Dirichlet series
| L(s) = 1 | + 2-s + 3-s + 3·4-s − 8·5-s + 6-s + 8·7-s + 2·8-s + 2·9-s − 8·10-s − 4·11-s + 3·12-s − 5·13-s + 8·14-s − 8·15-s + 4·16-s + 8·17-s + 2·18-s + 6·19-s − 24·20-s + 8·21-s − 4·22-s + 8·23-s + 2·24-s + 28·25-s − 5·26-s + 5·27-s + 24·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 3/2·4-s − 3.57·5-s + 0.408·6-s + 3.02·7-s + 0.707·8-s + 2/3·9-s − 2.52·10-s − 1.20·11-s + 0.866·12-s − 1.38·13-s + 2.13·14-s − 2.06·15-s + 16-s + 1.94·17-s + 0.471·18-s + 1.37·19-s − 5.36·20-s + 1.74·21-s − 0.852·22-s + 1.66·23-s + 0.408·24-s + 28/5·25-s − 0.980·26-s + 0.962·27-s + 4.53·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.56686\times 10^{6}\) |
| Root analytic conductor: | \(1.58596\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.553332165\) |
| \(L(\frac12)\) | \(\approx\) | \(4.553332165\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - T - T^{2} - 2 T^{3} + T^{4} + 2 p T^{5} + 2 p T^{6} - 5 p^{2} T^{7} + 10 p^{2} T^{8} - 5 p^{3} T^{9} + 2 p^{3} T^{10} + 2 p^{4} T^{11} + p^{4} T^{12} - 2 p^{5} T^{13} - p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T + T^{2} )^{8} \) | |
| 7 | \( ( 1 - T + T^{2} )^{8} \) | |
| good | 2 | \( 1 - T - p T^{2} + 3 T^{3} + T^{4} - 5 T^{5} - p T^{6} + 3 p T^{7} - p T^{9} + 13 T^{10} - 17 p T^{11} + 77 T^{12} + 23 p T^{13} - 33 p^{3} T^{14} - 3 p^{3} T^{15} + 125 p^{2} T^{16} - 3 p^{4} T^{17} - 33 p^{5} T^{18} + 23 p^{4} T^{19} + 77 p^{4} T^{20} - 17 p^{6} T^{21} + 13 p^{6} T^{22} - p^{8} T^{23} + 3 p^{10} T^{25} - p^{11} T^{26} - 5 p^{11} T^{27} + p^{12} T^{28} + 3 p^{13} T^{29} - p^{15} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) |
| 11 | \( 1 + 4 T - 8 T^{2} - 252 T^{3} - 884 T^{4} + 761 T^{5} + 29542 T^{6} + 109554 T^{7} + 21618 T^{8} - 2194876 T^{9} - 9004202 T^{10} - 8688626 T^{11} + 112135271 T^{12} + 523246502 T^{13} + 801407592 T^{14} - 4004853117 T^{15} - 22200406036 T^{16} - 4004853117 p T^{17} + 801407592 p^{2} T^{18} + 523246502 p^{3} T^{19} + 112135271 p^{4} T^{20} - 8688626 p^{5} T^{21} - 9004202 p^{6} T^{22} - 2194876 p^{7} T^{23} + 21618 p^{8} T^{24} + 109554 p^{9} T^{25} + 29542 p^{10} T^{26} + 761 p^{11} T^{27} - 884 p^{12} T^{28} - 252 p^{13} T^{29} - 8 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 5 T - 24 T^{2} - 129 T^{3} + 240 T^{4} + 1174 T^{5} - 1462 T^{6} - 4299 T^{7} - 36 T^{8} + 124186 T^{9} + 631224 T^{10} - 229852 p T^{11} - 15052803 T^{12} + 29465898 T^{13} + 184401460 T^{14} - 98215947 T^{15} - 1918362832 T^{16} - 98215947 p T^{17} + 184401460 p^{2} T^{18} + 29465898 p^{3} T^{19} - 15052803 p^{4} T^{20} - 229852 p^{6} T^{21} + 631224 p^{6} T^{22} + 124186 p^{7} T^{23} - 36 p^{8} T^{24} - 4299 p^{9} T^{25} - 1462 p^{10} T^{26} + 1174 p^{11} T^{27} + 240 p^{12} T^{28} - 129 p^{13} T^{29} - 24 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 - 4 T + 48 T^{2} - 60 T^{3} + 722 T^{4} + 693 T^{5} + 14099 T^{6} - 6718 T^{7} + 352974 T^{8} - 6718 p T^{9} + 14099 p^{2} T^{10} + 