Dirichlet series
| L(s) = 1 | + 2-s − 4·4-s − 3·7-s − 5·8-s − 8·9-s − 9·11-s + 11·13-s − 3·14-s + 6·16-s − 6·17-s − 8·18-s − 9·22-s + 23-s − 19·25-s + 11·26-s − 27-s + 12·28-s + 13·29-s + 16·31-s + 8·32-s − 6·34-s + 32·36-s + 18·37-s + 4·41-s − 43-s + 36·44-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2·4-s − 1.13·7-s − 1.76·8-s − 8/3·9-s − 2.71·11-s + 3.05·13-s − 0.801·14-s + 3/2·16-s − 1.45·17-s − 1.88·18-s − 1.91·22-s + 0.208·23-s − 3.79·25-s + 2.15·26-s − 0.192·27-s + 2.26·28-s + 2.41·29-s + 2.87·31-s + 1.41·32-s − 1.02·34-s + 16/3·36-s + 2.95·37-s + 0.624·41-s − 0.152·43-s + 5.42·44-s + 0.147·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(11^{9} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.24035\times 10^{13}\) |
| Root analytic conductor: | \(5.63103\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 11^{9} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.636381752\) |
| \(L(\frac12)\) | \(\approx\) | \(5.636381752\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + T )^{9} \) |
| 19 | \( 1 \) | |
| good | 2 | \( 1 - T + 5 T^{2} - p^{2} T^{3} + 13 T^{4} - 3 p T^{5} + 5 p^{2} T^{6} - p T^{7} + 23 T^{8} + 11 T^{9} + 23 p T^{10} - p^{3} T^{11} + 5 p^{5} T^{12} - 3 p^{5} T^{13} + 13 p^{5} T^{14} - p^{8} T^{15} + 5 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 3 | \( 1 + 8 T^{2} + T^{3} + 35 T^{4} + 103 T^{6} - 46 T^{7} + 283 T^{8} - 236 T^{9} + 283 p T^{10} - 46 p^{2} T^{11} + 103 p^{3} T^{12} + 35 p^{5} T^{14} + p^{6} T^{15} + 8 p^{7} T^{16} + p^{9} T^{18} \) | |
| 5 | \( 1 + 19 T^{2} + 7 T^{3} + 201 T^{4} + 81 T^{5} + 1589 T^{6} + 558 T^{7} + 9788 T^{8} + 3193 T^{9} + 9788 p T^{10} + 558 p^{2} T^{11} + 1589 p^{3} T^{12} + 81 p^{4} T^{13} + 201 p^{5} T^{14} + 7 p^{6} T^{15} + 19 p^{7} T^{16} + p^{9} T^{18} \) | |
| 7 | \( 1 + 3 T + 39 T^{2} + 104 T^{3} + 745 T^{4} + 1780 T^{5} + 9367 T^{6} + 19805 T^{7} + 86060 T^{8} + 159832 T^{9} + 86060 p T^{10} + 19805 p^{2} T^{11} + 9367 p^{3} T^{12} + 1780 p^{4} T^{13} + 745 p^{5} T^{14} + 104 p^{6} T^{15} + 39 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 11 T + 119 T^{2} - 813 T^{3} + 5400 T^{4} - 27993 T^{5} + 10943 p T^{6} - 605116 T^{7} + 2545975 T^{8} - 9210564 T^{9} + 2545975 p T^{10} - 605116 p^{2} T^{11} + 10943 p^{4} T^{12} - 27993 p^{4} T^{13} + 5400 p^{5} T^{14} - 813 p^{6} T^{15} + 119 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 6 T + 111 T^{2} + 617 T^{3} + 5790 T^{4} + 30432 T^{5} + 191081 T^{6} + 929784 T^{7} + 4435197 T^{8} + 19079666 T^{9} + 4435197 p T^{10} + 929784 p^{2} T^{11} + 191081 p^{3} T^{12} + 30432 p^{4} T^{13} + 5790 p^{5} T^{14} + 617 p^{6} T^{15} + 111 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - T + 147 T^{2} - 70 T^{3} + 10085 T^{4} + 248 T^{5} + 433001 T^{6} + 159707 T^{7} + 13203986 T^{8} + 5940710 T^{9} + 13203986 p T^{10} + 159707 p^{2} T^{11} + 433001 p^{3} T^{12} + 248 p^{4} T^{13} + 10085 p^{5} T^{14} - 70 p^{6} T^{15} + 147 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 13 T + 223 T^{2} - 1853 T^{3} + 19394 T^{4} - 126757 T^{5} + 1052735 T^{6} - 5884726 T^{7} + 41347885 T^{8} - 199424468 T^{9} + 41347885 p T^{10} - 5884726 p^{2} T^{11} + 1052735 p^{3} T^{12} - 126757 p^{4} T^{13} + 19394 p^{5} T^{14} - 1853 p^{6} T^{15} + 223 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 