Dirichlet series
| L(s) = 1 | + 9·2-s − 4·3-s + 45·4-s + 9·5-s − 36·6-s − 7·7-s + 165·8-s − 9·9-s + 81·10-s − 11·11-s − 180·12-s − 14·13-s − 63·14-s − 36·15-s + 495·16-s − 13·17-s − 81·18-s − 11·19-s + 405·20-s + 28·21-s − 99·22-s − 9·23-s − 660·24-s + 45·25-s − 126·26-s + 60·27-s − 315·28-s + ⋯ |
| L(s) = 1 | + 6.36·2-s − 2.30·3-s + 45/2·4-s + 4.02·5-s − 14.6·6-s − 2.64·7-s + 58.3·8-s − 3·9-s + 25.6·10-s − 3.31·11-s − 51.9·12-s − 3.88·13-s − 16.8·14-s − 9.29·15-s + 123.·16-s − 3.15·17-s − 19.0·18-s − 2.52·19-s + 90.5·20-s + 6.11·21-s − 21.1·22-s − 1.87·23-s − 134.·24-s + 9·25-s − 24.7·26-s + 11.5·27-s − 59.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 401^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 401^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 5^{9} \cdot 401^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.53828\times 10^{13}\) |
| Root analytic conductor: | \(5.65862\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{9} \cdot 5^{9} \cdot 401^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{9} \) |
| 5 | \( ( 1 - T )^{9} \) | |
| 401 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 + 4 T + 25 T^{2} + 76 T^{3} + 31 p^{2} T^{4} + 680 T^{5} + 1849 T^{6} + 3715 T^{7} + 898 p^{2} T^{8} + 13508 T^{9} + 898 p^{3} T^{10} + 3715 p^{2} T^{11} + 1849 p^{3} T^{12} + 680 p^{4} T^{13} + 31 p^{7} T^{14} + 76 p^{6} T^{15} + 25 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + p T + 65 T^{2} + 312 T^{3} + 239 p T^{4} + 6192 T^{5} + 24524 T^{6} + 74511 T^{7} + 34241 p T^{8} + 616764 T^{9} + 34241 p^{2} T^{10} + 74511 p^{2} T^{11} + 24524 p^{3} T^{12} + 6192 p^{4} T^{13} + 239 p^{6} T^{14} + 312 p^{6} T^{15} + 65 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + p T + 124 T^{2} + 871 T^{3} + 5868 T^{4} + 30847 T^{5} + 154017 T^{6} + 644266 T^{7} + 2555584 T^{8} + 8704990 T^{9} + 2555584 p T^{10} + 644266 p^{2} T^{11} + 154017 p^{3} T^{12} + 30847 p^{4} T^{13} + 5868 p^{5} T^{14} + 871 p^{6} T^{15} + 124 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 14 T + 154 T^{2} + 1195 T^{3} + 8168 T^{4} + 46697 T^{5} + 243049 T^{6} + 1109064 T^{7} + 4662406 T^{8} + 17476066 T^{9} + 4662406 p T^{10} + 1109064 p^{2} T^{11} + 243049 p^{3} T^{12} + 46697 p^{4} T^{13} + 8168 p^{5} T^{14} + 1195 p^{6} T^{15} + 154 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 13 T + 155 T^{2} + 1254 T^{3} + 9160 T^{4} + 3252 p T^{5} + 309024 T^{6} + 1522263 T^{7} + 7096052 T^{8} + 29952602 T^{9} + 7096052 p T^{10} + 1522263 p^{2} T^{11} + 309024 p^{3} T^{12} + 3252 p^{5} T^{13} + 9160 p^{5} T^{14} + 1254 p^{6} T^{15} + 155 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 11 T + 149 T^{2} + 1181 T^{3} + 9356 T^{4} + 58968 T^{5} + 348929 T^{6} + 1847834 T^{7} + 8968293 T^{8} + 40969788 T^{9} + 8968293 p T^{10} + 1847834 p^{2} T^{11} + 348929 p^{3} T^{12} + 58968 p^{4} T^{13} + 9356 p^{5} T^{14} + 1181 p^{6} T^{15} + 149 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 9 T + 204 T^{2} + 1428 T^{3} + 18095 T^{4} + 103135 T^{5} + 939715 T^{6} + 4443747 T^{7} + 31831811 T^{8} + 125006960 T^{9} + 31831811 p T^{10} + 4443747 p^{2} T^{11} + 939715 p^{3} T^{12} + 103135 p^{4} T^{13} + 18095 p^{5} T^{14} + 1428 p^{6} T^{15} + 204 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 20 T + 368 T^{2} + 4312 T^{3} + 46906 T^{4} + 399688 T^{5} + 3211699 T^{6} + 21630457 T^{7} + 138714394 T^{8} + 764244162 T^{9} + 