Dirichlet series
| L(s) = 1 | − 5·2-s + 9·4-s + 8·7-s − 5·8-s + 6·13-s − 40·14-s − 6·16-s − 18·17-s − 4·19-s − 16·23-s − 30·26-s + 72·28-s + 2·29-s − 6·31-s + 14·32-s + 90·34-s − 9·37-s + 20·38-s − 2·41-s + 14·43-s + 80·46-s − 34·47-s − 3·49-s + 54·52-s + 4·53-s − 40·56-s − 10·58-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 9/2·4-s + 3.02·7-s − 1.76·8-s + 1.66·13-s − 10.6·14-s − 3/2·16-s − 4.36·17-s − 0.917·19-s − 3.33·23-s − 5.88·26-s + 13.6·28-s + 0.371·29-s − 1.07·31-s + 2.47·32-s + 15.4·34-s − 1.47·37-s + 3.24·38-s − 0.312·41-s + 2.13·43-s + 11.7·46-s − 4.95·47-s − 3/7·49-s + 7.48·52-s + 0.549·53-s − 5.34·56-s − 1.31·58-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{18} \cdot 37^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 5^{18} \cdot 37^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{18} \cdot 5^{18} \cdot 37^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.53484\times 10^{16}\) |
| Root analytic conductor: | \(8.15324\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 3^{18} \cdot 5^{18} \cdot 37^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| 37 | \( ( 1 + T )^{9} \) | |
| good | 2 | \( 1 + 5 T + p^{4} T^{2} + 5 p^{3} T^{3} + 87 T^{4} + 171 T^{5} + 39 p^{3} T^{6} + 131 p^{2} T^{7} + 205 p^{2} T^{8} + 299 p^{2} T^{9} + 205 p^{3} T^{10} + 131 p^{4} T^{11} + 39 p^{6} T^{12} + 171 p^{4} T^{13} + 87 p^{5} T^{14} + 5 p^{9} T^{15} + p^{11} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 8 T + 67 T^{2} - 50 p T^{3} + 1751 T^{4} - 6864 T^{5} + 25791 T^{6} - 81794 T^{7} + 251836 T^{8} - 96302 p T^{9} + 251836 p T^{10} - 81794 p^{2} T^{11} + 25791 p^{3} T^{12} - 6864 p^{4} T^{13} + 1751 p^{5} T^{14} - 50 p^{7} T^{15} + 67 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 57 T^{2} + 24 T^{3} + 1603 T^{4} + 1128 T^{5} + 30009 T^{6} + 2296 p T^{7} + 420678 T^{8} + 345392 T^{9} + 420678 p T^{10} + 2296 p^{3} T^{11} + 30009 p^{3} T^{12} + 1128 p^{4} T^{13} + 1603 p^{5} T^{14} + 24 p^{6} T^{15} + 57 p^{7} T^{16} + p^{9} T^{18} \) | |
| 13 | \( 1 - 6 T + 77 T^{2} - 372 T^{3} + 2876 T^{4} - 11732 T^{5} + 5396 p T^{6} - 248108 T^{7} + 1230478 T^{8} - 3777212 T^{9} + 1230478 p T^{10} - 248108 p^{2} T^{11} + 5396 p^{4} T^{12} - 11732 p^{4} T^{13} + 2876 p^{5} T^{14} - 372 p^{6} T^{15} + 77 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 18 T + 251 T^{2} + 2452 T^{3} + 20402 T^{4} + 140812 T^{5} + 862542 T^{6} + 4621668 T^{7} + 1317988 p T^{8} + 96841700 T^{9} + 1317988 p^{2} T^{10} + 4621668 p^{2} T^{11} + 862542 p^{3} T^{12} + 140812 p^{4} T^{13} + 20402 p^{5} T^{14} + 2452 p^{6} T^{15} + 251 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 4 T + 89 T^{2} + 290 T^{3} + 4152 T^{4} + 11570 T^{5} + 133580 T^{6} + 329082 T^{7} + 3268744 T^{8} + 7150996 T^{9} + 3268744 p T^{10} + 329082 p^{2} T^{11} + 133580 p^{3} T^{12} + 11570 p^{4} T^{13} + 4152 p^{5} T^{14} + 290 p^{6} T^{15} + 89 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 16 T + 281 T^{2} + 2900 T^{3} + 29786 T^{4} + 228568 T^{5} + 1710722 T^{6} + 10342828 T^{7} + 2637880 p T^{8} + 295554928 T^{9} + 2637880 p^{2} T^{10} + 10342828 p^{2} T^{11} + 1710722 p^{3} T^{12} + 228568 p^{4} T^{13} + 29786 p^{5} T^{14} + 2900 p^{6} T^{15} + 281 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 2 T + 151 T^{2} - 192 T^{3} + 11086 