Dirichlet series
| L(s) = 1 | + 4·4-s + 8·7-s − 12·11-s + 6·16-s − 12·23-s + 16·25-s + 32·28-s + 12·31-s + 4·37-s + 48·41-s − 32·43-s − 48·44-s + 34·49-s + 24·59-s − 12·61-s − 4·67-s + 36·73-s − 96·77-s + 8·79-s + 72·83-s − 24·89-s − 48·92-s + 64·100-s + 60·101-s + 12·103-s − 24·107-s + 4·109-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3.02·7-s − 3.61·11-s + 3/2·16-s − 2.50·23-s + 16/5·25-s + 6.04·28-s + 2.15·31-s + 0.657·37-s + 7.49·41-s − 4.87·43-s − 7.23·44-s + 34/7·49-s + 3.12·59-s − 1.53·61-s − 0.488·67-s + 4.21·73-s − 10.9·77-s + 0.900·79-s + 7.90·83-s − 2.54·89-s − 5.00·92-s + 32/5·100-s + 5.97·101-s + 1.18·103-s − 2.32·107-s + 0.383·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{64} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{64} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.04287\times 10^{15}\) |
| Root analytic conductor: | \(3.00915\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{64} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(126.2520805\) |
| \(L(\frac12)\) | \(\approx\) | \(126.2520805\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 8 T + 30 T^{2} - 52 T^{3} + 8 T^{4} + 48 T^{5} + 808 T^{6} - 740 p T^{7} + 17235 T^{8} - 740 p^{2} T^{9} + 808 p^{2} T^{10} + 48 p^{3} T^{11} + 8 p^{4} T^{12} - 52 p^{5} T^{13} + 30 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 5 | \( 1 - 16 T^{2} - 48 T^{3} + 132 T^{4} + 696 T^{5} + 496 T^{6} - 5184 T^{7} - 13342 T^{8} + 3936 T^{9} + 103656 T^{10} + 173544 T^{11} - 194576 T^{12} - 1477224 T^{13} - 350432 p T^{14} + 3317472 T^{15} + 17218899 T^{16} + 3317472 p T^{17} - 350432 p^{3} T^{18} - 1477224 p^{3} T^{19} - 194576 p^{4} T^{20} + 173544 p^{5} T^{21} + 103656 p^{6} T^{22} + 3936 p^{7} T^{23} - 13342 p^{8} T^{24} - 5184 p^{9} T^{25} + 496 p^{10} T^{26} + 696 p^{11} T^{27} + 132 p^{12} T^{28} - 48 p^{13} T^{29} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3816 T^{4} + 18780 T^{5} + 82292 T^{6} + 339264 T^{7} + 1295504 T^{8} + 4684368 T^{9} + 16317810 T^{10} + 53860740 T^{11} + 174905272 T^{12} + 546261648 T^{13} + 1698235882 T^{14} + 5393970432 T^{15} + 17331860115 T^{16} + 5393970432 p T^{17} + 1698235882 p^{2} T^{18} + 546261648 p^{3} T^{19} + 174905272 p^{4} T^{20} + 53860740 p^{5} T^{21} + 16317810 p^{6} T^{22} + 4684368 p^{7} T^{23} + 1295504 p^{8} T^{24} + 339264 p^{9} T^{25} + 82292 p^{10} T^{26} + 18780 p^{11} T^{27} + 3816 p^{12} T^{28} + 696 p^{13} T^{29} + 106 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 - 124 T^{2} + 7476 T^{4} - 294668 T^{6} + 8606768 T^{8} - 199664268 T^{10} + 294981004 p T^{12} - 62479093756 T^{14} + 874574952354 T^{16} - 62479093756 p^{2} T^{18} + 294981004 p^{5} T^{20} - 199664268 p^{6} T^{22} + 8606768 p^{8} T^{24} - 294668 p^{10} T^{26} + 7476 p^{12} T^{28} - 124 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 40 T^{2} - 48 T^{3} + 456 T^{4} + 168 p T^{5} - 1952 T^{6} - 68976 T^{7} + 19358 T^{8} + 1462944 T^{9} + 1631856 T^{10} - 29438184 T^{11} - 85044992 T^{12} + 269449848 T^{13} + 2356271864 