Dirichlet series
| L(s) = 1 | + 3·2-s − 27·3-s − 9·4-s + 12·5-s − 81·6-s + 8·7-s − 27·8-s + 405·9-s + 36·10-s + 72·11-s + 243·12-s − 166·13-s + 24·14-s − 324·15-s + 14·16-s + 146·17-s + 1.21e3·18-s + 154·19-s − 108·20-s − 216·21-s + 216·22-s + 476·23-s + 729·24-s − 258·25-s − 498·26-s − 4.45e3·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 5.19·3-s − 9/8·4-s + 1.07·5-s − 5.51·6-s + 0.431·7-s − 1.19·8-s + 15·9-s + 1.13·10-s + 1.97·11-s + 5.84·12-s − 3.54·13-s + 0.458·14-s − 5.57·15-s + 7/32·16-s + 2.08·17-s + 15.9·18-s + 1.85·19-s − 1.20·20-s − 2.24·21-s + 2.09·22-s + 4.31·23-s + 6.20·24-s − 2.06·25-s − 3.75·26-s − 31.7·27-s − 0.485·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 67^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 67^{9}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 67^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.64043\times 10^{9}\) |
| Root analytic conductor: | \(3.44374\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 3^{9} \cdot 67^{9} ,\ ( \ : [3/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(3.652293752\) |
| \(L(\frac12)\) | \(\approx\) | \(3.652293752\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p T )^{9} \) |
| 67 | \( ( 1 + p T )^{9} \) | |
| good | 2 | \( 1 - 3 T + 9 p T^{2} - 27 p T^{3} + 229 T^{4} - 791 T^{5} + 365 p^{3} T^{6} - 527 p^{4} T^{7} + 883 p^{5} T^{8} - 555 p^{7} T^{9} + 883 p^{8} T^{10} - 527 p^{10} T^{11} + 365 p^{12} T^{12} - 791 p^{12} T^{13} + 229 p^{15} T^{14} - 27 p^{19} T^{15} + 9 p^{22} T^{16} - 3 p^{24} T^{17} + p^{27} T^{18} \) |
| 5 | \( 1 - 12 T + 402 T^{2} - 3878 T^{3} + 83536 T^{4} - 592678 T^{5} + 2198783 p T^{6} - 72898286 T^{7} + 1332330526 T^{8} - 8377438196 T^{9} + 1332330526 p^{3} T^{10} - 72898286 p^{6} T^{11} + 2198783 p^{10} T^{12} - 592678 p^{12} T^{13} + 83536 p^{15} T^{14} - 3878 p^{18} T^{15} + 402 p^{21} T^{16} - 12 p^{24} T^{17} + p^{27} T^{18} \) | |
| 7 | \( 1 - 8 T + 1360 T^{2} - 878 p T^{3} + 1030114 T^{4} - 2845284 T^{5} + 539455139 T^{6} - 86564652 p T^{7} + 222804332978 T^{8} - 163555664060 T^{9} + 222804332978 p^{3} T^{10} - 86564652 p^{7} T^{11} + 539455139 p^{9} T^{12} - 2845284 p^{12} T^{13} + 1030114 p^{15} T^{14} - 878 p^{19} T^{15} + 1360 p^{21} T^{16} - 8 p^{24} T^{17} + p^{27} T^{18} \) | |
| 11 | \( 1 - 72 T + 765 p T^{2} - 371036 T^{3} + 26502200 T^{4} - 881101256 T^{5} + 55609614160 T^{6} - 1700130869828 T^{7} + 98424777319954 T^{8} - 2696859907348576 T^{9} + 98424777319954 p^{3} T^{10} - 1700130869828 p^{6} T^{11} + 55609614160 p^{9} T^{12} - 881101256 p^{12} T^{13} + 26502200 p^{15} T^{14} - 371036 p^{18} T^{15} + 765 p^{22} T^{16} - 72 p^{24} T^{17} + p^{27} T^{18} \) | |
