Dirichlet series
| L(s) = 1 | + 4·2-s + 2·3-s + 10·4-s − 8·5-s + 8·6-s + 2·7-s + 20·8-s − 9-s − 32·10-s + 2·11-s + 20·12-s + 2·13-s + 8·14-s − 16·15-s + 33·16-s + 6·17-s − 4·18-s + 6·19-s − 80·20-s + 4·21-s + 8·22-s − 16·23-s + 40·24-s + 68·25-s + 8·26-s − 2·27-s + 20·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 5·4-s − 3.57·5-s + 3.26·6-s + 0.755·7-s + 7.07·8-s − 1/3·9-s − 10.1·10-s + 0.603·11-s + 5.77·12-s + 0.554·13-s + 2.13·14-s − 4.13·15-s + 33/4·16-s + 1.45·17-s − 0.942·18-s + 1.37·19-s − 17.8·20-s + 0.872·21-s + 1.70·22-s − 3.33·23-s + 8.16·24-s + 68/5·25-s + 1.56·26-s − 0.384·27-s + 3.77·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(31^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44546\times 10^{14}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 31^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8397481705\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8397481705\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 - p T + T^{2} + T^{4} - p^{4} T^{5} + 25 T^{6} - 3 p T^{7} - 3 T^{8} - 3 p^{2} T^{9} + 25 p^{2} T^{10} - p^{7} T^{11} + p^{4} T^{12} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \) |
| 3 | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 17 T^{4} - 58 T^{5} + 10 p^{2} T^{6} - 206 T^{7} + 110 p T^{8} - 550 T^{9} + 133 p^{2} T^{10} - 1550 T^{11} + 3206 T^{12} - 3404 T^{13} + 5630 T^{14} - 14596 T^{15} + 19633 T^{16} - 14596 p T^{17} + 5630 p^{2} T^{18} - 3404 p^{3} T^{19} + 3206 p^{4} T^{20} - 1550 p^{5} T^{21} + 133 p^{8} T^{22} - 550 p^{7} T^{23} + 110 p^{9} T^{24} - 206 p^{9} T^{25} + 10 p^{12} T^{26} - 58 p^{11} T^{27} + 17 p^{12} T^{28} - 10 p^{13} T^{29} + 5 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 5 | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{8} \) | |
| 7 | \( 1 - 2 T + 13 T^{2} - 6 p T^{3} + 15 p T^{4} - 282 T^{5} + 642 T^{6} - 162 p T^{7} + 1938 T^{8} - 1146 p T^{9} + 3141 T^{10} - 56838 T^{11} + 22314 p T^{12} - 45132 p T^{13} + 1613814 T^{14} - 4099436 T^{15} + 9160537 T^{16} - 4099436 p T^{17} + 1613814 p^{2} T^{18} - 45132 p^{4} T^{19} + 22314 p^{5} T^{20} - 56838 p^{5} T^{21} + 3141 p^{6} T^{22} - 1146 p^{8} T^{23} + 1938 p^{8} T^{24} - 162 p^{10} T^{25} + 642 p^{10} T^{26} - 282 p^{11} T^{27} + 15 p^{13} T^{28} - 6 p^{14} T^{29} + 13 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 - 2 T + 5 T^{2} + 2 p T^{3} + 3 p T^{4} - 1018 T^{5} + 2890 T^{6} - 30 p^{2} T^{7} - 20326 T^{8} + 2158 p T^{9} + 562509 T^{10} - 1752830 T^{11} + 295314 p T^{12} + 1215372 p T^{13} - 30565554 T^{14} - 176837172 T^{15} + 879349057 T^{16} - 176837172 p T^{17} - 30565554 p^{2} T^{18} + 1215372 p^{4} T^{19} + 295314 p^{5} T^{20} - 1752830 p^{5} T^{21} + 562509 p^{6} T^{22} + 2158 p^{8} T^{23} - 20326 p^{8} T^{24} - 30 p^{11} T^{25} + 2890 p^{10} T^{26} - 1018 p^{11} T^{27} + 3 p^{13} T^{28} + 2 p^{14} T^{29} + 5 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 2 T + 19 T^{2} - 54 T^{3} + 225 T^{4} - 1290 T^{5} + 4710 T^{6} - 24066 T^{7} + 87786 T^{8} - 412326 T^{9} + 1486395 T^{10} - 6418722 T^{11} + 23872638 T^{12} - 94373244 T^{13} + 367596522 T^{14} - 1027910924 T^{15} + 4810683361 T^{16} - 1027910924 p T^{17} + 367596522 p^{2} T^{18} - 94373244 p^{3} T^{19} + 23872638 p^{4} T^{20} - 6418722 p^{5} T^{21} + 1486395 p^{6} T^{22} - 