T^{14} - 374736096 T^{15} - 47644701309 T^{16} - 374736096 p T^{17} + 2356271864 p^{2} T^{18} + 269449848 p^{3} T^{19} - 85044992 p^{4} T^{20} - 29438184 p^{5} T^{21} + 1631856 p^{6} T^{22} + 1462944 p^{7} T^{23} + 19358 p^{8} T^{24} - 68976 p^{9} T^{25} - 1952 p^{10} T^{26} + 168 p^{12} T^{27} + 456 p^{12} T^{28} - 48 p^{13} T^{29} - 40 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 + 50 T^{2} + 852 T^{4} - 1656 T^{5} + 7084 T^{6} - 155844 T^{7} + 102248 T^{8} - 5063112 T^{9} + 2962722 T^{10} - 74091240 T^{11} + 157206496 T^{12} - 627828084 T^{13} + 7911711962 T^{14} - 7705470168 T^{15} + 211534900083 T^{16} - 7705470168 p T^{17} + 7911711962 p^{2} T^{18} - 627828084 p^{3} T^{19} + 157206496 p^{4} T^{20} - 74091240 p^{5} T^{21} + 2962722 p^{6} T^{22} - 5063112 p^{7} T^{23} + 102248 p^{8} T^{24} - 155844 p^{9} T^{25} + 7084 p^{10} T^{26} - 1656 p^{11} T^{27} + 852 p^{12} T^{28} + 50 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( 1 + 12 T + 130 T^{2} + 984 T^{3} + 5616 T^{4} + 30192 T^{5} + 139280 T^{6} + 772452 T^{7} + 5182949 T^{8} + 32712708 T^{9} + 200130048 T^{10} + 1048916976 T^{11} + 5003487172 T^{12} + 24467647128 T^{13} + 119933762350 T^{14} + 26501305716 p T^{15} + 132680909520 p T^{16} + 26501305716 p^{2} T^{17} + 119933762350 p^{2} T^{18} + 24467647128 p^{3} T^{19} + 5003487172 p^{4} T^{20} + 1048916976 p^{5} T^{21} + 200130048 p^{6} T^{22} + 32712708 p^{7} T^{23} + 5182949 p^{8} T^{24} + 772452 p^{9} T^{25} + 139280 p^{10} T^{26} + 30192 p^{11} T^{27} + 5616 p^{12} T^{28} + 984 p^{13} T^{29} + 130 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 - 284 T^{2} + 39756 T^{4} - 3679108 T^{6} + 8746288 p T^{8} - 13850369988 T^{10} + 619661396932 T^{12} - 23152786976252 T^{14} + 729818999322498 T^{16} - 23152786976252 p^{2} T^{18} + 619661396932 p^{4} T^{20} - 13850369988 p^{6} T^{22} + 8746288 p^{9} T^{24} - 3679108 p^{10} T^{26} + 39756 p^{12} T^{28} - 284 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 13596 T^{4} - 94236 T^{5} + 595792 T^{6} - 3309624 T^{7} + 18249701 T^{8} - 82862508 T^{9} + 433124652 T^{10} - 1856453880 T^{11} + 10447382488 T^{12} - 57141936780 T^{13} + 389100682022 T^{14} - 2245519883304 T^{15} + 14416090883712 T^{16} - 2245519883304 p T^{17} + 389100682022 p^{2} T^{18} - 57141936780 p^{3} T^{19} + 10447382488 p^{4} T^{20} - 1856453880 p^{5} T^{21} + 433124652 p^{6} T^{22} - 82862508 p^{7} T^{23} + 18249701 p^{8} T^{24} - 3309624 p^{9} T^{25} + 595792 p^{10} T^{26} - 94236 p^{11} T^{27} + 13596 p^{12} T^{28} - 1608 p^{13} T^{29} + 182 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 - 4 T - 116 T^{2} + 424 T^{3} + 4172 T^{4} - 7892 T^{5} - 135576 T^{6} - 320508 T^{7} + 14036346 T^{8} - 14997408 T^{9} - 636416940 T^{10} + 40246644 p T^{11} + 11685311472 T^{12} + 8583713124 T^{13} - 774236845932 T^{14} - 1160254635264 T^{15} + 49595741554995 T^{16} - 1160254635264 p T^{17} - 774236845932 p^{2} T^{18} + 8583713124 p^{3} T^{19} + 11685311472 p^{4} T^{20} + 40246644 p^{6} T^{21} - 636416940 p^{6} T^{22} - 