| 13 | \( 1 + 166 T + 1479 p T^{2} + 1700084 T^{3} + 130220014 T^{4} + 664360392 p T^{5} + 524787877386 T^{6} + 29001087157964 T^{7} + 1506839569534976 T^{8} + 72656475062787556 T^{9} + 1506839569534976 p^{3} T^{10} + 29001087157964 p^{6} T^{11} + 524787877386 p^{9} T^{12} + 664360392 p^{13} T^{13} + 130220014 p^{15} T^{14} + 1700084 p^{18} T^{15} + 1479 p^{22} T^{16} + 166 p^{24} T^{17} + p^{27} T^{18} \) | |
| 17 | \( 1 - 146 T + 25393 T^{2} - 2988060 T^{3} + 312043948 T^{4} - 28772370960 T^{5} + 2416866553132 T^{6} - 184170173953316 T^{7} + 13896304963125990 T^{8} - 961479618638920764 T^{9} + 13896304963125990 p^{3} T^{10} - 184170173953316 p^{6} T^{11} + 2416866553132 p^{9} T^{12} - 28772370960 p^{12} T^{13} + 312043948 p^{15} T^{14} - 2988060 p^{18} T^{15} + 25393 p^{21} T^{16} - 146 p^{24} T^{17} + p^{27} T^{18} \) | |
| 19 | \( 1 - 154 T + 25071 T^{2} - 3294396 T^{3} + 427051432 T^{4} - 40197104720 T^{5} + 4147025596304 T^{6} - 377879862349732 T^{7} + 34156542879579354 T^{8} - 2626769419926025836 T^{9} + 34156542879579354 p^{3} T^{10} - 377879862349732 p^{6} T^{11} + 4147025596304 p^{9} T^{12} - 40197104720 p^{12} T^{13} + 427051432 p^{15} T^{14} - 3294396 p^{18} T^{15} + 25071 p^{21} T^{16} - 154 p^{24} T^{17} + p^{27} T^{18} \) | |
| 23 | \( 1 - 476 T + 184170 T^{2} - 47979806 T^{3} + 10912553666 T^{4} - 2004773142082 T^{5} + 332699111283739 T^{6} - 47337566744182662 T^{7} + 6192689463326208976 T^{8} - \)\(71\!\cdots\!04\)\( T^{9} + 6192689463326208976 p^{3} T^{10} - 47337566744182662 p^{6} T^{11} + 332699111283739 p^{9} T^{12} - 2004773142082 p^{12} T^{13} + 10912553666 p^{15} T^{14} - 47979806 p^{18} T^{15} + 184170 p^{21} T^{16} - 476 p^{24} T^{17} + p^{27} T^{18} \) | |
| 29 | \( 1 - 432 T + 163239 T^{2} - 37685132 T^{3} + 7516592690 T^{4} - 996267521000 T^{5} + 96567376225078 T^{6} + 928165322969836 T^{7} - 65157939409453852 p T^{8} + \)\(44\!\cdots\!36\)\( T^{9} - 65157939409453852 p^{4} T^{10} + 928165322969836 p^{6} T^{11} + 96567376225078 p^{9} T^{12} - 996267521000 p^{12} T^{13} + 7516592690 p^{15} T^{14} - 37685132 p^{18} T^{15} + 163239 p^{21} T^{16} - 432 p^{24} T^{17} + p^{27} T^{18} \) | |
| 31 | \( 1 - 8 p T + 116084 T^{2} - 26140994 T^{3} + 6510341658 T^{4} - 1387341258952 T^{5} + 289781191846555 T^{6} - 57893685453153312 T^{7} + 11470480361208499942 T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + 11470480361208499942 p^{3} T^{10} - 57893685453153312 p^{6} T^{11} + 289781191846555 p^{9} T^{12} - 1387341258952 p^{12} T^{13} + 6510341658 p^{15} T^{14} - 26140994 p^{18} T^{15} + 116084 p^{21} T^{16} - 8 p^{25} T^{17} + p^{27} T^{18} \) | |