412326 p^{7} T^{23} + 87786 p^{8} T^{24} - 24066 p^{9} T^{25} + 4710 p^{10} T^{26} - 1290 p^{11} T^{27} + 225 p^{12} T^{28} - 54 p^{13} T^{29} + 19 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 6 T + 35 T^{2} - 210 T^{3} + 937 T^{4} - 5214 T^{5} + 20590 T^{6} - 87222 T^{7} + 335090 T^{8} - 923010 T^{9} + 2216123 T^{10} + 3200850 T^{11} - 46067074 T^{12} + 378514548 T^{13} - 2113510430 T^{14} + 9586438068 T^{15} - 44495982647 T^{16} + 9586438068 p T^{17} - 2113510430 p^{2} T^{18} + 378514548 p^{3} T^{19} - 46067074 p^{4} T^{20} + 3200850 p^{5} T^{21} + 2216123 p^{6} T^{22} - 923010 p^{7} T^{23} + 335090 p^{8} T^{24} - 87222 p^{9} T^{25} + 20590 p^{10} T^{26} - 5214 p^{11} T^{27} + 937 p^{12} T^{28} - 210 p^{13} T^{29} + 35 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 - 6 T + 45 T^{2} - 366 T^{3} + 1873 T^{4} - 7374 T^{5} + 40410 T^{6} - 125370 T^{7} + 144954 T^{8} - 222546 T^{9} - 3423339 T^{10} + 49722390 T^{11} - 171136666 T^{12} + 611150316 T^{13} - 3398847426 T^{14} + 11296666596 T^{15} - 16856794783 T^{16} + 11296666596 p T^{17} - 3398847426 p^{2} T^{18} + 611150316 p^{3} T^{19} - 171136666 p^{4} T^{20} + 49722390 p^{5} T^{21} - 3423339 p^{6} T^{22} - 222546 p^{7} T^{23} + 144954 p^{8} T^{24} - 125370 p^{9} T^{25} + 40410 p^{10} T^{26} - 7374 p^{11} T^{27} + 1873 p^{12} T^{28} - 366 p^{13} T^{29} + 45 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 + 4 T - 7 T^{2} - 120 T^{3} - 319 T^{4} - 120 p T^{5} - 7 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 29 | \( ( 1 + 8 T - 2 T^{2} - 312 T^{3} - 1349 T^{4} + 520 T^{5} + 22492 T^{6} + 95040 T^{7} + 530997 T^{8} + 95040 p T^{9} + 22492 p^{2} T^{10} + 520 p^{3} T^{11} - 1349 p^{4} T^{12} - 312 p^{5} T^{13} - 2 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 11 T + p T^{2} )^{8}( 1 + 10 T + p T^{2} )^{8} \) | |
| 41 | \( 1 - 2 T + 11 T^{2} + 106 T^{3} + 1305 T^{4} - 10 p^{2} T^{5} + 76894 T^{6} - 18 p T^{7} - 1400974 T^{8} - 55718 p T^{9} + 115639683 T^{10} - 16512730 p T^{11} - 1046126754 T^{12} + 38572228860 T^{13} - 228209243598 T^{14} + 100427856732 T^{15} + 3568233222985 T^{16} + 100427856732 p T^{17} - 228209243598 p^{2} T^{18} + 38572228860 p^{3} T^{19} - 1046126754 p^{4} T^{20} - 16512730 p^{6} T^{21} + 115639683 p^{6} T^{22} - 55718 p^{8} T^{23} - 1400974 p^{8} T^{24} - 18 p^{10} T^{25} + 76894 p^{10} T^{26} - 10 p^{13} T^{27} + 1305 p^{12} T^{28} + 106 p^{13} T^{29} + 11 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 - 2 T - 11 T^{2} + 246 T^{3} + 1185 T^{4} - 14010 T^{5} - 7350 T^{6} + 726834 T^{7} - 2553414 T^{8} - 22191366 T^{9} + 191665725 T^{10} + 132958338 T^{11} - 7853905722 T^{12} + 16339825956 T^{13} + 270337696782 T^{14} + 1392379050796 T^{15} - 10540676407199 T^{16} + 1392379050796 p T^{17} + 270337696782 p^{2} T^{18} + 16339825956 p^{3} T^{19} - 7853905722 p^{4} T^{20} + 132958338 p^{5} T^{21} + 191665725 p^{6} T^{22} - 22191366 p^{7} T^{23} - 2553414 p^{8} T^{24} + 726834 p^{9} T^{25} - 7350 p^{10} T^{26} - 14010 p^{11} T^{27} + 1185 p^{12} T^{28} + 246 p^{13} T^{29} - 11 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( ( 1 + 8 T - 14 T^{2} - 360 T^{3} - 989 T^{4} + 35464 T^{5} + 256420 T^{6} - 291456 T^{7} - 6813243 T^{8} - 291456 p T^{9} + 256420 p^{2} T^{10} + 35464 p^{3} T^{11} - 989 p^{4} T^{12} - 