14997408 p^{7} T^{23} + 14036346 p^{8} T^{24} - 320508 p^{9} T^{25} - 135576 p^{10} T^{26} - 7892 p^{11} T^{27} + 4172 p^{12} T^{28} + 424 p^{13} T^{29} - 116 p^{14} T^{30} - 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 24 T + 484 T^{2} - 6648 T^{3} + 80530 T^{4} - 790176 T^{5} + 7005544 T^{6} - 52926192 T^{7} + 364770283 T^{8} - 52926192 p T^{9} + 7005544 p^{2} T^{10} - 790176 p^{3} T^{11} + 80530 p^{4} T^{12} - 6648 p^{5} T^{13} + 484 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 + 16 T + 240 T^{2} + 2312 T^{3} + 23396 T^{4} + 186216 T^{5} + 1540840 T^{6} + 10456672 T^{7} + 74455902 T^{8} + 10456672 p T^{9} + 1540840 p^{2} T^{10} + 186216 p^{3} T^{11} + 23396 p^{4} T^{12} + 2312 p^{5} T^{13} + 240 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 - 226 T^{2} + 240 T^{3} + 24432 T^{4} - 45132 T^{5} - 1872944 T^{6} + 3589488 T^{7} + 126346421 T^{8} - 162622320 T^{9} - 7848078576 T^{10} + 106116060 p T^{11} + 448293258436 T^{12} - 111771555696 T^{13} - 24155896089886 T^{14} + 1387833966576 T^{15} + 1200737631395088 T^{16} + 1387833966576 p T^{17} - 24155896089886 p^{2} T^{18} - 111771555696 p^{3} T^{19} + 448293258436 p^{4} T^{20} + 106116060 p^{6} T^{21} - 7848078576 p^{6} T^{22} - 162622320 p^{7} T^{23} + 126346421 p^{8} T^{24} + 3589488 p^{9} T^{25} - 1872944 p^{10} T^{26} - 45132 p^{11} T^{27} + 24432 p^{12} T^{28} + 240 p^{13} T^{29} - 226 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 118 T^{2} + 4392 T^{4} + 9396 T^{5} - 58252 T^{6} + 1486188 T^{7} - 11291008 T^{8} + 85807512 T^{9} - 464906718 T^{10} + 3404217132 T^{11} + 1528866784 T^{12} - 60287254128 T^{13} + 1371678087142 T^{14} - 22771366017768 T^{15} + 89253588137715 T^{16} - 22771366017768 p T^{17} + 1371678087142 p^{2} T^{18} - 60287254128 p^{3} T^{19} + 1528866784 p^{4} T^{20} + 3404217132 p^{5} T^{21} - 464906718 p^{6} T^{22} + 85807512 p^{7} T^{23} - 11291008 p^{8} T^{24} + 1486188 p^{9} T^{25} - 58252 p^{10} T^{26} + 9396 p^{11} T^{27} + 4392 p^{12} T^{28} + 118 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 24 T + 128 T^{2} + 912 T^{3} - 7500 T^{4} - 18984 T^{5} + 77824 T^{6} + 1219944 T^{7} - 11193718 T^{8} + 99638880 T^{9} - 434827392 T^{10} + 6850017624 T^{11} - 27089592368 T^{12} - 1010052031416 T^{13} + 7549935181952 T^{14} + 34562005017696 T^{15} - 648392529438477 T^{16} + 34562005017696 p T^{17} + 7549935181952 p^{2} T^{18} - 1010052031416 p^{3} T^{19} - 27089592368 p^{4} T^{20} + 6850017624 p^{5} T^{21} - 434827392 p^{6} T^{22} + 99638880 p^{7} T^{23} - 11193718 p^{8} T^{24} + 1219944 p^{9} T^{25} + 77824 p^{10} T^{26} - 18984 p^{11} T^{27} - 7500 p^{12} T^{28} + 912 p^{13} T^{29} + 128 p^{14} T^{30} - 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 + 12 T + 326 T^{2} + 3336 T^{3} + 48420 T^{4} + 463632 T^{5} + 5112220 T^{6} + 48293784 T^{7} + 481638104 T^{8} + 4344825936 T^{9} + 40778839242 T^{10} + 347025700344 T^{11} + 3071030921704 T^{12} + 25577008533204 T^{13} + 214133582320958 T^{14} + 1748216964657312 T^{15} + 13756218890275539 T^{16} + 