Dirichlet series
| L(s) = 1 | − 9·5-s + 2·7-s − 7·11-s + 6·13-s − 7·17-s + 2·19-s − 19·23-s + 29·25-s − 13·29-s + 12·31-s − 18·35-s + 15·37-s − 18·41-s − 6·43-s − 25·47-s − 20·49-s − 17·53-s + 63·55-s − 3·59-s + 14·61-s − 54·65-s − 4·67-s − 17·71-s − 20·73-s − 14·77-s − 8·79-s + 83-s + ⋯ |
| L(s) = 1 | − 4.02·5-s + 0.755·7-s − 2.11·11-s + 1.66·13-s − 1.69·17-s + 0.458·19-s − 3.96·23-s + 29/5·25-s − 2.41·29-s + 2.15·31-s − 3.04·35-s + 2.46·37-s − 2.81·41-s − 0.914·43-s − 3.64·47-s − 2.85·49-s − 2.33·53-s + 8.49·55-s − 0.390·59-s + 1.79·61-s − 6.69·65-s − 0.488·67-s − 2.01·71-s − 2.34·73-s − 1.59·77-s − 0.900·79-s + 0.109·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 3^{18} \cdot 167^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.35414\times 10^{15}\) |
| Root analytic conductor: | \(6.92864\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{18} \cdot 3^{18} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( 1 \) |
| 3 | \( 1 \) | |
| 167 | \( ( 1 + T )^{9} \) | |
| good | 5 | \( 1 + 9 T + 52 T^{2} + 234 T^{3} + 886 T^{4} + 117 p^{2} T^{5} + 8664 T^{6} + 23344 T^{7} + 58229 T^{8} + 134906 T^{9} + 58229 p T^{10} + 23344 p^{2} T^{11} + 8664 p^{3} T^{12} + 117 p^{6} T^{13} + 886 p^{5} T^{14} + 234 p^{6} T^{15} + 52 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 2 T + 24 T^{2} - 3 T^{3} + 204 T^{4} + 67 p T^{5} + 1446 T^{6} + 5119 T^{7} + 15489 T^{8} + 32962 T^{9} + 15489 p T^{10} + 5119 p^{2} T^{11} + 1446 p^{3} T^{12} + 67 p^{5} T^{13} + 204 p^{5} T^{14} - 3 p^{6} T^{15} + 24 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 7 T + 76 T^{2} + 448 T^{3} + 2798 T^{4} + 13617 T^{5} + 64556 T^{6} + 259240 T^{7} + 1013497 T^{8} + 3395532 T^{9} + 1013497 p T^{10} + 259240 p^{2} T^{11} + 64556 p^{3} T^{12} + 13617 p^{4} T^{13} + 2798 p^{5} T^{14} + 448 p^{6} T^{15} + 76 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 6 T + 72 T^{2} - 290 T^{3} + 2079 T^{4} - 5583 T^{5} + 2501 p T^{6} - 50538 T^{7} + 355779 T^{8} - 363982 T^{9} + 355779 p T^{10} - 50538 p^{2} T^{11} + 2501 p^{4} T^{12} - 5583 p^{4} T^{13} + 2079 p^{5} T^{14} - 290 p^{6} T^{15} + 72 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 7 T + 95 T^{2} + 412 T^{3} + 3239 T^{4} + 7217 T^{5} + 46207 T^{6} - 42146 T^{7} + 182922 T^{8} - 2723810 T^{9} + 182922 p T^{10} - 42146 p^{2} T^{11} + 46207 p^{3} T^{12} + 7217 p^{4} T^{13} + 3239 p^{5} T^{14} + 412 p^{6} T^{15} + 95 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 2 T + 102 T^{2} - 170 T^{3} + 237 p T^{4} - 4717 T^{5} + 117347 T^{6} - 26098 T^{7} + 2285365 T^{8} + 603238 T^{9} + 2285365 p T^{10} - 26098 p^{2} T^{11} + 117347 p^{3} T^{12} - 4717 p^{4} T^{13} + 237 p^{6} T^{14} - 170 p^{6} T^{15} + 102 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 19 T + 254 T^{2} + 2284 T^{3} + 17484 T^{4} + 110157 T^{5} + 659522 T^{6} + 3574832 T^{7} + 19162755 T^{8} + 4049112 p T^{9} + 19162755 p T^{10} + 3574832 p^{2} T^{11} + 659522 p^{3} T^{12} + 110157 p^{4} T^{13} + 17484 p^{5} T^{14} + 2284 p^{6} T^{15} + 254 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 13 T + 154 T^{2} + 1330 T^{3} + 10268 T^{4} + 65265 T^{5} + 395422 T^{6} + 2088442 T^{7} + 11323883 T^{8} + 58112060 T^{9} + 11323883 p T^{10} + 2088442 p^{2} T^{11} + 395422 p^{3} T^{12} + 65265 p^{4} T^{13} + 10268 p^{5} T^{14} + 1330 p^{6} T^{15} + 154 