16 T + 204 T^{2} - 1637 T^{3} + 12380 T^{4} - 74671 T^{5} + 16592 p T^{6} - 3228381 T^{7} + 22145447 T^{8} - 124293758 T^{9} + 22145447 p T^{10} - 3228381 p^{2} T^{11} + 16592 p^{4} T^{12} - 74671 p^{4} T^{13} + 12380 p^{5} T^{14} - 1637 p^{6} T^{15} + 204 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 18 T + 255 T^{2} - 2371 T^{3} + 19420 T^{4} - 126102 T^{5} + 817729 T^{6} - 4747378 T^{7} + 30516247 T^{8} - 177678966 T^{9} + 30516247 p T^{10} - 4747378 p^{2} T^{11} + 817729 p^{3} T^{12} - 126102 p^{4} T^{13} + 19420 p^{5} T^{14} - 2371 p^{6} T^{15} + 255 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 4 T + 102 T^{2} - 466 T^{3} + 7319 T^{4} - 40452 T^{5} + 399061 T^{6} - 2450030 T^{7} + 19484933 T^{8} - 117806128 T^{9} + 19484933 p T^{10} - 2450030 p^{2} T^{11} + 399061 p^{3} T^{12} - 40452 p^{4} T^{13} + 7319 p^{5} T^{14} - 466 p^{6} T^{15} + 102 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + T + 245 T^{2} + 336 T^{3} + 29619 T^{4} + 49988 T^{5} + 54415 p T^{6} + 4238487 T^{7} + 133607998 T^{8} + 226251824 T^{9} + 133607998 p T^{10} + 4238487 p^{2} T^{11} + 54415 p^{4} T^{12} + 49988 p^{4} T^{13} + 29619 p^{5} T^{14} + 336 p^{6} T^{15} + 245 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 14 T + 389 T^{2} - 4450 T^{3} + 67899 T^{4} - 643673 T^{5} + 7043097 T^{6} - 55779851 T^{7} + 481927342 T^{8} - 3186952630 T^{9} + 481927342 p T^{10} - 55779851 p^{2} T^{11} + 7043097 p^{3} T^{12} - 643673 p^{4} T^{13} + 67899 p^{5} T^{14} - 4450 p^{6} T^{15} + 389 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 4 T + 262 T^{2} + 992 T^{3} + 29948 T^{4} + 90003 T^{5} + 2045897 T^{6} + 3553202 T^{7} + 106752274 T^{8} + 103234313 T^{9} + 106752274 p T^{10} + 3553202 p^{2} T^{11} + 2045897 p^{3} T^{12} + 90003 p^{4} T^{13} + 29948 p^{5} T^{14} + 992 p^{6} T^{15} + 262 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 31 T + 733 T^{2} - 12305 T^{3} + 178108 T^{4} - 2169823 T^{5} + 23798005 T^{6} - 231234820 T^{7} + 2056073739 T^{8} - 16473750500 T^{9} + 2056073739 p T^{10} - 231234820 p^{2} T^{11} + 23798005 p^{3} T^{12} - 2169823 p^{4} T^{13} + 178108 p^{5} T^{14} - 12305 p^{6} T^{15} + 733 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 3 T + 333 T^{2} + 810 T^{3} + 54985 T^{4} + 106406 T^{5} + 5915573 T^{6} + 9104805 T^{7} + 466642044 T^{8} + 607593492 T^{9} + 466642044 p T^{10} + 9104805 p^{2} T^{11} + 5915573 p^{3} T^{12} + 106406 p^{4} T^{13} + 54985 p^{5} T^{14} + 810 p^{6} T^{15} + 333 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 2 T + 173 T^{2} - 900 T^{3} + 16941 T^{4} - 154631 T^{5} + 1909993 T^{6} - 13954719 T^{7} + 169271824 T^{8} - 1036918952 T^{9} + 169271824 p T^{10} - 13954719 p^{2} T^{11} + 1909993 p^{3} T^{12} - 154631 p^{4} T^{13} + 16941 p^{5} T^{14} - 900 p^{6} T^{15} + 173 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 12 T + 482 T^{2} + 4271 T^{3} + 101775 T^{4} + 706390 T^{5} + 13179487 T^{6} + 75020186 T^{7} + 1217547613 T^{8} + 5975256326 T^{9} + 1217547613 p T^{10} + 75020186 p^{2} T^{11} + 13179487 p^{3} T^{12} + 706390 p^{4} T^{13} + 101775 p^{5} T^{14} + 4271 p^{6} T^{15} + 482 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 26 T + 735 T^{2} - 12394 T^{3} + 209699 T^{4} - 2688291 T^{5} + 34149005 T^{6} - 354861739 T^{7} + 3639401356 