693 p^{3} T^{11} + 722 p^{4} T^{12} - 60 p^{5} T^{13} + 48 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 3 T + 51 T^{2} - 156 T^{3} + 1837 T^{4} - 5706 T^{5} + 46629 T^{6} - 140085 T^{7} + 980820 T^{8} - 140085 p T^{9} + 46629 p^{2} T^{10} - 5706 p^{3} T^{11} + 1837 p^{4} T^{12} - 156 p^{5} T^{13} + 51 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 8 T - 56 T^{2} + 636 T^{3} + 781 T^{4} - 20716 T^{5} + 6724 T^{6} + 401256 T^{7} - 165759 T^{8} - 5155168 T^{9} - 11298644 T^{10} + 13905856 T^{11} + 907093574 T^{12} + 431304176 T^{13} - 30428900436 T^{14} - 606821352 T^{15} + 718124779886 T^{16} - 606821352 p T^{17} - 30428900436 p^{2} T^{18} + 431304176 p^{3} T^{19} + 907093574 p^{4} T^{20} + 13905856 p^{5} T^{21} - 11298644 p^{6} T^{22} - 5155168 p^{7} T^{23} - 165759 p^{8} T^{24} + 401256 p^{9} T^{25} + 6724 p^{10} T^{26} - 20716 p^{11} T^{27} + 781 p^{12} T^{28} + 636 p^{13} T^{29} - 56 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 19 T + 128 T^{2} + 803 T^{3} + 316 p T^{4} + 53572 T^{5} + 135245 T^{6} + 1457003 T^{7} + 11306929 T^{8} + 8567345 T^{9} + 77536775 T^{10} + 2101449919 T^{11} + 3528685908 T^{12} - 3024827910 T^{13} + 382687393437 T^{14} + 1623090782861 T^{15} - 344149034518 T^{16} + 1623090782861 p T^{17} + 382687393437 p^{2} T^{18} - 3024827910 p^{3} T^{19} + 3528685908 p^{4} T^{20} + 2101449919 p^{5} T^{21} + 77536775 p^{6} T^{22} + 8567345 p^{7} T^{23} + 11306929 p^{8} T^{24} + 1457003 p^{9} T^{25} + 135245 p^{10} T^{26} + 53572 p^{11} T^{27} + 316 p^{13} T^{28} + 803 p^{13} T^{29} + 128 p^{14} T^{30} + 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - 140 T^{2} + 126 T^{3} + 9063 T^{4} - 13239 T^{5} - 402475 T^{6} + 353781 T^{7} + 16648256 T^{8} + 3630141 T^{9} - 737222427 T^{10} - 31707297 T^{11} + 29744064802 T^{12} - 15980009589 T^{13} - 962711725271 T^{14} + 358072466157 T^{15} + 28652584678461 T^{16} + 358072466157 p T^{17} - 962711725271 p^{2} T^{18} - 15980009589 p^{3} T^{19} + 29744064802 p^{4} T^{20} - 31707297 p^{5} T^{21} - 737222427 p^{6} T^{22} + 3630141 p^{7} T^{23} + 16648256 p^{8} T^{24} + 353781 p^{9} T^{25} - 402475 p^{10} T^{26} - 13239 p^{11} T^{27} + 9063 p^{12} T^{28} + 126 p^{13} T^{29} - 140 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - 21 T + 401 T^{2} - 5016 T^{3} + 57403 T^{4} - 523584 T^{5} + 4405763 T^{6} - 31274655 T^{7} + 205649488 T^{8} - 31274655 p T^{9} + 4405763 p^{2} T^{10} - 523584 p^{3} T^{11} + 57403 p^{4} T^{12} - 5016 p^{5} T^{13} + 401 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 + 20 T + 118 T^{2} - 150 T^{3} - 4985 T^{4} - 42197 T^{5} - 220145 T^{6} + 1270731 T^{7} + 23848134 T^{8} + 110309977 T^{9} + 134825125 T^{10} - 2724509833 T^{11} - 47046106654 T^{12} - 288572614127 T^{13} - 46076715579 T^{14} + 8512166036907 T^{15} + 65404558839995 T^{16} + 8512166036907 p T^{17} - 46076715579 p^{2} T^{18} - 288572614127 p^{3} T^{19} - 47046106654 p^{4} T^{20} - 2724509833 p^{5} T^{21} + 