Dirichlet series
| L(s) = 1 | + 12·5-s + 8·7-s + 12·11-s + 2·16-s + 36·17-s + 4·23-s + 78·25-s + 24·31-s + 96·35-s + 4·37-s − 8·43-s − 12·47-s + 32·49-s + 28·53-s + 144·55-s − 12·61-s + 32·67-s − 16·71-s − 12·73-s + 96·77-s + 24·80-s + 432·85-s − 60·101-s + 24·103-s − 20·107-s + 16·112-s + 8·113-s + ⋯ |
| L(s) = 1 | + 5.36·5-s + 3.02·7-s + 3.61·11-s + 1/2·16-s + 8.73·17-s + 0.834·23-s + 78/5·25-s + 4.31·31-s + 16.2·35-s + 0.657·37-s − 1.21·43-s − 1.75·47-s + 32/7·49-s + 3.84·53-s + 19.4·55-s − 1.53·61-s + 3.90·67-s − 1.89·71-s − 1.40·73-s + 10.9·77-s + 2.68·80-s + 46.8·85-s − 5.97·101-s + 2.36·103-s − 1.93·107-s + 1.51·112-s + 0.752·113-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.68221\times 10^{11}\) |
| Root analytic conductor: | \(2.24289\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1105.113289\) |
| \(L(\frac12)\) | \(\approx\) | \(1105.113289\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 12 T + 66 T^{2} - 216 T^{3} + 18 p^{2} T^{4} - 468 T^{5} - 864 T^{6} + 6252 T^{7} - 18241 T^{8} + 6252 p T^{9} - 864 p^{2} T^{10} - 468 p^{3} T^{11} + 18 p^{6} T^{12} - 216 p^{5} T^{13} + 66 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - 8 T + 32 T^{2} - 80 T^{3} + 73 T^{4} + 160 T^{5} - 416 T^{6} - 696 T^{7} + 4944 T^{8} - 696 p T^{9} - 416 p^{2} T^{10} + 160 p^{3} T^{11} + 73 p^{4} T^{12} - 80 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 11 | \( ( 1 - 6 T + 5 T^{2} + 18 T^{3} - 29 T^{4} + 252 T^{5} - 544 T^{6} + 516 T^{7} - 6650 T^{8} + 516 p T^{9} - 544 p^{2} T^{10} + 252 p^{3} T^{11} - 29 p^{4} T^{12} + 18 p^{5} T^{13} + 5 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 13 | \( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 68653746 p^{4} T^{20} + 266977 p^{8} T^{24} + 714 p^{12} T^{28} + p^{16} T^{32} \) | |
| 17 | \( 1 - 36 T + 648 T^{2} - 7776 T^{3} + 70080 T^{4} - 507420 T^{5} + 3088368 T^{6} - 16357284 T^{7} + 77768530 T^{8} - 343436904 T^{9} + 1468660248 T^{10} - 6351048828 T^{11} + 28493424864 T^{12} - 131692528476 T^{13} + 608521516920 T^{14} - 2723900163624 T^{15} + 11569567164915 T^{16} - 2723900163624 p T^{17} + 608521516920 p^{2} T^{18} - 131692528476 p^{3} T^{19} + 28493424864 p^{4} T^{20} - 6351048828 p^{5} T^{21} + 1468660248 p^{6} T^{22} - 343436904 p^{7} T^{23} + 77768530 p^{8} T^{24} - 16357284 p^{9} T^{25} + 3088368 p^{10} T^{26} - 507420 p^{11} T^{27} + 70080 p^{12} T^{28} - 7776 p^{13} T^{29} + 648 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 18780034728 p^{2} T^{18} + 1094493006 p^{4} T^{20} - 61998180 p^{6} T^{22} + 3034057 p^{8} T^{24} - 123414 p^{10} T^{26} + 4011 p^{12} T^{28} - 90 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 - 4 T + 8 T^{2} + 128 T^{3} - 198 T^{4} + 292 T^{5} + 8608 T^{6} - 3564 T^{7} - 17335 T^{8} - 1017836 T^{9} + 5500848 T^{10} + 18484 p T^{11} + 20115594 T^{12} - 929862264 T^{13} + 4560471720 T^{14} + 14572270428 T^{15} - 102219015884 T^{16} + 14572270428 p T^{17} + 4560471720 p^{2} T^{18} - 929862264 p^{3} T^{19} + 20115594 p^{4} T^{20} + 18484 p^{6} T^{21} + 5500848 p^{6} T^{22} - 1017836 p^{7} T^{23} - 17335 p^{8} T^{24} - 3564 p^{9} T^{25} + 8608 p^{10} T^{26} + 292 p^{11} T^{27} - 198 p^{12} T^{28} + 128 p^{13} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 80906 p^{2} T^{10} + 2685 p^{4} T^{12} - 70 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 