Dirichlet series
| L(s) = 1 | + 9·2-s − 9·3-s + 45·4-s − 4·5-s − 81·6-s − 4·7-s + 165·8-s + 45·9-s − 36·10-s − 5·11-s − 405·12-s + 9·13-s − 36·14-s + 36·15-s + 495·16-s − 6·17-s + 405·18-s − 4·19-s − 180·20-s + 36·21-s − 45·22-s − 6·23-s − 1.48e3·24-s − 20·25-s + 81·26-s − 165·27-s − 180·28-s + ⋯ |
| L(s) = 1 | + 6.36·2-s − 5.19·3-s + 45/2·4-s − 1.78·5-s − 33.0·6-s − 1.51·7-s + 58.3·8-s + 15·9-s − 11.3·10-s − 1.50·11-s − 116.·12-s + 2.49·13-s − 9.62·14-s + 9.29·15-s + 123.·16-s − 1.45·17-s + 95.4·18-s − 0.917·19-s − 40.2·20-s + 7.85·21-s − 9.59·22-s − 1.25·23-s − 303.·24-s − 4·25-s + 15.8·26-s − 31.7·27-s − 34.0·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 13^{9} \cdot 103^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 13^{9} \cdot 103^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 3^{9} \cdot 13^{9} \cdot 103^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.84026\times 10^{16}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{9} \cdot 3^{9} \cdot 13^{9} \cdot 103^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( ( 1 - T )^{9} \) |
| 3 | \( ( 1 + T )^{9} \) | |
| 13 | \( ( 1 - T )^{9} \) | |
| 103 | \( ( 1 + T )^{9} \) | |
| good | 5 | \( 1 + 4 T + 36 T^{2} + 23 p T^{3} + 592 T^{4} + 1573 T^{5} + 5987 T^{6} + 13498 T^{7} + 41626 T^{8} + 80072 T^{9} + 41626 p T^{10} + 13498 p^{2} T^{11} + 5987 p^{3} T^{12} + 1573 p^{4} T^{13} + 592 p^{5} T^{14} + 23 p^{7} T^{15} + 36 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + 4 T + 41 T^{2} + 150 T^{3} + 825 T^{4} + 2836 T^{5} + 221 p^{2} T^{6} + 34281 T^{7} + 101917 T^{8} + 285898 T^{9} + 101917 p T^{10} + 34281 p^{2} T^{11} + 221 p^{5} T^{12} + 2836 p^{4} T^{13} + 825 p^{5} T^{14} + 150 p^{6} T^{15} + 41 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 5 T + 92 T^{2} + 380 T^{3} + 3797 T^{4} + 13159 T^{5} + 93013 T^{6} + 271467 T^{7} + 1497576 T^{8} + 3653008 T^{9} + 1497576 p T^{10} + 271467 p^{2} T^{11} + 93013 p^{3} T^{12} + 13159 p^{4} T^{13} + 3797 p^{5} T^{14} + 380 p^{6} T^{15} + 92 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 6 T + 105 T^{2} + 535 T^{3} + 5230 T^{4} + 23865 T^{5} + 168151 T^{6} + 689302 T^{7} + 3862362 T^{8} + 13892394 T^{9} + 3862362 p T^{10} + 689302 p^{2} T^{11} + 168151 p^{3} T^{12} + 23865 p^{4} T^{13} + 5230 p^{5} T^{14} + 535 p^{6} T^{15} + 105 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 4 T + 89 T^{2} + 116 T^{3} + 167 p T^{4} - 2110 T^{5} + 87057 T^{6} - 93359 T^{7} + 2295634 T^{8} - 1501406 T^{9} + 2295634 p T^{10} - 93359 p^{2} T^{11} + 87057 p^{3} T^{12} - 2110 p^{4} T^{13} + 167 p^{6} T^{14} + 116 p^{6} T^{15} + 89 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 6 T + 122 T^{2} + 663 T^{3} + 7508 T^{4} + 35794 T^{5} + 303460 T^{6} + 1273572 T^{7} + 9005105 T^{8} + 33522378 T^{9} + 9005105 p T^{10} + 1273572 p^{2} T^{11} + 303460 p^{3} T^{12} + 35794 p^{4} T^{13} + 7508 p^{5} T^{14} + 663 p^{6} T^{15} + 122 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 19 T + 306 T^{2} + 3152 T^{3} + 27740 T^{4} + 185659 T^{5} + 1075537 T^{6} + 5043049 T^{7} + 22812464 T^{8} + 107108802 T^{9} + 22812464 p T^{10} + 5043049 p^{2} T^{11} + 1075537 p^{3} T^{12} + 185659 p^{4} T^{13} + 27740 p^{5} T^{14} + 3152 p^{6} T^{15} + 306 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 6 T + 173 T^{2} + 812 T^{3} + 13665 T^{4} + 51436 T^{5} + 675453 T^{6} + 2100801 T^{7} + 24902436 T^{8} + 68887654 T^{9} + 24902436 p T^{10} + 2100801 p^{2} T^{11} + 675453 p^{3} T^{12} + 51436 p^{4} T^{13} + 13665 p^{5} T^{14} + 812 p^{6} T^{15} + 173 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 13 T + 292 T^{2} + 2370 T^{3} + 30442 T^{4} + 164673 T^{5} + 1660233 T^{6} + 6131065 T^{7} + 63235570 T^{8} + 194561306 T^{9} + 63235570 p T^{10} + 6131065 p^{2} T^{11} + 1660233 p^{3} T^{12} + 164673 p^{4} T^{13} + 30442 p^{5} T^{14} + 2370 p^{6} T^{15} + 292 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 18 T + 470 T^{2} + 5998 T^{3} + 88496 T^{4} + 866722 T^{5} + 9144477 T^{6} + 71249051 T^{7} + 582284380 T^{8} + 3653553638 T^{9} + 582284380 p T^{10} + 71249051 p^{2} T^{11} + 9144477 p^{3} T^{12} + 866722 p^{4} T^{13} + 88496 p^{5} T^{14} + 5998 p^{6} T^{15} + 470 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 20 T + 341 T^{2} + 4024 T^{3} + 42622 T^{4} + 368278 T^{5} + 2977167 T^{6} + 21109515 T^{7} + 147645727 T^{8} + 959590918 T^{9} + 147645727 p T^{10} + 21109515 p^{2} T^{11} + 2977167 p^{3} T^{12} + 368278 p^{4} T^{13} + 42622 p^{5} T^{14} + 4024 p^{6} T^{15} + 341 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 14 T + 269 T^{2} - 1986 T^{3} + 22311 T^{4} - 81366 T^{5} + 835281 T^{6} + 691675 T^{7} + 16401744 T^{8} + 159981022 T^{9} + 16401744 p T^{10} + 691675 p^{2} T^{11} + 835281 p^{3} T^{12} - 81366 p^{4} T^{13} + 22311 p^{5} T^{14} - 1986 p^{6} T^{15} + 269 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 3 T + 319 T^{2} + 565 T^{3} + 47654 T^{4} + 34022 T^{5} + 4489511 T^{6} - 127050 T^{7} + 306209494 T^{8} - 83666246 T^{9} + 306209494 p T^{10} - 127050 p^{2} T^{11} + 4489511 p^{3} T^{12} + 34022 p^{4} T^{13} + 47654 p^{5} T^{14} + 565 p^{6} T^{15} + 319 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 9 T + 280 T^{2} + 2715 T^{3} + 43245 T^{4} + 413654 T^{5} + 4593623 T^{6} + 40696966 T^{7} + 359358817 T^{8} + 2826164840 T^{9} + 359358817 p T^{10} + 40696966 p^{2} T^{11} + 4593623 p^{3} T^{12} + 413654 p^{4} T^{13} + 43245 p^{5} T^{14} + 2715 p^{6} T^{15} + 280 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 24 T + 703 T^{2} + 11630 T^{3} + 196570 T^{4} + 2464330 T^{5} + 30146861 T^{6} + 298996083 T^{7} + 2852048483 T^{8} + 22752947118 T^{9} + 2852048483 p T^{10} + 298996083 p^{2} T^{11} + 30146861 p^{3} T^{12} + 2464330 p^{4} T^{13} + 196570 p^{5} T^{14} + 11630 p^{6} T^{15} + 703 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 4 T + 252 T^{2} + 1098 T^{3} + 35440 T^{4} + 133165 T^{5} + 3487683 T^{6} + 11366594 T^{7} + 275587775 T^{8} + 787775524 T^{9} + 275587775 p T^{10} + 11366594 p^{2} T^{11} + 3487683 p^{3} T^{12} + 133165 p^{4} T^{13} + 35440 p^{5} T^{14} + 1098 p^{6} T^{15} + 252 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 9 T + 330 T^{2} + 1637 T^{3} + 43158 T^{4} + 29483 T^{5} + 2830734 T^{6} - 18603230 T^{7} + 102528747 T^{8} - 2231638070 T^{9} + 102528747 p T^{10} - 18603230 p^{2} T^{11} + 2830734 p^{3} T^{12} + 29483 p^{4} T^{13} + 43158 p^{5} T^{14} + 1637 p^{6} T^{15} + 330 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 24 T + 545 T^{2} + 7036 T^{3} + 90751 T^{4} + 845619 T^{5} + 9305813 T^{6} + 83904070 T^{7} + 901932383 T^{8} + 7390785746 