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 12 T + 237 T^{2} - 2081 T^{3} + 24736 T^{4} - 175357 T^{5} + 1571692 T^{6} - 9333683 T^{7} + 68244056 T^{8} - 343757830 T^{9} + 68244056 p T^{10} - 9333683 p^{2} T^{11} + 1571692 p^{3} T^{12} - 175357 p^{4} T^{13} + 24736 p^{5} T^{14} - 2081 p^{6} T^{15} + 237 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 15 T + 242 T^{2} - 2138 T^{3} + 21014 T^{4} - 145273 T^{5} + 1219856 T^{6} - 212410 p T^{7} + 60067423 T^{8} - 344194216 T^{9} + 60067423 p T^{10} - 212410 p^{3} T^{11} + 1219856 p^{3} T^{12} - 145273 p^{4} T^{13} + 21014 p^{5} T^{14} - 2138 p^{6} T^{15} + 242 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 18 T + 391 T^{2} + 4825 T^{3} + 61116 T^{4} + 575535 T^{5} + 5406478 T^{6} + 41258085 T^{7} + 314515424 T^{8} + 2011563376 T^{9} + 314515424 p T^{10} + 41258085 p^{2} T^{11} + 5406478 p^{3} T^{12} + 575535 p^{4} T^{13} + 61116 p^{5} T^{14} + 4825 p^{6} T^{15} + 391 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 6 T + 97 T^{2} + 675 T^{3} + 5484 T^{4} + 22849 T^{5} + 166420 T^{6} + 187403 T^{7} + 1595534 T^{8} - 10660100 T^{9} + 1595534 p T^{10} + 187403 p^{2} T^{11} + 166420 p^{3} T^{12} + 22849 p^{4} T^{13} + 5484 p^{5} T^{14} + 675 p^{6} T^{15} + 97 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 25 T + 448 T^{2} + 121 p T^{3} + 1367 p T^{4} + 630769 T^{5} + 5789701 T^{6} + 47606857 T^{7} + 365856543 T^{8} + 2574158528 T^{9} + 365856543 p T^{10} + 47606857 p^{2} T^{11} + 5789701 p^{3} T^{12} + 630769 p^{4} T^{13} + 1367 p^{6} T^{14} + 121 p^{7} T^{15} + 448 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 17 T + 6 p T^{2} + 3079 T^{3} + 33015 T^{4} + 222369 T^{5} + 1931189 T^{6} + 10783739 T^{7} + 96671639 T^{8} + 529914954 T^{9} + 96671639 p T^{10} + 10783739 p^{2} T^{11} + 1931189 p^{3} T^{12} + 222369 p^{4} T^{13} + 33015 p^{5} T^{14} + 3079 p^{6} T^{15} + 6 p^{8} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 3 T + 248 T^{2} + 587 T^{3} + 35733 T^{4} + 67161 T^{5} + 3580941 T^{6} + 97955 p T^{7} + 272819099 T^{8} + 381917456 T^{9} + 272819099 p T^{10} + 97955 p^{3} T^{11} + 3580941 p^{3} T^{12} + 67161 p^{4} T^{13} + 35733 p^{5} T^{14} + 587 p^{6} T^{15} + 248 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 14 T + 286 T^{2} - 2662 T^{3} + 34623 T^{4} - 221617 T^{5} + 2297933 T^{6} - 9978522 T^{7} + 111413285 T^{8} - 393019054 T^{9} + 111413285 p T^{10} - 9978522 p^{2} T^{11} + 2297933 p^{3} T^{12} - 221617 p^{4} T^{13} + 34623 p^{5} T^{14} - 2662 p^{6} T^{15} + 286 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 4 T + 6 p T^{2} + 1172 T^{3} + 77435 T^{4} + 162163 T^{5} + 9543569 T^{6} + 14628924 T^{7} + 845476133 T^{8} + 1048306460 T^{9} + 845476133 p T^{10} + 14628924 p^{2} T^{11} + 9543569 p^{3} T^{12} + 162163 p^{4} T^{13} + 77435 p^{5} T^{14} + 1172 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 17 T + 319 T^{2} + 2841 T^{3} + 39999 T^{4} + 355099 T^{5} + 69615 p T^{6} + 38075859 T^{7} + 407671928 T^{8} + 2653132208 T^{9} + 407671928 p T^{10} + 38075859 p^{2} T^{11} + 69615 p^{4} T^{12} + 355099 p^{4} T^{13} + 39999 p^{5} T^{14} + 2841 p^{6} T^{15} + 319 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 20 T + 498 T^{2} + 7086 T^{3} + 112237 T^{4} + 1284803 T^{5} + 15841175 T^{6} + 152923942 T^{7} + 1573749217 T^{8} + 13055799498 T^{9} + 1573749217 p T^{10} + 152923942 