134825125 p^{6} T^{22} + 110309977 p^{7} T^{23} + 23848134 p^{8} T^{24} + 1270731 p^{9} T^{25} - 220145 p^{10} T^{26} - 42197 p^{11} T^{27} - 4985 p^{12} T^{28} - 150 p^{13} T^{29} + 118 p^{14} T^{30} + 20 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 13 T - 93 T^{2} - 1248 T^{3} + 9810 T^{4} + 69431 T^{5} - 827740 T^{6} - 2844144 T^{7} + 45303354 T^{8} + 67606073 T^{9} - 1695183309 T^{10} + 999801391 T^{11} + 52470440688 T^{12} - 81486657567 T^{13} - 1021133493305 T^{14} + 1151633415525 T^{15} + 13207491736631 T^{16} + 1151633415525 p T^{17} - 1021133493305 p^{2} T^{18} - 81486657567 p^{3} T^{19} + 52470440688 p^{4} T^{20} + 999801391 p^{5} T^{21} - 1695183309 p^{6} T^{22} + 67606073 p^{7} T^{23} + 45303354 p^{8} T^{24} - 2844144 p^{9} T^{25} - 827740 p^{10} T^{26} + 69431 p^{11} T^{27} + 9810 p^{12} T^{28} - 1248 p^{13} T^{29} - 93 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 11 T - 184 T^{2} + 2519 T^{3} + 18578 T^{4} - 328940 T^{5} - 1113262 T^{6} + 30708371 T^{7} + 18431464 T^{8} - 2157799474 T^{9} + 3826585394 T^{10} + 115855042036 T^{11} - 510755934975 T^{12} - 4424444319954 T^{13} + 38393050326972 T^{14} + 80181278073827 T^{15} - 2076366795569416 T^{16} + 80181278073827 p T^{17} + 38393050326972 p^{2} T^{18} - 4424444319954 p^{3} T^{19} - 510755934975 p^{4} T^{20} + 115855042036 p^{5} T^{21} + 3826585394 p^{6} T^{22} - 2157799474 p^{7} T^{23} + 18431464 p^{8} T^{24} + 30708371 p^{9} T^{25} - 1113262 p^{10} T^{26} - 328940 p^{11} T^{27} + 18578 p^{12} T^{28} + 2519 p^{13} T^{29} - 184 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( ( 1 + 8 T + 306 T^{2} + 2229 T^{3} + 44063 T^{4} + 295215 T^{5} + 3976442 T^{6} + 23818424 T^{7} + 249123624 T^{8} + 23818424 p T^{9} + 3976442 p^{2} T^{10} + 295215 p^{3} T^{11} + 44063 p^{4} T^{12} + 2229 p^{5} T^{13} + 306 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 7 T - 253 T^{2} - 1432 T^{3} + 34994 T^{4} + 154249 T^{5} - 3340312 T^{6} - 13153948 T^{7} + 235615798 T^{8} + 1028459735 T^{9} - 13624035805 T^{10} - 71911153079 T^{11} + 749305929396 T^{12} + 3788827420407 T^{13} - 43280647002525 T^{14} - 92100685540933 T^{15} + 2571044091919703 T^{16} - 92100685540933 p T^{17} - 43280647002525 p^{2} T^{18} + 3788827420407 p^{3} T^{19} + 749305929396 p^{4} T^{20} - 71911153079 p^{5} T^{21} - 13624035805 p^{6} T^{22} + 1028459735 p^{7} T^{23} + 235615798 p^{8} T^{24} - 13153948 p^{9} T^{25} - 3340312 p^{10} T^{26} + 154249 p^{11} T^{27} + 34994 p^{12} T^{28} - 1432 p^{13} T^{29} - 253 p^{14} T^{30} + 7 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 24 T + 19 T^{2} - 3276 T^{3} - 8256 T^{4} + 259110 T^{5} - 422515 T^{6} - 22871148 T^{7} + 156657818 T^{8} + 2168029494 T^{9} - 15387409002 T^{10} - 138383636634 T^{11} + 1199692712956 T^{12} + 4463711557584 T^{13} - 105028925164004 T^{14} - 58126091327262 T^{15} + 7596603299571471 T^{16} - 58126091327262 p T^{17} - 105028925164004 p^{2} T^{18} + 4463711557584 