2919012 p T^{9} + 489600 p^{2} T^{10} - 72996 p^{3} T^{11} + 10686 p^{4} T^{12} - 1296 p^{5} T^{13} + 156 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 4 T + 8 T^{2} + 240 T^{3} - 3049 T^{4} + 13544 T^{5} - 984 T^{6} - 514972 T^{7} + 3800209 T^{8} - 12256568 T^{9} - 16248928 T^{10} + 7999752 p T^{11} + 119383314 T^{12} - 6433305488 T^{13} + 16607500784 T^{14} + 478842888760 T^{15} - 3828375485890 T^{16} + 478842888760 p T^{17} + 16607500784 p^{2} T^{18} - 6433305488 p^{3} T^{19} + 119383314 p^{4} T^{20} + 7999752 p^{6} T^{21} - 16248928 p^{6} T^{22} - 12256568 p^{7} T^{23} + 3800209 p^{8} T^{24} - 514972 p^{9} T^{25} - 984 p^{10} T^{26} + 13544 p^{11} T^{27} - 3049 p^{12} T^{28} + 240 p^{13} T^{29} + 8 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1146544 p^{2} T^{10} + 17802 p^{4} T^{12} - 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 651228 p T^{9} + 55896 p^{2} T^{10} + 20696 p^{3} T^{11} + 3425 p^{4} T^{12} + 32 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 12 T + 72 T^{2} + 288 T^{3} - 449 T^{4} - 2880 T^{5} + 39240 T^{6} + 500820 T^{7} + 44673 T^{8} - 37644384 T^{9} - 288946944 T^{10} - 1723474464 T^{11} + 9984376994 T^{12} + 179746348152 T^{13} + 1044056404368 T^{14} + 4096130163936 T^{15} + 8350203627806 T^{16} + 4096130163936 p T^{17} + 1044056404368 p^{2} T^{18} + 179746348152 p^{3} T^{19} + 9984376994 p^{4} T^{20} - 1723474464 p^{5} T^{21} - 288946944 p^{6} T^{22} - 37644384 p^{7} T^{23} + 44673 p^{8} T^{24} + 500820 p^{9} T^{25} + 39240 p^{10} T^{26} - 2880 p^{11} T^{27} - 449 p^{12} T^{28} + 288 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 - 28 T + 392 T^{2} - 1808 T^{3} - 25321 T^{4} + 567704 T^{5} - 4335448 T^{6} - 1435172 T^{7} + 392677969 T^{8} - 4105212392 T^{9} + 14977011104 T^{10} + 124625519288 T^{11} - 2189877391278 T^{12} + 14281594759536 T^{13} - 14623477702416 T^{14} - 648224117593688 T^{15} + 7103959219709822 T^{16} - 648224117593688 p T^{17} - 14623477702416 p^{2} T^{18} + 14281594759536 p^{3} T^{19} - 2189877391278 p^{4} T^{20} + 124625519288 p^{5} T^{21} + 14977011104 p^{6} T^{22} - 4105212392 p^{7} T^{23} + 392677969 p^{8} T^{24} - 1435172 p^{9} T^{25} - 4335448 p^{10} T^{26} + 567704 p^{11} T^{27} - 25321 p^{12} T^{28} - 1808 p^{13} T^{29} + 392 p^{14} T^{30} - 28 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 105925693186240 p^{2} T^{18} + 2045329177232 p^{4} T^{20} - 36082424000 p^{6} T^{22} + 532238186 p^{8} T^{24} - 6209920 p^{10} T^{26} + 54244 p^{12} T^{28} - 320 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 9244740 p T^{9} + 1969284 p^{2} T^{10} + 119796 p^{3} T^{11} + 23079 p^{4} T^{12} + 1170 p^{5} T^{13} + 207 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 32 T + 512 T^{2} - 3896 T^{3} - 10509 T^{4} + 663104 T^{5} - 8249312 T^{6} + 44834240 T^{7} + 165963557 T^{8} - 5505231928 T^{9} + 49681319072 T^{10} - 159136747544 T^{11} - 1412717145882 T^{12} + 21627811976512 T^{13} - 103082665870656 T^{14} - 335467607852040 T^{15} + 7552610469365206 T^{16} - 335467607852040 p T^{17} - 103082665870656 p^{2} T^{18} + 21627811976512 p^{3} T^{19} - 1412717145882 p^{4} T^{20} - 159136747544 p^{5} T^{21} + 49681319072 p^{6} T^{22} - 5505231928 p^{7} T^{23} + 165963557 p^{8} T^{24} + 44834240 p^{9} T^{25} - 8249312 p^{10} T^{26} + 663104 p^{11} T^{27} - 10509 p^{12} T^{28} - 3896 