360 p^{5} T^{13} - 14 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( 1 + 6 T + 107 T^{2} + 1074 T^{3} + 7345 T^{4} + 42366 T^{5} + 263158 T^{6} - 475242 T^{7} - 268094 p T^{8} - 137855694 T^{9} - 1456093261 T^{10} - 8790488874 T^{11} - 29076625762 T^{12} - 77795814348 T^{13} + 1839839494426 T^{14} + 24268138485252 T^{15} + 167291794983457 T^{16} + 24268138485252 p T^{17} + 1839839494426 p^{2} T^{18} - 77795814348 p^{3} T^{19} - 29076625762 p^{4} T^{20} - 8790488874 p^{5} T^{21} - 1456093261 p^{6} T^{22} - 137855694 p^{7} T^{23} - 268094 p^{9} T^{24} - 475242 p^{9} T^{25} + 263158 p^{10} T^{26} + 42366 p^{11} T^{27} + 7345 p^{12} T^{28} + 1074 p^{13} T^{29} + 107 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 6 T + 77 T^{2} + 462 T^{3} + 4129 T^{4} + 82830 T^{5} + 583306 T^{6} + 6701466 T^{7} + 41970698 T^{8} + 384443826 T^{9} + 4167138773 T^{10} + 31114278522 T^{11} + 287388283430 T^{12} + 1912728206628 T^{13} + 15855218618446 T^{14} + 144755800897452 T^{15} + 1063265432584561 T^{16} + 144755800897452 p T^{17} + 15855218618446 p^{2} T^{18} + 1912728206628 p^{3} T^{19} + 287388283430 p^{4} T^{20} + 31114278522 p^{5} T^{21} + 4167138773 p^{6} T^{22} + 384443826 p^{7} T^{23} + 41970698 p^{8} T^{24} + 6701466 p^{9} T^{25} + 583306 p^{10} T^{26} + 82830 p^{11} T^{27} + 4129 p^{12} T^{28} + 462 p^{13} T^{29} + 77 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{8} \) | |
| 67 | \( ( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 34 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 71 | \( 1 + 14 T + 141 T^{2} + 1190 T^{3} + 7785 T^{4} + 112182 T^{5} + 704258 T^{6} + 3470082 T^{7} + 5546130 T^{8} - 213643430 T^{9} - 494207419 T^{10} - 21486750166 T^{11} - 110188555034 T^{12} - 446023123700 T^{13} - 1260474127050 T^{14} + 40147513000308 T^{15} - 313622042595623 T^{16} + 40147513000308 p T^{17} - 1260474127050 p^{2} T^{18} - 446023123700 p^{3} T^{19} - 110188555034 p^{4} T^{20} - 21486750166 p^{5} T^{21} - 494207419 p^{6} T^{22} - 213643430 p^{7} T^{23} + 5546130 p^{8} T^{24} + 3470082 p^{9} T^{25} + 704258 p^{10} T^{26} + 112182 p^{11} T^{27} + 7785 p^{12} T^{28} + 1190 p^{13} T^{29} + 141 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 + 2 T + 139 T^{2} + 534 T^{3} + 6105 T^{4} + 18090 T^{5} - 334050 T^{6} - 3552414 T^{7} - 63482574 T^{8} - 420418794 T^{9} - 2974704765 T^{10} - 10208265558 T^{11} + 57608812638 T^{12} + 1274077954884 T^{13} + 12703602179442 T^{14} + 225801005799524 T^{15} + 705589419983881 T^{16} + 225801005799524 p T^{17} + 12703602179442 p^{2} T^{18} + 1274077954884 p^{3} T^{19} + 57608812638 p^{4} T^{20} - 10208265558 p^{5} T^{21} - 2974704765 p^{6} T^{22} - 420418794 p^{7} T^{23} - 63482574 p^{8} T^{24} - 3552414 p^{9} T^{25} - 334050 p^{10} T^{26} + 18090 p^{11} T^{27} + 6105 p^{12} T^{28} + 534 p^{13} T^{29} + 139 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 22 T + 261 T^{2} - 3102 T^{3} + 36249 T^{4} - 353342 T^{5} + 3094610 T^{6} - 25033866 T^{7} + 195985938 T^{8} - 1533126386 T^{9} + 14069345373 T^{10} - 154423685746 T^{11} + 1695289636998 T^{12} - 18539401088604 T^{13} + 199778213883046 T^{14} - 2014370580148740 T^{15} + 18751490544662489 T^{16} - 2014370580148740 p T^{17} + 199778213883046 p^{2} T^{18} - 18539401088604 p^{3} T^{19} + 1695289636998 