p^{3} T^{19} + 1199692712956 p^{4} T^{20} - 138383636634 p^{5} T^{21} - 15387409002 p^{6} T^{22} + 2168029494 p^{7} T^{23} + 156657818 p^{8} T^{24} - 22871148 p^{9} T^{25} - 422515 p^{10} T^{26} + 259110 p^{11} T^{27} - 8256 p^{12} T^{28} - 3276 p^{13} T^{29} + 19 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 16 T - 303 T^{2} - 5196 T^{3} + 65028 T^{4} + 1020572 T^{5} - 10522045 T^{6} - 137230818 T^{7} + 1450070274 T^{8} + 14061798986 T^{9} - 167697342162 T^{10} - 1081437670724 T^{11} + 16651518835248 T^{12} + 59637358262580 T^{13} - 1404138817994528 T^{14} - 1533447489642366 T^{15} + 101851891881704651 T^{16} - 1533447489642366 p T^{17} - 1404138817994528 p^{2} T^{18} + 59637358262580 p^{3} T^{19} + 16651518835248 p^{4} T^{20} - 1081437670724 p^{5} T^{21} - 167697342162 p^{6} T^{22} + 14061798986 p^{7} T^{23} + 1450070274 p^{8} T^{24} - 137230818 p^{9} T^{25} - 10522045 p^{10} T^{26} + 1020572 p^{11} T^{27} + 65028 p^{12} T^{28} - 5196 p^{13} T^{29} - 303 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 + 5 T + 278 T^{2} + 15 p T^{3} + 37904 T^{4} + 109648 T^{5} + 3610954 T^{6} + 7649988 T^{7} + 277532887 T^{8} + 7649988 p T^{9} + 3610954 p^{2} T^{10} + 109648 p^{3} T^{11} + 37904 p^{4} T^{12} + 15 p^{6} T^{13} + 278 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 10 T + 496 T^{2} - 3839 T^{3} + 108764 T^{4} - 681709 T^{5} + 14326022 T^{6} - 74456369 T^{7} + 1261625443 T^{8} - 74456369 p T^{9} + 14326022 p^{2} T^{10} - 681709 p^{3} T^{11} + 108764 p^{4} T^{12} - 3839 p^{5} T^{13} + 496 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 + 27 T - 135 T^{2} - 7902 T^{3} + 39995 T^{4} + 1914831 T^{5} - 6362838 T^{6} - 289531215 T^{7} + 1468553911 T^{8} + 37012370742 T^{9} - 224794814526 T^{10} - 3282480268776 T^{11} + 32122316535770 T^{12} + 227052617209830 T^{13} - 3341971702380213 T^{14} - 6208633901935233 T^{15} + 304269296557139782 T^{16} - 6208633901935233 p T^{17} - 3341971702380213 p^{2} T^{18} + 227052617209830 p^{3} T^{19} + 32122316535770 p^{4} T^{20} - 3282480268776 p^{5} T^{21} - 224794814526 p^{6} T^{22} + 37012370742 p^{7} T^{23} + 1468553911 p^{8} T^{24} - 289531215 p^{9} T^{25} - 6362838 p^{10} T^{26} + 1914831 p^{11} T^{27} + 39995 p^{12} T^{28} - 7902 p^{13} T^{29} - 135 p^{14} T^{30} + 27 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 + 5 T - 412 T^{2} - 1919 T^{3} + 85319 T^{4} + 351005 T^{5} - 12811975 T^{6} - 43639307 T^{7} + 1623479161 T^{8} + 4403947840 T^{9} - 179310486826 T^{10} - 369180098536 T^{11} + 17626003433352 T^{12} + 22874863773276 T^{13} - 1609250765486967 T^{14} - 692777265967880 T^{15} + 138259235603031206 T^{16} - 692777265967880 p T^{17} - 1609250765486967 p^{2} T^{18} + 22874863773276 p^{3} T^{19} + 17626003433352 p^{4} T^{20} - 369180098536 p^{5} T^{21} - 179310486826 p^{6} T^{22} + 4403947840 p^{7} T^{23} + 1623479161 p^{8} T^{24} - 43639307 p^{9} T^{25} - 12811975 p^{10} T^{26} + 351005 p^{11} T^{27} + 85319 p^{12} T^{28} - 