| 37 | \( 1 + 240 T + 222132 T^{2} + 51591910 T^{3} + 29345935528 T^{4} + 6379672578708 T^{5} + 2654244823067435 T^{6} + 515928894717883916 T^{7} + \)\(17\!\cdots\!20\)\( T^{8} + \)\(30\!\cdots\!52\)\( T^{9} + \)\(17\!\cdots\!20\)\( p^{3} T^{10} + 515928894717883916 p^{6} T^{11} + 2654244823067435 p^{9} T^{12} + 6379672578708 p^{12} T^{13} + 29345935528 p^{15} T^{14} + 51591910 p^{18} T^{15} + 222132 p^{21} T^{16} + 240 p^{24} T^{17} + p^{27} T^{18} \) | |
| 41 | \( 1 - 406 T + 443078 T^{2} - 169007214 T^{3} + 95833068084 T^{4} - 33603463540144 T^{5} + 13170271004672971 T^{6} - 4135158820090704122 T^{7} + \)\(12\!\cdots\!38\)\( T^{8} - \)\(34\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!38\)\( p^{3} T^{10} - 4135158820090704122 p^{6} T^{11} + 13170271004672971 p^{9} T^{12} - 33603463540144 p^{12} T^{13} + 95833068084 p^{15} T^{14} - 169007214 p^{18} T^{15} + 443078 p^{21} T^{16} - 406 p^{24} T^{17} + p^{27} T^{18} \) | |
| 43 | \( 1 - 154 T + 500942 T^{2} - 1822120 p T^{3} + 119493446954 T^{4} - 18638281797594 T^{5} + 18123972363860097 T^{6} - 2694706711311237100 T^{7} + \)\(19\!\cdots\!88\)\( T^{8} - \)\(25\!\cdots\!24\)\( T^{9} + \)\(19\!\cdots\!88\)\( p^{3} T^{10} - 2694706711311237100 p^{6} T^{11} + 18123972363860097 p^{9} T^{12} - 18638281797594 p^{12} T^{13} + 119493446954 p^{15} T^{14} - 1822120 p^{19} T^{15} + 500942 p^{21} T^{16} - 154 p^{24} T^{17} + p^{27} T^{18} \) | |
| 47 | \( 1 - 494 T + 671533 T^{2} - 282294988 T^{3} + 219372065914 T^{4} - 79086646685320 T^{5} + 45200567442112378 T^{6} - 14064789473590519764 T^{7} + \)\(64\!\cdots\!20\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{9} + \)\(64\!\cdots\!20\)\( p^{3} T^{10} - 14064789473590519764 p^{6} T^{11} + 45200567442112378 p^{9} T^{12} - 79086646685320 p^{12} T^{13} + 219372065914 p^{15} T^{14} - 282294988 p^{18} T^{15} + 671533 p^{21} T^{16} - 494 p^{24} T^{17} + p^{27} T^{18} \) | |
| 53 | \( 1 + 450 T + 769898 T^{2} + 289396366 T^{3} + 273430432984 T^{4} + 88115920345792 T^{5} + 60836319355642539 T^{6} + 17453599155816128270 T^{7} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(27\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!26\)\( p^{3} T^{10} + 17453599155816128270 p^{6} T^{11} + 60836319355642539 p^{9} T^{12} + 88115920345792 p^{12} T^{13} + 273430432984 p^{15} T^{14} + 289396366 p^{18} T^{15} + 769898 p^{21} T^{16} + 450 p^{24} T^{17} + p^{27} T^{18} \) | |
| 59 | \( 1 - 732 T + 945340 T^{2} - 612879532 T^{3} + 513207430894 T^{4} - 282220552064066 T^{5} + 184384670746115453 T^{6} - 88565394268901556874 T^{7} + \)\(49\!\cdots\!34\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{9} + \)\(49\!\cdots\!34\)\( p^{3} T^{10} - 88565394268901556874 p^{6} T^{11} + 184384670746115453 p^{9} T^{12} - 282220552064066 p^{12} T^{13} + 513207430894 p^{15} T^{14} - 612879532 p^{18} T^{15} + 945340 p^{21} T^{16} - 732 p^{24} T^{17} + p^{27} T^{18} \) | |