p^{4} T^{20} - 154423685746 p^{5} T^{21} + 14069345373 p^{6} T^{22} - 1533126386 p^{7} T^{23} + 195985938 p^{8} T^{24} - 25033866 p^{9} T^{25} + 3094610 p^{10} T^{26} - 353342 p^{11} T^{27} + 36249 p^{12} T^{28} - 3102 p^{13} T^{29} + 261 p^{14} T^{30} - 22 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 6 T + 125 T^{2} - 1038 T^{3} + 10129 T^{4} - 154254 T^{5} + 16190 p T^{6} - 17890266 T^{7} + 159341402 T^{8} - 1904758674 T^{9} + 202851175 p T^{10} - 181139490090 T^{11} + 1707940970726 T^{12} - 16283622781620 T^{13} + 164778776820286 T^{14} - 1236787322499708 T^{15} + 13860037309386625 T^{16} - 1236787322499708 p T^{17} + 164778776820286 p^{2} T^{18} - 16283622781620 p^{3} T^{19} + 1707940970726 p^{4} T^{20} - 181139490090 p^{5} T^{21} + 202851175 p^{7} T^{22} - 1904758674 p^{7} T^{23} + 159341402 p^{8} T^{24} - 17890266 p^{9} T^{25} + 16190 p^{11} T^{26} - 154254 p^{11} T^{27} + 10129 p^{12} T^{28} - 1038 p^{13} T^{29} + 125 p^{14} T^{30} - 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 8 T - 58 T^{2} - 728 T^{3} - 973 T^{4} + 183880 T^{5} + 1530828 T^{6} - 5113024 T^{7} - 89035515 T^{8} - 5113024 p T^{9} + 1530828 p^{2} T^{10} + 183880 p^{3} T^{11} - 973 p^{4} T^{12} - 728 p^{5} T^{13} - 58 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 16 T + 6 T^{2} - 2352 T^{3} - 23709 T^{4} - 154992 T^{5} - 259380 T^{6} + 19931520 T^{7} + 366279717 T^{8} + 19931520 p T^{9} - 259380 p^{2} T^{10} - 154992 p^{3} T^{11} - 23709 p^{4} T^{12} - 2352 p^{5} T^{13} + 6 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.70427115972603332027513398030, −2.68242914531904933034964934591, −2.64234826365496096504261525769, −2.62571948921564751579302967555, −2.47343750991601483633847056129, −2.25625899737550967468626433092, −2.14603057231935480984367287060, −2.14354153499231762132222457436, −2.06359497221553230285083519516, −1.94122285192654718175878084428, −1.93493410811428612381824708104, −1.63104009543543077475178856047, −1.60814767798976006022292674068, −1.50216808947489980969764360286, −1.43756970815815464751776036167, −1.30689861872536448396890436470, −1.18143795757590436180459853829, −1.13791636131447118159080228923, −0.941521985897740002218886074829, −0.930298195712364226488538080932, −0.887120776185448112581947046236, −0.863006264726747021175235481165, −0.60183205185511512718310528073, −0.07116674740366241643913521156, −0.06802976357963801664518405716, 0.06802976357963801664518405716, 0.07116674740366241643913521156, 0.60183205185511512718310528073, 0.863006264726747021175235481165, 0.887120776185448112581947046236, 0.930298195712364226488538080932, 0.941521985897740002218886074829, 1.13791636131447118159080228923, 1.18143795757590436180459853829, 1.30689861872536448396890436470, 1.43756970815815464751776036167, 1.50216808947489980969764360286, 1.60814767798976006022292674068, 1.63104009543543077475178856047, 1.93493410811428612381824708104, 1.94122285192654718175878084428, 2.06359497221553230285083519516, 2.14354153499231762132222457436, 2.14603057231935480984367287060, 2.25625899737550967468626433092, 2.47343750991601483633847056129, 2.62571948921564751579302967555, 2.64234826365496096504261525769, 2.68242914531904933034964934591, 2.70427115972603332027513398030
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.