782870 T^{5} + 24864615 T^{6} + 128648625 T^{7} - 2455749012 T^{8} - 32523730400 T^{9} + 198546895590 T^{10} + 6107136391690 T^{11} + 22790842050674 T^{12} - 467484966261655 T^{13} - 5116691236864650 T^{14} + 25645040516147735 T^{15} + 768501785039960845 T^{16} + 25645040516147735 p T^{17} - 5116691236864650 p^{2} T^{18} - 467484966261655 p^{3} T^{19} + 22790842050674 p^{4} T^{20} + 6107136391690 p^{5} T^{21} + 198546895590 p^{6} T^{22} - 32523730400 p^{7} T^{23} - 2455749012 p^{8} T^{24} + 128648625 p^{9} T^{25} + 24864615 p^{10} T^{26} + 782870 p^{11} T^{27} - 85598 p^{12} T^{28} - 5615 p^{13} T^{29} + 340 p^{14} T^{30} + 40 p^{15} T^{31} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.98412188015498957520617083238, −2.97752177659461677340225271355, −2.83903829578738914482484025586, −2.75526581529901661945563982762, −2.67786384444709217339106639135, −2.67357861739520735130646561124, −2.62592484420435067692710133639, −2.60295716894992582147628759666, −2.56553509681758768600576010165, −2.21675778125151810214622795758, −1.76857592165036992731996127612, −1.69459655730708215976404754718, −1.66380560844967162105892649339, −1.65773722714577394668865808362, −1.64580685529066488018464928618, −1.62442298547384013015034431777, −1.52471445834741546028738859973, −1.38575423585305901543249236859, −1.17494026836269908683895571392, −1.14759162953092402370397420764, −0.879908455928177496957124038651, −0.56728155851466010225408949013, −0.49665956742229486794641439132, −0.49637319074180791366100491298, −0.22215528853858164029356427333, 0.22215528853858164029356427333, 0.49637319074180791366100491298, 0.49665956742229486794641439132, 0.56728155851466010225408949013, 0.879908455928177496957124038651, 1.14759162953092402370397420764, 1.17494026836269908683895571392, 1.38575423585305901543249236859, 1.52471445834741546028738859973, 1.62442298547384013015034431777, 1.64580685529066488018464928618, 1.65773722714577394668865808362, 1.66380560844967162105892649339, 1.69459655730708215976404754718, 1.76857592165036992731996127612, 2.21675778125151810214622795758, 2.56553509681758768600576010165, 2.60295716894992582147628759666, 2.62592484420435067692710133639, 2.67357861739520735130646561124, 2.67786384444709217339106639135, 2.75526581529901661945563982762, 2.83903829578738914482484025586, 2.97752177659461677340225271355, 2.98412188015498957520617083238

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

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