138714394 p T^{10} + 21630457 p^{2} T^{11} + 3211699 p^{3} T^{12} + 399688 p^{4} T^{13} + 46906 p^{5} T^{14} + 4312 p^{6} T^{15} + 368 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 11 T + 207 T^{2} + 1833 T^{3} + 20926 T^{4} + 154797 T^{5} + 1326048 T^{6} + 8304128 T^{7} + 57773680 T^{8} + 307226774 T^{9} + 57773680 p T^{10} + 8304128 p^{2} T^{11} + 1326048 p^{3} T^{12} + 154797 p^{4} T^{13} + 20926 p^{5} T^{14} + 1833 p^{6} T^{15} + 207 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 25 T + 527 T^{2} + 7600 T^{3} + 95916 T^{4} + 991756 T^{5} + 9178026 T^{6} + 73462489 T^{7} + 532196738 T^{8} + 3397008442 T^{9} + 532196738 p T^{10} + 73462489 p^{2} T^{11} + 9178026 p^{3} T^{12} + 991756 p^{4} T^{13} + 95916 p^{5} T^{14} + 7600 p^{6} T^{15} + 527 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 29 T + 563 T^{2} + 8225 T^{3} + 99790 T^{4} + 1036815 T^{5} + 9513930 T^{6} + 77954978 T^{7} + 577358404 T^{8} + 3876393614 T^{9} + 577358404 p T^{10} + 77954978 p^{2} T^{11} + 9513930 p^{3} T^{12} + 1036815 p^{4} T^{13} + 99790 p^{5} T^{14} + 8225 p^{6} T^{15} + 563 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 11 T + 388 T^{2} + 3506 T^{3} + 66348 T^{4} + 499753 T^{5} + 6611727 T^{6} + 41605289 T^{7} + 424683678 T^{8} + 2212249306 T^{9} + 424683678 p T^{10} + 41605289 p^{2} T^{11} + 6611727 p^{3} T^{12} + 499753 p^{4} T^{13} + 66348 p^{5} T^{14} + 3506 p^{6} T^{15} + 388 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 3 T + 263 T^{2} + 906 T^{3} + 31922 T^{4} + 118986 T^{5} + 2468778 T^{6} + 9354601 T^{7} + 142747902 T^{8} + 512493344 T^{9} + 142747902 p T^{10} + 9354601 p^{2} T^{11} + 2468778 p^{3} T^{12} + 118986 p^{4} T^{13} + 31922 p^{5} T^{14} + 906 p^{6} T^{15} + 263 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 9 T + 412 T^{2} + 3087 T^{3} + 77174 T^{4} + 489029 T^{5} + 8728308 T^{6} + 47009594 T^{7} + 662187581 T^{8} + 3012842556 T^{9} + 662187581 p T^{10} + 47009594 p^{2} T^{11} + 8728308 p^{3} T^{12} + 489029 p^{4} T^{13} + 77174 p^{5} T^{14} + 3087 p^{6} T^{15} + 412 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 10 T + 383 T^{2} + 3454 T^{3} + 70659 T^{4} + 570721 T^{5} + 8282388 T^{6} + 58937689 T^{7} + 677699543 T^{8} + 4158111660 T^{9} + 677699543 p T^{10} + 58937689 p^{2} T^{11} + 8282388 p^{3} T^{12} + 570721 p^{4} T^{13} + 70659 p^{5} T^{14} + 3454 p^{6} T^{15} + 383 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 10 T + 265 T^{2} + 2420 T^{3} + 38202 T^{4} + 294768 T^{5} + 3824699 T^{6} + 26084475 T^{7} + 291676029 T^{8} + 1807841310 T^{9} + 291676029 p T^{10} + 26084475 p^{2} T^{11} + 3824699 p^{3} T^{12} + 294768 p^{4} T^{13} + 38202 p^{5} T^{14} + 2420 p^{6} T^{15} + 265 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 16 T + 404 T^{2} + 4911 T^{3} + 77694 T^{4} + 806081 T^{5} + 9931975 T^{6} + 88694538 T^{7} + 900379838 T^{8} + 6936558622 T^{9} + 900379838 p T^{10} + 88694538 p^{2} T^{11} + 9931975 p^{3} T^{12} + 806081 p^{4} T^{13} + 77694 p^{5} T^{14} + 4911 p^{6} T^{15} + 404 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 8 T + 427 T^{2} + 2671 T^{3} + 81571 T^{4} + 382243 T^{5} + 9559831 T^{6} + 33024958 T^{7} + 821306390 T^{8} + 2341152544 T^{9} + 821306390 p T^{10} + 33024958 p^{2} T^{11} + 9559831 p^{3} T^{12} + 382243 p^{4} T^{13} + 81571 p^{5} T^{14} + 2671 p^{6} T^{15} + 427 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 22 T + 445 