1919 p^{13} T^{29} - 412 p^{14} T^{30} + 5 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 27 T + 659 T^{2} - 11088 T^{3} + 177613 T^{4} - 2313648 T^{5} + 28600757 T^{6} - 303824349 T^{7} + 3074549044 T^{8} - 303824349 p T^{9} + 28600757 p^{2} T^{10} - 2313648 p^{3} T^{11} + 177613 p^{4} T^{12} - 11088 p^{5} T^{13} + 659 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 27 T + 15 T^{2} - 9096 T^{3} - 108991 T^{4} + 887943 T^{5} + 28936428 T^{6} + 99887319 T^{7} - 3490694579 T^{8} - 39893502474 T^{9} + 151428852180 T^{10} + 5743284567084 T^{11} + 21023505312788 T^{12} - 484348205982834 T^{13} - 5169233095099629 T^{14} + 18335501604402927 T^{15} + 619275518773189270 T^{16} + 18335501604402927 p T^{17} - 5169233095099629 p^{2} T^{18} - 484348205982834 p^{3} T^{19} + 21023505312788 p^{4} T^{20} + 5743284567084 p^{5} T^{21} + 151428852180 p^{6} T^{22} - 39893502474 p^{7} T^{23} - 3490694579 p^{8} T^{24} + 99887319 p^{9} T^{25} + 28936428 p^{10} T^{26} + 887943 p^{11} T^{27} - 108991 p^{12} T^{28} - 9096 p^{13} T^{29} + 15 p^{14} T^{30} + 27 p^{15} T^{31} + p^{16} T^{32} \) | |
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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
−3.29768806550002899434182306883, −3.16963800554142448498586520209, −3.16060463734822689001068104319, −3.00382605937250501703355324431, −2.92951599400670932751166921353, −2.91641075734738881109185083393, −2.85306029586688379401149233907, −2.70013145683981747727626763024, −2.60126898871472014201299932480, −2.54596026367671654381809706839, −2.42827566217118307690145466759, −2.32535085296588022991597840837, −2.09074875150293608343684261772, −1.95563545243913091910465272150, −1.82806945629748203288000594548, −1.80321855774758697661331961645, −1.70687077936748302190932745042, −1.63048048261139953603470622943, −1.40559943128066702048112651663, −1.39241502448521232981972147789, −1.08642959768917448214648727422, −1.03839799123368340146022468766, −0.77817113777242596726804053584, −0.57546271130084498119140163393, −0.22429190384785843888969560176, 0.22429190384785843888969560176, 0.57546271130084498119140163393, 0.77817113777242596726804053584, 1.03839799123368340146022468766, 1.08642959768917448214648727422, 1.39241502448521232981972147789, 1.40559943128066702048112651663, 1.63048048261139953603470622943, 1.70687077936748302190932745042, 1.80321855774758697661331961645, 1.82806945629748203288000594548, 1.95563545243913091910465272150, 2.09074875150293608343684261772, 2.32535085296588022991597840837, 2.42827566217118307690145466759, 2.54596026367671654381809706839, 2.60126898871472014201299932480, 2.70013145683981747727626763024, 2.85306029586688379401149233907, 2.91641075734738881109185083393, 2.92951599400670932751166921353, 3.00382605937250501703355324431, 3.16060463734822689001068104319, 3.16963800554142448498586520209, 3.29768806550002899434182306883
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.