p^{2} T^{11} + 15841175 p^{3} T^{12} + 1284803 p^{4} T^{13} + 112237 p^{5} T^{14} + 7086 p^{6} T^{15} + 498 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 8 T + 471 T^{2} + 3237 T^{3} + 106842 T^{4} + 664023 T^{5} + 15704132 T^{6} + 88577445 T^{7} + 1658985638 T^{8} + 8277689232 T^{9} + 1658985638 p T^{10} + 88577445 p^{2} T^{11} + 15704132 p^{3} T^{12} + 664023 p^{4} T^{13} + 106842 p^{5} T^{14} + 3237 p^{6} T^{15} + 471 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - T + 488 T^{2} - 1161 T^{3} + 113615 T^{4} - 399973 T^{5} + 16903319 T^{6} - 70850779 T^{7} + 1823276283 T^{8} - 7479201516 T^{9} + 1823276283 p T^{10} - 70850779 p^{2} T^{11} + 16903319 p^{3} T^{12} - 399973 p^{4} T^{13} + 113615 p^{5} T^{14} - 1161 p^{6} T^{15} + 488 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 36 T + 1226 T^{2} + 26142 T^{3} + 5835 p T^{4} + 7977763 T^{5} + 115284093 T^{6} + 1381129790 T^{7} + 15698593261 T^{8} + 151798632666 T^{9} + 15698593261 p T^{10} + 1381129790 p^{2} T^{11} + 115284093 p^{3} T^{12} + 7977763 p^{4} T^{13} + 5835 p^{6} T^{14} + 26142 p^{6} T^{15} + 1226 p^{7} T^{16} + 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 31 T + 713 T^{2} - 13605 T^{3} + 225439 T^{4} - 3322243 T^{5} + 44076123 T^{6} - 534128975 T^{7} + 5960297404 T^{8} - 60929995076 T^{9} + 5960297404 p T^{10} - 534128975 p^{2} T^{11} + 44076123 p^{3} T^{12} - 3322243 p^{4} T^{13} + 225439 p^{5} T^{14} - 13605 p^{6} T^{15} + 713 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 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
−3.48903622527512118185334490043, −3.34353499346442773267489647438, −3.34051342292403646661244352301, −3.10688384720875646502197199658, −3.09967390065760243041997944671, −3.07617808417482556439763211057, −2.93906338995643284974202538921, −2.79885480296349288015927023563, −2.65013601507624858897581298123, −2.42045187340929748678073705131, −2.41260547439023453010659548692, −2.29102729514005691075343787437, −2.22010683110475705970483451928, −2.21465059854270164325989712150, −2.16427006480293518396605144915, −2.09468240940035645490671304077, −1.66819370041233017337482848148, −1.52236553805898768386129461975, −1.43829705372707307696583679140, −1.38592505325482015043604847330, −1.32236105359509128837276337066, −1.29515050064276626236407756855, −1.22226549761570637353054128299, −1.03493322925017414395140923279, −0.931786599871272815703546304316, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.931786599871272815703546304316, 1.03493322925017414395140923279, 1.22226549761570637353054128299, 1.29515050064276626236407756855, 1.32236105359509128837276337066, 1.38592505325482015043604847330, 1.43829705372707307696583679140, 1.52236553805898768386129461975, 1.66819370041233017337482848148, 2.09468240940035645490671304077, 2.16427006480293518396605144915, 2.21465059854270164325989712150, 2.22010683110475705970483451928, 2.29102729514005691075343787437, 2.41260547439023453010659548692, 2.42045187340929748678073705131, 2.65013601507624858897581298123, 2.79885480296349288015927023563, 2.93906338995643284974202538921, 3.07617808417482556439763211057, 3.09967390065760243041997944671, 3.10688384720875646502197199658, 3.34051342292403646661244352301, 3.34353499346442773267489647438, 3.48903622527512118185334490043
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.