1748216964657312 p T^{17} + 214133582320958 p^{2} T^{18} + 25577008533204 p^{3} T^{19} + 3071030921704 p^{4} T^{20} + 347025700344 p^{5} T^{21} + 40778839242 p^{6} T^{22} + 4344825936 p^{7} T^{23} + 481638104 p^{8} T^{24} + 48293784 p^{9} T^{25} + 5112220 p^{10} T^{26} + 463632 p^{11} T^{27} + 48420 p^{12} T^{28} + 3336 p^{13} T^{29} + 326 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 + 4 T - 266 T^{2} - 208 T^{3} + 36428 T^{4} - 71968 T^{5} - 2876292 T^{6} + 13972392 T^{7} + 140025096 T^{8} - 848587872 T^{9} - 5550553470 T^{10} - 14918998008 T^{11} + 663442794120 T^{12} + 5379274746876 T^{13} - 90955362013602 T^{14} - 206160027852600 T^{15} + 7873724845416051 T^{16} - 206160027852600 p T^{17} - 90955362013602 p^{2} T^{18} + 5379274746876 p^{3} T^{19} + 663442794120 p^{4} T^{20} - 14918998008 p^{5} T^{21} - 5550553470 p^{6} T^{22} - 848587872 p^{7} T^{23} + 140025096 p^{8} T^{24} + 13972392 p^{9} T^{25} - 2876292 p^{10} T^{26} - 71968 p^{11} T^{27} + 36428 p^{12} T^{28} - 208 p^{13} T^{29} - 266 p^{14} T^{30} + 4 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( 1 - 812 T^{2} + 313356 T^{4} - 76672744 T^{6} + 13415231558 T^{8} - 1799113382700 T^{10} + 194000356249216 T^{12} - 17424298416095396 T^{14} + 1334243967038097363 T^{16} - 17424298416095396 p^{2} T^{18} + 194000356249216 p^{4} T^{20} - 1799113382700 p^{6} T^{22} + 13415231558 p^{8} T^{24} - 76672744 p^{10} T^{26} + 313356 p^{12} T^{28} - 812 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 36 T + 884 T^{2} - 16272 T^{3} + 246378 T^{4} - 3227076 T^{5} + 37489192 T^{6} - 391668588 T^{7} + 3702303425 T^{8} - 31160643372 T^{9} + 227555064024 T^{10} - 1328188947516 T^{11} + 4243099305274 T^{12} + 29321606943216 T^{13} - 762446635195924 T^{14} + 9565941747927996 T^{15} - 91003246620260796 T^{16} + 9565941747927996 p T^{17} - 762446635195924 p^{2} T^{18} + 29321606943216 p^{3} T^{19} + 4243099305274 p^{4} T^{20} - 1328188947516 p^{5} T^{21} + 227555064024 p^{6} T^{22} - 31160643372 p^{7} T^{23} + 3702303425 p^{8} T^{24} - 391668588 p^{9} T^{25} + 37489192 p^{10} T^{26} - 3227076 p^{11} T^{27} + 246378 p^{12} T^{28} - 16272 p^{13} T^{29} + 884 p^{14} T^{30} - 36 p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 8 T - 326 T^{2} + 2168 T^{3} + 58148 T^{4} - 268720 T^{5} - 7215408 T^{6} + 11433060 T^{7} + 713003085 T^{8} + 1183710984 T^{9} - 54300221004 T^{10} - 250559349276 T^{11} + 2909701245216 T^{12} + 20579846269764 T^{13} - 79208223705366 T^{14} - 700296927224508 T^{15} + 942293697775632 T^{16} - 700296927224508 p T^{17} - 79208223705366 p^{2} T^{18} + 20579846269764 p^{3} T^{19} + 2909701245216 p^{4} T^{20} - 250559349276 p^{5} T^{21} - 54300221004 p^{6} T^{22} + 1183710984 p^{7} T^{23} + 713003085 p^{8} T^{24} + 11433060 p^{9} T^{25} - 7215408 p^{10} T^{26} - 268720 p^{11} T^{27} + 58148 p^{12} T^{28} + 2168 p^{13} T^{29} - 326 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( ( 1 - 36 T + 1054 T^{2} - 21000 T^{3} + 366268 T^{4} - 5167524 T^{5} + 65590558 T^{6} - 708249864 T^{7} + 