| 61 | \( 1 + 914 T + 1384791 T^{2} + 944091564 T^{3} + 820017012594 T^{4} + 466805168615128 T^{5} + 303065212323191990 T^{6} + \)\(15\!\cdots\!96\)\( T^{7} + \)\(83\!\cdots\!04\)\( T^{8} + \)\(38\!\cdots\!76\)\( T^{9} + \)\(83\!\cdots\!04\)\( p^{3} T^{10} + \)\(15\!\cdots\!96\)\( p^{6} T^{11} + 303065212323191990 p^{9} T^{12} + 466805168615128 p^{12} T^{13} + 820017012594 p^{15} T^{14} + 944091564 p^{18} T^{15} + 1384791 p^{21} T^{16} + 914 p^{24} T^{17} + p^{27} T^{18} \) | |
| 71 | \( 1 - 2990 T + 6605845 T^{2} - 10318863444 T^{3} + 13414035645594 T^{4} - 14430255958701560 T^{5} + 13495843927433373994 T^{6} - \)\(10\!\cdots\!32\)\( T^{7} + \)\(78\!\cdots\!16\)\( T^{8} - \)\(49\!\cdots\!48\)\( T^{9} + \)\(78\!\cdots\!16\)\( p^{3} T^{10} - \)\(10\!\cdots\!32\)\( p^{6} T^{11} + 13495843927433373994 p^{9} T^{12} - 14430255958701560 p^{12} T^{13} + 13414035645594 p^{15} T^{14} - 10318863444 p^{18} T^{15} + 6605845 p^{21} T^{16} - 2990 p^{24} T^{17} + p^{27} T^{18} \) | |
| 73 | \( 1 + 1384 T + 3034984 T^{2} + 2812890514 T^{3} + 3642043401336 T^{4} + 2629469468687432 T^{5} + 2653195541634116599 T^{6} + \)\(16\!\cdots\!40\)\( T^{7} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(74\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!40\)\( p^{3} T^{10} + \)\(16\!\cdots\!40\)\( p^{6} T^{11} + 2653195541634116599 p^{9} T^{12} + 2629469468687432 p^{12} T^{13} + 3642043401336 p^{15} T^{14} + 2812890514 p^{18} T^{15} + 3034984 p^{21} T^{16} + 1384 p^{24} T^{17} + p^{27} T^{18} \) | |
| 79 | \( 1 - 2438 T + 5772541 T^{2} - 8299448616 T^{3} + 11415392087258 T^{4} - 11837332411608056 T^{5} + 11976906545341109578 T^{6} - \)\(98\!\cdots\!48\)\( T^{7} + \)\(81\!\cdots\!76\)\( T^{8} - \)\(71\!\cdots\!76\)\( p T^{9} + \)\(81\!\cdots\!76\)\( p^{3} T^{10} - \)\(98\!\cdots\!48\)\( p^{6} T^{11} + 11976906545341109578 p^{9} T^{12} - 11837332411608056 p^{12} T^{13} + 11415392087258 p^{15} T^{14} - 8299448616 p^{18} T^{15} + 5772541 p^{21} T^{16} - 2438 p^{24} T^{17} + p^{27} T^{18} \) | |
| 83 | \( 1 - 972 T + 3617968 T^{2} - 3355067444 T^{3} + 6562635805982 T^{4} - 5506694628270722 T^{5} + 7609104727353806845 T^{6} - \)\(55\!\cdots\!18\)\( T^{7} + \)\(61\!\cdots\!58\)\( T^{8} - \)\(38\!\cdots\!72\)\( T^{9} + \)\(61\!\cdots\!58\)\( p^{3} T^{10} - \)\(55\!\cdots\!18\)\( p^{6} T^{11} + 7609104727353806845 p^{9} T^{12} - 5506694628270722 p^{12} T^{13} + 6562635805982 p^{15} T^{14} - 3355067444 p^{18} T^{15} + 3617968 p^{21} T^{16} - 972 p^{24} T^{17} + p^{27} T^{18} \) | |