T^{9} + 901932383 p T^{10} + 83904070 p^{2} T^{11} + 9305813 p^{3} T^{12} + 845619 p^{4} T^{13} + 90751 p^{5} T^{14} + 7036 p^{6} T^{15} + 545 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 15 T + 382 T^{2} + 3782 T^{3} + 58990 T^{4} + 483541 T^{5} + 6600977 T^{6} + 54987597 T^{7} + 678518210 T^{8} + 5226799074 T^{9} + 678518210 p T^{10} + 54987597 p^{2} T^{11} + 6600977 p^{3} T^{12} + 483541 p^{4} T^{13} + 58990 p^{5} T^{14} + 3782 p^{6} T^{15} + 382 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 20 T + 604 T^{2} - 7955 T^{3} + 133514 T^{4} - 1257238 T^{5} + 15305398 T^{6} - 109977758 T^{7} + 1204553977 T^{8} - 8133004818 T^{9} + 1204553977 p T^{10} - 109977758 p^{2} T^{11} + 15305398 p^{3} T^{12} - 1257238 p^{4} T^{13} + 133514 p^{5} T^{14} - 7955 p^{6} T^{15} + 604 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 3 T + 350 T^{2} - 1263 T^{3} + 58339 T^{4} - 206631 T^{5} + 6828624 T^{6} - 16185020 T^{7} + 686865500 T^{8} - 1029803626 T^{9} + 686865500 p T^{10} - 16185020 p^{2} T^{11} + 6828624 p^{3} T^{12} - 206631 p^{4} T^{13} + 58339 p^{5} T^{14} - 1263 p^{6} T^{15} + 350 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 19 T + 793 T^{2} + 12520 T^{3} + 287601 T^{4} + 3773718 T^{5} + 62226404 T^{6} + 681492277 T^{7} + 8847368525 T^{8} + 80684784892 T^{9} + 8847368525 p T^{10} + 681492277 p^{2} T^{11} + 62226404 p^{3} T^{12} + 3773718 p^{4} T^{13} + 287601 p^{5} T^{14} + 12520 p^{6} T^{15} + 793 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.36667974088197718461359505387, −3.34458142798970251110341024522, −3.34004744358743922683777822681, −3.24892561459762713951666272009, −3.14778001069062404703597110003, −3.14635970046549897597116141163, −3.12681005487854165774623078699, −2.56621027395369406083038388327, −2.51258197097722731059887072870, −2.48643565088574851748182589865, −2.28334478940358831545298111656, −2.25805062126849823484771108438, −2.21887292966317093482488498579, −2.14477704181566443501137935634, −2.06039872397322528046770878509, −2.02543963369548393549655203228, −1.54738789717763171414539532389, −1.53298569203725853134364850259, −1.52145066216255556474930602595, −1.48286346400189866853131050179, −1.45115796584244731893326838365, −1.43577433646403294880035473240, −1.11446913555108819694796087138, −1.07905186996643163416606528431, −1.07306544798497275066455203342, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.07306544798497275066455203342, 1.07905186996643163416606528431, 1.11446913555108819694796087138, 1.43577433646403294880035473240, 1.45115796584244731893326838365, 1.48286346400189866853131050179, 1.52145066216255556474930602595, 1.53298569203725853134364850259, 1.54738789717763171414539532389, 2.02543963369548393549655203228, 2.06039872397322528046770878509, 2.14477704181566443501137935634, 2.21887292966317093482488498579, 2.25805062126849823484771108438, 2.28334478940358831545298111656, 2.48643565088574851748182589865, 2.51258197097722731059887072870, 2.56621027395369406083038388327, 3.12681005487854165774623078699, 3.14635970046549897597116141163, 3.14778001069062404703597110003, 3.24892561459762713951666272009, 3.34004744358743922683777822681, 3.34458142798970251110341024522, 3.36667974088197718461359505387
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.