6952323004 T^{8} - 708249864 p T^{9} + 65590558 p^{2} T^{10} - 5167524 p^{3} T^{11} + 366268 p^{4} T^{12} - 21000 p^{5} T^{13} + 1054 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 + 24 T - 136 T^{2} - 7632 T^{3} - 5166 T^{4} + 1342512 T^{5} + 5014480 T^{6} - 146505792 T^{7} - 847336495 T^{8} + 8332393608 T^{9} + 48032828208 T^{10} + 48296931624 T^{11} + 5796628215346 T^{12} - 49353786800040 T^{13} - 1651483987190200 T^{14} + 2587999813873344 T^{15} + 195120202249032228 T^{16} + 2587999813873344 p T^{17} - 1651483987190200 p^{2} T^{18} - 49353786800040 p^{3} T^{19} + 5796628215346 p^{4} T^{20} + 48296931624 p^{5} T^{21} + 48032828208 p^{6} T^{22} + 8332393608 p^{7} T^{23} - 847336495 p^{8} T^{24} - 146505792 p^{9} T^{25} + 5014480 p^{10} T^{26} + 1342512 p^{11} T^{27} - 5166 p^{12} T^{28} - 7632 p^{13} T^{29} - 136 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 97 | \( 1 - 568 T^{2} + 139056 T^{4} - 17617832 T^{6} + 923613860 T^{8} + 31748686824 T^{10} - 2089703902640 T^{12} - 1325450547734920 T^{14} + 222082744905287622 T^{16} - 1325450547734920 p^{2} T^{18} - 2089703902640 p^{4} T^{20} + 31748686824 p^{6} T^{22} + 923613860 p^{8} T^{24} - 17617832 p^{10} T^{26} + 139056 p^{12} T^{28} - 568 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.51973931061024706715730234955, −2.47587971793793722495401481207, −2.32006869351335170811356034019, −2.26684409219741247989526825242, −2.22179823790831670746866091927, −2.11509625332664375584995238506, −2.04447901084612146698786755496, −2.04130949627519389155360643141, −1.91447980250568572229752781679, −1.85539657053619865543120726103, −1.78838393921982300327984646247, −1.73898217309048224880899227751, −1.71951048314237182701815080214, −1.49139087475772856367278179888, −1.41007057972973329524827859369, −1.35462166848178457911300158806, −1.00601636776239015735004265159, −0.974548884073076628261355121287, −0.860563539525968035561158564846, −0.792450983460555241994094803320, −0.790443847870108281787902039344, −0.63933299752059972166933218157, −0.60212595104686745696440803918, −0.45827428602353563983753775852, −0.25553969852331027358558311858, 0.25553969852331027358558311858, 0.45827428602353563983753775852, 0.60212595104686745696440803918, 0.63933299752059972166933218157, 0.790443847870108281787902039344, 0.792450983460555241994094803320, 0.860563539525968035561158564846, 0.974548884073076628261355121287, 1.00601636776239015735004265159, 1.35462166848178457911300158806, 1.41007057972973329524827859369, 1.49139087475772856367278179888, 1.71951048314237182701815080214, 1.73898217309048224880899227751, 1.78838393921982300327984646247, 1.85539657053619865543120726103, 1.91447980250568572229752781679, 2.04130949627519389155360643141, 2.04447901084612146698786755496, 2.11509625332664375584995238506, 2.22179823790831670746866091927, 2.26684409219741247989526825242, 2.32006869351335170811356034019, 2.47587971793793722495401481207, 2.51973931061024706715730234955
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.