T^{8} - 31323905962 T^{9} + 3639401356 p T^{10} - 354861739 p^{2} T^{11} + 34149005 p^{3} T^{12} - 2688291 p^{4} T^{13} + 209699 p^{5} T^{14} - 12394 p^{6} T^{15} + 735 p^{7} T^{16} - 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 31 T + 742 T^{2} - 14342 T^{3} + 229347 T^{4} - 3223040 T^{5} + 40513046 T^{6} - 455576998 T^{7} + 4654775821 T^{8} - 43478831993 T^{9} + 4654775821 p T^{10} - 455576998 p^{2} T^{11} + 40513046 p^{3} T^{12} - 3223040 p^{4} T^{13} + 229347 p^{5} T^{14} - 14342 p^{6} T^{15} + 742 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 18 T + 592 T^{2} - 8721 T^{3} + 156875 T^{4} - 1984688 T^{5} + 25392575 T^{6} - 281639548 T^{7} + 2859446887 T^{8} - 27675516522 T^{9} + 2859446887 p T^{10} - 281639548 p^{2} T^{11} + 25392575 p^{3} T^{12} - 1984688 p^{4} T^{13} + 156875 p^{5} T^{14} - 8721 p^{6} T^{15} + 592 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 11 T + 549 T^{2} + 4995 T^{3} + 1502 p T^{4} + 981629 T^{5} + 19768773 T^{6} + 117305546 T^{7} + 23914739 p T^{8} + 10991786538 T^{9} + 23914739 p^{2} T^{10} + 117305546 p^{2} T^{11} + 19768773 p^{3} T^{12} + 981629 p^{4} T^{13} + 1502 p^{6} T^{14} + 4995 p^{6} T^{15} + 549 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 16 T + 302 T^{2} - 2744 T^{3} + 37192 T^{4} - 307461 T^{5} + 5192125 T^{6} - 42248018 T^{7} + 570256950 T^{8} - 4194757543 T^{9} + 570256950 p T^{10} - 42248018 p^{2} T^{11} + 5192125 p^{3} T^{12} - 307461 p^{4} T^{13} + 37192 p^{5} T^{14} - 2744 p^{6} T^{15} + 302 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.13889567592382093711666147100, −3.07932697147533834143305337925, −3.07732368638354055432364002204, −2.97040640940708571810762122708, −2.94261263769254497762922203794, −2.73106870086836540342027545088, −2.47162303233855025253627227893, −2.40921043000739861268056007269, −2.37617446364266621506029660228, −2.26449448988682815268792259238, −2.22640354267102376513955543162, −2.15596250493065475270497708474, −1.99492257291340873635942229756, −1.73598120307268340635087218015, −1.65029667241330541301024418459, −1.64184389458348514774280727959, −1.20221735992101812742114563677, −1.07406684116710787277961086867, −1.05696608622015937956569481003, −0.797850546832835060022784538220, −0.64901126752227560035742641770, −0.44026495043847370435503332121, −0.43589813136329832599549602497, −0.42287732364286712625452112062, −0.28706210617433026823864249467, 0.28706210617433026823864249467, 0.42287732364286712625452112062, 0.43589813136329832599549602497, 0.44026495043847370435503332121, 0.64901126752227560035742641770, 0.797850546832835060022784538220, 1.05696608622015937956569481003, 1.07406684116710787277961086867, 1.20221735992101812742114563677, 1.64184389458348514774280727959, 1.65029667241330541301024418459, 1.73598120307268340635087218015, 1.99492257291340873635942229756, 2.15596250493065475270497708474, 2.22640354267102376513955543162, 2.26449448988682815268792259238, 2.37617446364266621506029660228, 2.40921043000739861268056007269, 2.47162303233855025253627227893, 2.73106870086836540342027545088, 2.94261263769254497762922203794, 2.97040640940708571810762122708, 3.07732368638354055432364002204, 3.07932697147533834143305337925, 3.13889567592382093711666147100
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.