T^{2} + 5498 T^{3} + 63365 T^{4} + 546992 T^{5} + 4813565 T^{6} + 35010851 T^{7} + 303068460 T^{8} + 2291105682 T^{9} + 303068460 p T^{10} + 35010851 p^{2} T^{11} + 4813565 p^{3} T^{12} + 546992 p^{4} T^{13} + 63365 p^{5} T^{14} + 5498 p^{6} T^{15} + 445 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 9 T + 306 T^{2} + 2670 T^{3} + 49718 T^{4} + 425525 T^{5} + 6075105 T^{6} + 46257105 T^{7} + 590046696 T^{8} + 3928870030 T^{9} + 590046696 p T^{10} + 46257105 p^{2} T^{11} + 6075105 p^{3} T^{12} + 425525 p^{4} T^{13} + 49718 p^{5} T^{14} + 2670 p^{6} T^{15} + 306 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 11 T + 474 T^{2} - 3846 T^{3} + 97672 T^{4} - 547461 T^{5} + 11848717 T^{6} - 42502105 T^{7} + 1056612310 T^{8} - 2868090410 T^{9} + 1056612310 p T^{10} - 42502105 p^{2} T^{11} + 11848717 p^{3} T^{12} - 547461 p^{4} T^{13} + 97672 p^{5} T^{14} - 3846 p^{6} T^{15} + 474 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 28 T + 502 T^{2} + 6163 T^{3} + 75886 T^{4} + 967153 T^{5} + 12492667 T^{6} + 135318502 T^{7} + 1302126084 T^{8} + 11889195164 T^{9} + 1302126084 p T^{10} + 135318502 p^{2} T^{11} + 12492667 p^{3} T^{12} + 967153 p^{4} T^{13} + 75886 p^{5} T^{14} + 6163 p^{6} T^{15} + 502 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 28 T + 915 T^{2} + 17360 T^{3} + 342597 T^{4} + 5017891 T^{5} + 74248348 T^{6} + 883806883 T^{7} + 108495073 p T^{8} + 103749957330 T^{9} + 108495073 p^{2} T^{10} + 883806883 p^{2} T^{11} + 74248348 p^{3} T^{12} + 5017891 p^{4} T^{13} + 342597 p^{5} T^{14} + 17360 p^{6} T^{15} + 915 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.62584467271749170877557655588, −3.35879912011386665408072376316, −3.30641418638923445592721100779, −3.29375456169382467853291312091, −3.26473566722828181532576868142, −3.18959234610140552352245790644, −3.09911781141934324790826952024, −2.86951895118045451424872676434, −2.82707248507748980088446471343, −2.72856373318853933302494166791, −2.67975019184401300911856209571, −2.51933275589975312491599921745, −2.48606114850531776007979262503, −2.42087438207413086066587009775, −2.40499120838037847733085599115, −2.36695183766416101479600491853, −2.06635126548649286542377262228, −1.91628649896114209265902253566, −1.88381660225939977553472376374, −1.83063766644060458114333581584, −1.73304216701051429477529393921, −1.49578716787361820520992146960, −1.47179685160561711330623566148, −1.43723995056140250044135516472, −1.43217437445407826362081816543, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.43217437445407826362081816543, 1.43723995056140250044135516472, 1.47179685160561711330623566148, 1.49578716787361820520992146960, 1.73304216701051429477529393921, 1.83063766644060458114333581584, 1.88381660225939977553472376374, 1.91628649896114209265902253566, 2.06635126548649286542377262228, 2.36695183766416101479600491853, 2.40499120838037847733085599115, 2.42087438207413086066587009775, 2.48606114850531776007979262503, 2.51933275589975312491599921745, 2.67975019184401300911856209571, 2.72856373318853933302494166791, 2.82707248507748980088446471343, 2.86951895118045451424872676434, 3.09911781141934324790826952024, 3.18959234610140552352245790644, 3.26473566722828181532576868142, 3.29375456169382467853291312091, 3.30641418638923445592721100779, 3.35879912011386665408072376316, 3.62584467271749170877557655588
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.