Dirichlet series
| L(s) = 1 | − 9·2-s − 5·3-s + 45·4-s − 5-s + 45·6-s + 13·7-s − 165·8-s + 4·9-s + 9·10-s + 13·11-s − 225·12-s − 117·14-s + 5·15-s + 495·16-s − 12·17-s − 36·18-s − 9·19-s − 45·20-s − 65·21-s − 117·22-s − 22·23-s + 825·24-s − 20·25-s + 18·27-s + 585·28-s + 12·29-s − 45·30-s + ⋯ |
| L(s) = 1 | − 6.36·2-s − 2.88·3-s + 45/2·4-s − 0.447·5-s + 18.3·6-s + 4.91·7-s − 58.3·8-s + 4/3·9-s + 2.84·10-s + 3.91·11-s − 64.9·12-s − 31.2·14-s + 1.29·15-s + 123.·16-s − 2.91·17-s − 8.48·18-s − 2.06·19-s − 10.0·20-s − 14.1·21-s − 24.9·22-s − 4.58·23-s + 168.·24-s − 4·25-s + 3.46·27-s + 110.·28-s + 2.22·29-s − 8.21·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{18} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{18} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 13^{18} \cdot 19^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45203\times 10^{15}\) |
| Root analytic conductor: | \(7.16100\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 13^{18} \cdot 19^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07790097630\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07790097630\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{9} \) |
| 13 | \( 1 \) | |
| 19 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 + 5 T + 7 p T^{2} + 67 T^{3} + 187 T^{4} + 442 T^{5} + 968 T^{6} + 637 p T^{7} + 400 p^{2} T^{8} + 6311 T^{9} + 400 p^{3} T^{10} + 637 p^{3} T^{11} + 968 p^{3} T^{12} + 442 p^{4} T^{13} + 187 p^{5} T^{14} + 67 p^{6} T^{15} + 7 p^{8} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + T + 21 T^{2} + 19 T^{3} + 178 T^{4} + 134 T^{5} + 27 p^{2} T^{6} + 373 T^{7} + 129 p T^{8} + 529 T^{9} + 129 p^{2} T^{10} + 373 p^{2} T^{11} + 27 p^{5} T^{12} + 134 p^{4} T^{13} + 178 p^{5} T^{14} + 19 p^{6} T^{15} + 21 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 - 13 T + 115 T^{2} - 106 p T^{3} + 3932 T^{4} - 17561 T^{5} + 68433 T^{6} - 33689 p T^{7} + 14864 p^{2} T^{8} - 289591 p T^{9} + 14864 p^{3} T^{10} - 33689 p^{3} T^{11} + 68433 p^{3} T^{12} - 17561 p^{4} T^{13} + 3932 p^{5} T^{14} - 106 p^{7} T^{15} + 115 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 13 T + 118 T^{2} - 725 T^{3} + 3628 T^{4} - 13856 T^{5} + 44079 T^{6} - 105121 T^{7} + 225932 T^{8} - 504335 T^{9} + 225932 p T^{10} - 105121 p^{2} T^{11} + 44079 p^{3} T^{12} - 13856 p^{4} T^{13} + 3628 p^{5} T^{14} - 725 p^{6} T^{15} + 118 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 12 T + 113 T^{2} + 820 T^{3} + 5292 T^{4} + 30033 T^{5} + 160627 T^{6} + 778066 T^{7} + 3578622 T^{8} + 15195769 T^{9} + 3578622 p T^{10} + 778066 p^{2} T^{11} + 160627 p^{3} T^{12} + 30033 p^{4} T^{13} + 5292 p^{5} T^{14} + 820 p^{6} T^{15} + 113 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 22 T + 310 T^{2} + 3272 T^{3} + 29305 T^{4} + 225483 T^{5} + 1539765 T^{6} + 9383073 T^{7} + 52011739 T^{8} + 261164297 T^{9} + 52011739 p T^{10} + 9383073 p^{2} T^{11} + 1539765 p^{3} T^{12} + 225483 p^{4} T^{13} + 29305 p^{5} T^{14} + 3272 p^{6} T^{15} + 310 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 12 T + 180 T^{2} - 1594 T^{3} + 14826 T^{4} - 112089 T^{5} + 812656 T^{6} - 5286719 T^{7} + 31755262 T^{8} - 178245323 T^{9} + 31755262 p T^{10} - 5286719 p^{2} T^{11} + 812656 p^{3} T^{12} - 112089 p^{4} T^{13} + 14826 p^{5} T^{14} - 1594 p^{6} T^{15} + 180 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + T + 135 T^{2} + 101 T^{3} + 9072 T^{4} + 3481 T^{5} + 420195 T^{6} - 13241 T^{7} + 15372662 T^{8} - 3306377 T^{9} + 15372662 p T^{10} - 13241 p^{2} T^{11} + 420195 p^{3} T^{12} + 3481 p^{4} T^{13} + 9072 p^{5} T^{14} + 101 p^{6} T^{15} + 135 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 25 T + 562 T^{2} - 8233 T^{3} + 108663 T^{4} - 1136556 T^{5} + 10839476 T^{6} - 87034883 T^{7} + 641794907 T^{8} - 4069038161 T^{9} + 641794907 p T^{10} - 87034883 p^{2} T^{11} + 10839476 p^{3} T^{12} - 1136556 p^{4} T^{13} + 108663 p^{5} T^{14} - 8233 p^{6} T^{15} + 562 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 11 T + 219 T^{2} + 1634 T^{3} + 21246 T^{4} + 128835 T^{5} + 1401890 T^{6} + 177539 p T^{7} + 70048476 T^{8} + 323514265 T^{9} + 70048476 p T^{10} + 177539 p^{3} T^{11} + 1401890 p^{3} T^{12} + 128835 p^{4} T^{13} + 21246 p^{5} T^{14} + 1634 p^{6} T^{15} + 219 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 10 T + 259 T^{2} + 1914 T^{3} + 31030 T^{4} + 188859 T^{5} + 2420292 T^{6} + 12659674 T^{7} + 137661972 T^{8} + 627670231 T^{9} + 137661972 p T^{10} + 12659674 p^{2} T^{11} + 2420292 p^{3} T^{12} + 188859 p^{4} T^{13} + 31030 p^{5} T^{14} + 1914 p^{6} T^{15} + 259 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 12 T + 313 T^{2} - 3035 T^{3} + 45516 T^{4} - 364388 T^{5} + 4113073 T^{6} - 595645 p T^{7} + 260760486 T^{8} - 1534786273 T^{9} + 260760486 p T^{10} - 595645 p^{3} T^{11} + 4113073 p^{3} T^{12} - 364388 p^{4} T^{13} + 45516 p^{5} T^{14} - 3035 p^{6} T^{15} + 313 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 9 T + 240 T^{2} - 1607 T^{3} + 23547 T^{4} - 117969 T^{5} + 1236063 T^{6} - 4492884 T^{7} + 43810099 T^{8} - 150409425 T^{9} + 43810099 p T^{10} - 4492884 p^{2} T^{11} + 1236063 p^{3} T^{12} - 117969 p^{4} T^{13} + 23547 p^{5} T^{14} - 1607 p^{6} T^{15} + 240 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 10 T + 437 T^{2} + 3303 T^{3} + 84318 T^{4} + 504683 T^{5} + 9860397 T^{6} + 820227 p T^{7} + 795920121 T^{8} + 3309982211 T^{9} + 795920121 p T^{10} + 820227 p^{3} T^{11} + 9860397 p^{3} T^{12} + 504683 p^{4} T^{13} + 84318 p^{5} T^{14} + 3303 p^{6} T^{15} + 437 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 32 T + 714 T^{2} - 11175 T^{3} + 145071 T^{4} - 1547115 T^{5} + 14600924 T^{6} - 122250303 T^{7} + 979184361 T^{8} - 7584870765 T^{9} + 979184361 p T^{10} - 122250303 p^{2} T^{11} + 14600924 p^{3} T^{12} - 1547115 p^{4} T^{13} + 145071 p^{5} T^{14} - 11175 p^{6} T^{15} + 714 p^{7} T^{16} - 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 73 T + 2902 T^{2} - 80000 T^{3} + 1690169 T^{4} - 28780040 T^{5} + 407075371 T^{6} - 4871057891 T^{7} + 49872487909 T^{8} - 439549952007 T^{9} + 49872487909 p T^{10} - 4871057891 p^{2} T^{11} + 407075371 p^{3} T^{12} - 28780040 p^{4} T^{13} + 1690169 p^{5} T^{14} - 80000 p^{6} T^{15} + 2902 p^{7} T^{16} - 73 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 51 T + 1426 T^{2} - 26665 T^{3} + 359184 T^{4} - 3437597 T^{5} + 20264179 T^{6} - 1963657 T^{7} - 1482129355 T^{8} + 17932294083 T^{9} - 1482129355 p T^{10} - 1963657 p^{2} T^{11} + 20264179 p^{3} T^{12} - 3437597 p^{4} T^{13} + 359184 p^{5} T^{14} - 26665 p^{6} T^{15} + 1426 p^{7} T^{16} - 51 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 14 T + 468 T^{2} - 6114 T^{3} + 107725 T^{4} - 1237871 T^{5} + 15791240 T^{6} - 