| 89 | \( 1 - 1034 T + 4128177 T^{2} - 2752260700 T^{3} + 7010341794268 T^{4} - 2580866299893328 T^{5} + 6687942370385771884 T^{6} - \)\(59\!\cdots\!80\)\( T^{7} + \)\(46\!\cdots\!22\)\( T^{8} + \)\(29\!\cdots\!24\)\( T^{9} + \)\(46\!\cdots\!22\)\( p^{3} T^{10} - \)\(59\!\cdots\!80\)\( p^{6} T^{11} + 6687942370385771884 p^{9} T^{12} - 2580866299893328 p^{12} T^{13} + 7010341794268 p^{15} T^{14} - 2752260700 p^{18} T^{15} + 4128177 p^{21} T^{16} - 1034 p^{24} T^{17} + p^{27} T^{18} \) | |
| 97 | \( 1 + 1516 T + 4926113 T^{2} + 4305288356 T^{3} + 86054541988 p T^{4} + 2319749075206136 T^{5} + 4784892495212791012 T^{6} - \)\(57\!\cdots\!44\)\( T^{7} - \)\(17\!\cdots\!82\)\( T^{8} - \)\(99\!\cdots\!64\)\( T^{9} - \)\(17\!\cdots\!82\)\( p^{3} T^{10} - \)\(57\!\cdots\!44\)\( p^{6} T^{11} + 4784892495212791012 p^{9} T^{12} + 2319749075206136 p^{12} T^{13} + 86054541988 p^{16} T^{14} + 4305288356 p^{18} T^{15} + 4926113 p^{21} T^{16} + 1516 p^{24} T^{17} + p^{27} T^{18} \) | |
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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
−4.70199437257010689463673031791, −4.60860942424592922118440456632, −4.48473378956745378139099133084, −4.45653359172799844874209115980, −4.41468222799394947727010464102, −4.36686243459045494530257355360, −4.34910582381110595844634259193, −3.59817686684584379582114146609, −3.54269061203177871226833563278, −3.39750626850341652453716240374, −3.23864445734698344126175567773, −3.21116287807370812272821104246, −3.04797106615702272713446492939, −2.50787740734343226356563184810, −2.35002736449192852512770394634, −2.08844346814207714993777503057, −1.98580820387924721582934854355, −1.64003413789932136998779153610, −1.40883012043942386201571903980, −1.05122350055731889298979897337, −0.913636958694115708997193936223, −0.867566535901196381350473369327, −0.807963912093876088455584687934, −0.39066350586561155376896886135, −0.37223223968697484651500368614, 0.37223223968697484651500368614, 0.39066350586561155376896886135, 0.807963912093876088455584687934, 0.867566535901196381350473369327, 0.913636958694115708997193936223, 1.05122350055731889298979897337, 1.40883012043942386201571903980, 1.64003413789932136998779153610, 1.98580820387924721582934854355, 2.08844346814207714993777503057, 2.35002736449192852512770394634, 2.50787740734343226356563184810, 3.04797106615702272713446492939, 3.21116287807370812272821104246, 3.23864445734698344126175567773, 3.39750626850341652453716240374, 3.54269061203177871226833563278, 3.59817686684584379582114146609, 4.34910582381110595844634259193, 4.36686243459045494530257355360, 4.41468222799394947727010464102, 4.45653359172799844874209115980, 4.48473378956745378139099133084, 4.60860942424592922118440456632, 4.70199437257010689463673031791
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.