T^{4} - 9472 T^{5} + 547218 T^{6} - 367456 T^{7} + 20374196 T^{8} - 11917948 T^{9} + 20374196 p T^{10} - 367456 p^{2} T^{11} + 547218 p^{3} T^{12} - 9472 p^{4} T^{13} + 11086 p^{5} T^{14} - 192 p^{6} T^{15} + 151 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 6 T + 117 T^{2} + 652 T^{3} + 7900 T^{4} + 42482 T^{5} + 382924 T^{6} + 1922952 T^{7} + 14422916 T^{8} + 67506176 T^{9} + 14422916 p T^{10} + 1922952 p^{2} T^{11} + 382924 p^{3} T^{12} + 42482 p^{4} T^{13} + 7900 p^{5} T^{14} + 652 p^{6} T^{15} + 117 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 2 T + 207 T^{2} + 308 T^{3} + 22135 T^{4} + 23834 T^{5} + 1590167 T^{6} + 1260212 T^{7} + 84684498 T^{8} + 54351400 T^{9} + 84684498 p T^{10} + 1260212 p^{2} T^{11} + 1590167 p^{3} T^{12} + 23834 p^{4} T^{13} + 22135 p^{5} T^{14} + 308 p^{6} T^{15} + 207 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 14 T + 209 T^{2} - 1776 T^{3} + 13590 T^{4} - 62808 T^{5} + 164694 T^{6} + 1659984 T^{7} - 20033612 T^{8} + 186487436 T^{9} - 20033612 p T^{10} + 1659984 p^{2} T^{11} + 164694 p^{3} T^{12} - 62808 p^{4} T^{13} + 13590 p^{5} T^{14} - 1776 p^{6} T^{15} + 209 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 34 T + 691 T^{2} + 10098 T^{3} + 119667 T^{4} + 1203098 T^{5} + 10717955 T^{6} + 86467054 T^{7} + 648355640 T^{8} + 4565820786 T^{9} + 648355640 p T^{10} + 86467054 p^{2} T^{11} + 10717955 p^{3} T^{12} + 1203098 p^{4} T^{13} + 119667 p^{5} T^{14} + 10098 p^{6} T^{15} + 691 p^{7} T^{16} + 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 4 T + 219 T^{2} - 714 T^{3} + 27303 T^{4} - 82394 T^{5} + 2419235 T^{6} - 6546958 T^{7} + 162792322 T^{8} - 393594900 T^{9} + 162792322 p T^{10} - 6546958 p^{2} T^{11} + 2419235 p^{3} T^{12} - 82394 p^{4} T^{13} + 27303 p^{5} T^{14} - 714 p^{6} T^{15} + 219 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 10 T + 363 T^{2} + 2748 T^{3} + 56906 T^{4} + 329642 T^{5} + 5349206 T^{6} + 24228380 T^{7} + 368128354 T^{8} + 1449707336 T^{9} + 368128354 p T^{10} + 24228380 p^{2} T^{11} + 5349206 p^{3} T^{12} + 329642 p^{4} T^{13} + 56906 p^{5} T^{14} + 2748 p^{6} T^{15} + 363 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 8 T + 287 T^{2} - 2428 T^{3} + 46078 T^{4} - 364856 T^{5} + 5023538 T^{6} - 36330404 T^{7} + 402838300 T^{8} - 2591580928 T^{9} + 402838300 p T^{10} - 36330404 p^{2} T^{11} + 5023538 p^{3} T^{12} - 364856 p^{4} T^{13} + 46078 p^{5} T^{14} - 2428 p^{6} T^{15} + 287 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 16 T + 457 T^{2} - 5334 T^{3} + 91400 T^{4} - 863542 T^{5} + 11351860 T^{6} - 90969834 T^{7} + 1002999192 T^{8} - 6994279188 T^{9} + 1002999192 p T^{10} - 90969834 p^{2} T^{11} + 11351860 p^{3} T^{12} - 863542 p^{4} T^{13} + 91400 p^{5} T^{14} - 5334 p^{6} T^{15} + 457 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 8 T + 461 T^{2} + 3084 T^{3} + 101579 T^{4} + 577916 T^{5} + 14118333 T^{6} + 68868932 T^{7} + 1373824126 T^{8} + 5759364312 T^{9} + 1373824126 p T^{10} + 68868932 p^{2} T^{11} + 14118333 p^{3} T^{12} + 577916 p^{4} T^{13} + 101579 p^{5} T^{14} + 3084 p^{6} T^{15} + 461 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 8 T + 463 T^{2} - 3322 T^{3} + 104127 T^{4} - 660254 T^{5} + 14888495 T^{6} - 82668926 T^{7} + 1491866866 T^{8} - 7163969732 T^{9} + 1491866866 p T^{10} - 82668926 