155325148 T^{7} + 1607566890 T^{8} - 13446211601 T^{9} + 1607566890 p T^{10} - 155325148 p^{2} T^{11} + 15791240 p^{3} T^{12} - 1237871 p^{4} T^{13} + 107725 p^{5} T^{14} - 6114 p^{6} T^{15} + 468 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 28 T + 735 T^{2} + 171 p T^{3} + 220633 T^{4} + 3053806 T^{5} + 38302244 T^{6} + 427340522 T^{7} + 4378584196 T^{8} + 40541907937 T^{9} + 4378584196 p T^{10} + 427340522 p^{2} T^{11} + 38302244 p^{3} T^{12} + 3053806 p^{4} T^{13} + 220633 p^{5} T^{14} + 171 p^{7} T^{15} + 735 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 22 T + 558 T^{2} - 7469 T^{3} + 100271 T^{4} - 858471 T^{5} + 6559033 T^{6} - 23933769 T^{7} - 24298885 T^{8} + 1402330001 T^{9} - 24298885 p T^{10} - 23933769 p^{2} T^{11} + 6559033 p^{3} T^{12} - 858471 p^{4} T^{13} + 100271 p^{5} T^{14} - 7469 p^{6} T^{15} + 558 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 3 T + 525 T^{2} + 277 T^{3} + 124456 T^{4} - 181814 T^{5} + 18817829 T^{6} - 49978054 T^{7} + 2121541234 T^{8} - 6001725747 T^{9} + 2121541234 p T^{10} - 49978054 p^{2} T^{11} + 18817829 p^{3} T^{12} - 181814 p^{4} T^{13} + 124456 p^{5} T^{14} + 277 p^{6} T^{15} + 525 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 677 T^{2} + 39 T^{3} + 217610 T^{4} + 9629 T^{5} + 43488651 T^{6} + 982720 T^{7} + 5946774468 T^{8} + 79008975 T^{9} + 5946774468 p T^{10} + 982720 p^{2} T^{11} + 43488651 p^{3} T^{12} + 9629 p^{4} T^{13} + 217610 p^{5} T^{14} + 39 p^{6} T^{15} + 677 p^{7} T^{16} + p^{9} T^{18} \) | |
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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
−2.61278140389754421161750070646, −2.59738165829296035501766939330, −2.54859439976361192921578396856, −2.50504409084609303873193771514, −2.41210968793806971633398338458, −2.38985478310481990747742873579, −2.11020443859212368580174604099, −1.91303544280726526224181415961, −1.91218320940937260353201691018, −1.91084887865059109720303693177, −1.86485249442428366946363900108, −1.66274337717372438135750900397, −1.62346978669955624041468221394, −1.57928258812714877612128554250, −1.55176214367110650230804405893, −1.36616255891480066072784154385, −1.11226274979288374967981199420, −0.853383987332098862444282876612, −0.77537666972459292751234395026, −0.74742339023143757087240570668, −0.70103452342245111583604486696, −0.54914313781671101280063581572, −0.39000774034106127285054252488, −0.26783133035553759533874846146, −0.13188106797117813218323571584, 0.13188106797117813218323571584, 0.26783133035553759533874846146, 0.39000774034106127285054252488, 0.54914313781671101280063581572, 0.70103452342245111583604486696, 0.74742339023143757087240570668, 0.77537666972459292751234395026, 0.853383987332098862444282876612, 1.11226274979288374967981199420, 1.36616255891480066072784154385, 1.55176214367110650230804405893, 1.57928258812714877612128554250, 1.62346978669955624041468221394, 1.66274337717372438135750900397, 1.86485249442428366946363900108, 1.91084887865059109720303693177, 1.91218320940937260353201691018, 1.91303544280726526224181415961, 2.11020443859212368580174604099, 2.38985478310481990747742873579, 2.41210968793806971633398338458, 2.50504409084609303873193771514, 2.54859439976361192921578396856, 2.59738165829296035501766939330, 2.61278140389754421161750070646
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.