p^{13} T^{29} + 512 p^{14} T^{30} - 32 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 4 T + 94 T^{2} - 964 T^{3} - 1158 T^{4} - 964 p T^{5} + 94 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 12064 T^{4} + 83700 T^{5} + 177264 T^{6} - 3613524 T^{7} + 19255890 T^{8} - 96865032 T^{9} - 2225167272 T^{10} - 39441388620 T^{11} + 6056513120 T^{12} - 1050679100940 T^{13} - 2551121523144 T^{14} - 71385024146568 T^{15} + 865631164724531 T^{16} - 71385024146568 p T^{17} - 2551121523144 p^{2} T^{18} - 1050679100940 p^{3} T^{19} + 6056513120 p^{4} T^{20} - 39441388620 p^{5} T^{21} - 2225167272 p^{6} T^{22} - 96865032 p^{7} T^{23} + 19255890 p^{8} T^{24} - 3613524 p^{9} T^{25} + 177264 p^{10} T^{26} + 83700 p^{11} T^{27} + 12064 p^{12} T^{28} + 288 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 54520165812136 p^{2} T^{18} + 2052833339792 p^{4} T^{20} + 45125337896 p^{6} T^{22} + 630756554 p^{8} T^{24} + 6614224 p^{10} T^{26} + 56836 p^{12} T^{28} + 344 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - 2957648820318 p^{4} T^{20} + 318852241 p^{8} T^{24} - 25638 p^{12} T^{28} + p^{16} T^{32} \) | |
| 89 | \( 1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 4926383904774360 p^{2} T^{18} + 51768593923950 p^{4} T^{20} - 494840689332 p^{6} T^{22} + 4091335849 p^{8} T^{24} - 27754278 p^{10} T^{26} + 144915 p^{12} T^{28} - 522 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 + 3868 T^{4} + 90482118 T^{8} + 3868 p^{4} T^{12} + p^{8} T^{16} )^{2} \) | |
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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
−2.75841717720257866238299995902, −2.72640420678823942159950493606, −2.63484634073159485401117583238, −2.60421717837181026570780493793, −2.52391863492926282631861118635, −2.51927265441559047764650928932, −2.25499030152367403283208101337, −2.16919217681946424645019821924, −2.11812145018474310328442749794, −1.99735540798851866062670092075, −1.82530149626850066439324096428, −1.79629812043631793362247417639, −1.62668413371298413934532213010, −1.60815948809842461146701914419, −1.47045975906220467538209455184, −1.44983713360727171187961998959, −1.38823176069122681008577388684, −1.37076850971702774257066682277, −1.12892687118478778412766127055, −1.11659055317809220023476540030, −1.00997864481205690381397040070, −0.981609851928948372742980243870, −0.882854889155919811856378915011, −0.74917322387410588225036565539, −0.63912672160233422288413676949, 0.63912672160233422288413676949, 0.74917322387410588225036565539, 0.882854889155919811856378915011, 0.981609851928948372742980243870, 1.00997864481205690381397040070, 1.11659055317809220023476540030, 1.12892687118478778412766127055, 1.37076850971702774257066682277, 1.38823176069122681008577388684, 1.44983713360727171187961998959, 1.47045975906220467538209455184, 1.60815948809842461146701914419, 1.62668413371298413934532213010, 1.79629812043631793362247417639, 1.82530149626850066439324096428, 1.99735540798851866062670092075, 2.11812145018474310328442749794, 2.16919217681946424645019821924, 2.25499030152367403283208101337, 2.51927265441559047764650928932, 2.52391863492926282631861118635, 2.60421717837181026570780493793, 2.63484634073159485401117583238, 2.72640420678823942159950493606, 2.75841717720257866238299995902
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.