p^{2} T^{11} + 14888495 p^{3} T^{12} - 660254 p^{4} T^{13} + 104127 p^{5} T^{14} - 3322 p^{6} T^{15} + 463 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 26 T + 763 T^{2} + 13592 T^{3} + 240966 T^{4} + 3294462 T^{5} + 43709806 T^{6} + 481618160 T^{7} + 5120271010 T^{8} + 46351492472 T^{9} + 5120271010 p T^{10} + 481618160 p^{2} T^{11} + 43709806 p^{3} T^{12} + 3294462 p^{4} T^{13} + 240966 p^{5} T^{14} + 13592 p^{6} T^{15} + 763 p^{7} T^{16} + 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 70 T + 33 p T^{2} + 75488 T^{3} + 1620083 T^{4} + 28461806 T^{5} + 422117107 T^{6} + 5384607998 T^{7} + 59783071428 T^{8} + 581342998750 T^{9} + 59783071428 p T^{10} + 5384607998 p^{2} T^{11} + 422117107 p^{3} T^{12} + 28461806 p^{4} T^{13} + 1620083 p^{5} T^{14} + 75488 p^{6} T^{15} + 33 p^{8} T^{16} + 70 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 8 T + 241 T^{2} + 444 T^{3} + 10408 T^{4} + 333920 T^{5} + 750216 T^{6} + 23712740 T^{7} + 328869494 T^{8} + 588300176 T^{9} + 328869494 p T^{10} + 23712740 p^{2} T^{11} + 750216 p^{3} T^{12} + 333920 p^{4} T^{13} + 10408 p^{5} T^{14} + 444 p^{6} T^{15} + 241 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 2 T + 435 T^{2} + 2304 T^{3} + 103150 T^{4} + 691808 T^{5} + 17711706 T^{6} + 115122720 T^{7} + 2305098788 T^{8} + 13060706492 T^{9} + 2305098788 p T^{10} + 115122720 p^{2} T^{11} + 17711706 p^{3} T^{12} + 691808 p^{4} T^{13} + 103150 p^{5} T^{14} + 2304 p^{6} T^{15} + 435 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.21327964372879248555389703063, −3.17624042654767420360944609198, −3.16198458468167796972038708716, −2.99484189931832468366977751476, −2.70533778052655227887904850071, −2.68247958812858492185135784173, −2.59222685217719980656864365007, −2.53315730304655511998285659918, −2.50389836848244009782925857716, −2.16095698522699484116472486511, −2.05258319529922631645663924431, −2.05181301409679534604481705509, −1.98641553239599751300513896639, −1.96797592598804578437127923017, −1.95734278662095822258155067533, −1.93881922078108502414534963812, −1.60609440381623621803030560143, −1.45716526107551279154172329566, −1.44116481278538967880634684259, −1.19956272790734176649947699057, −1.18059856355490195863974377261, −1.15144864719017574484768844365, −1.08560787030496148194796095461, −1.01828581119107798973882922415, −0.918272729024876785292568136123, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.918272729024876785292568136123, 1.01828581119107798973882922415, 1.08560787030496148194796095461, 1.15144864719017574484768844365, 1.18059856355490195863974377261, 1.19956272790734176649947699057, 1.44116481278538967880634684259, 1.45716526107551279154172329566, 1.60609440381623621803030560143, 1.93881922078108502414534963812, 1.95734278662095822258155067533, 1.96797592598804578437127923017, 1.98641553239599751300513896639, 2.05181301409679534604481705509, 2.05258319529922631645663924431, 2.16095698522699484116472486511, 2.50389836848244009782925857716, 2.53315730304655511998285659918, 2.59222685217719980656864365007, 2.68247958812858492185135784173, 2.70533778052655227887904850071, 2.99484189931832468366977751476, 3.16198458468167796972038708716, 3.17624042654767420360944609198, 3.21327964372879248555389703063
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.