Dirichlet series
| L(s) = 1 | + 3·3-s − 45·5-s − 25·7-s − 86·9-s − 22·11-s + 23·13-s − 135·15-s − 135·17-s − 102·19-s − 75·21-s + 207·23-s + 1.12e3·25-s − 173·27-s + 280·29-s − 168·31-s − 66·33-s + 1.12e3·35-s + 153·37-s + 69·39-s − 502·41-s − 110·43-s + 3.87e3·45-s + 153·47-s − 849·49-s − 405·51-s − 273·53-s + 990·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4.02·5-s − 1.34·7-s − 3.18·9-s − 0.603·11-s + 0.490·13-s − 2.32·15-s − 1.92·17-s − 1.23·19-s − 0.779·21-s + 1.87·23-s + 9·25-s − 1.23·27-s + 1.79·29-s − 0.973·31-s − 0.348·33-s + 5.43·35-s + 0.679·37-s + 0.283·39-s − 1.91·41-s − 0.390·43-s + 12.8·45-s + 0.474·47-s − 2.47·49-s − 1.11·51-s − 0.707·53-s + 2.42·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{9} \cdot 23^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{9} \cdot 23^{9}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{36} \cdot 5^{9} \cdot 23^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.09486\times 10^{18}\) |
| Root analytic conductor: | \(10.4193\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{36} \cdot 5^{9} \cdot 23^{9} ,\ ( \ : [3/2]^{9} ),\ -1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T )^{9} \) | |
| 23 | \( ( 1 - p T )^{9} \) | |
| good | 3 | \( 1 - p T + 95 T^{2} - 370 T^{3} + 4774 T^{4} - 21128 T^{5} + 178067 T^{6} - 756301 T^{7} + 1871741 p T^{8} - 21359300 T^{9} + 1871741 p^{4} T^{10} - 756301 p^{6} T^{11} + 178067 p^{9} T^{12} - 21128 p^{12} T^{13} + 4774 p^{15} T^{14} - 370 p^{18} T^{15} + 95 p^{21} T^{16} - p^{25} T^{17} + p^{27} T^{18} \) |
| 7 | \( 1 + 25 T + 1474 T^{2} + 21015 T^{3} + 126360 p T^{4} + 7263721 T^{5} + 351296420 T^{6} + 3826691 p^{3} T^{7} + 119279934045 T^{8} + 221297270724 T^{9} + 119279934045 p^{3} T^{10} + 3826691 p^{9} T^{11} + 351296420 p^{9} T^{12} + 7263721 p^{12} T^{13} + 126360 p^{16} T^{14} + 21015 p^{18} T^{15} + 1474 p^{21} T^{16} + 25 p^{24} T^{17} + p^{27} T^{18} \) | |
| 11 | \( 1 + 2 p T + 5596 T^{2} + 52898 T^{3} + 14396714 T^{4} + 186370 T^{5} + 27120032828 T^{6} - 85540904346 T^{7} + 44968205825277 T^{8} - 114223804452480 T^{9} + 44968205825277 p^{3} T^{10} - 85540904346 p^{6} T^{11} + 27120032828 p^{9} T^{12} + 186370 p^{12} T^{13} + 14396714 p^{15} T^{14} + 52898 p^{18} T^{15} + 5596 p^{21} T^{16} + 2 p^{25} T^{17} + p^{27} T^{18} \) | |
| 13 | \( 1 - 23 T + 9233 T^{2} - 231856 T^{3} + 47827432 T^{4} - 1242658892 T^{5} + 171924729303 T^{6} - 4359656842317 T^{7} + 473120829223879 T^{8} - 11194929718987080 T^{9} + 473120829223879 p^{3} T^{10} - 4359656842317 p^{6} T^{11} + 171924729303 p^{9} T^{12} - 1242658892 p^{12} T^{13} + 47827432 p^{15} T^{14} - 231856 p^{18} T^{15} + 9233 p^{21} T^{16} - 23 p^{24} T^{17} + p^{27} T^{18} \) | |
| 17 | \( 1 + 135 T + 30080 T^{2} + 2854635 T^{3} + 20798308 p T^{4} + 25215019709 T^{5} + 2277428059164 T^{6} + 129993902567433 T^{7} + 10530289224953303 T^{8} + 579516659269453664 T^{9} + 10530289224953303 p^{3} T^{10} + 129993902567433 p^{6} T^{11} + 2277428059164 p^{9} T^{12} + 25215019709 p^{12} T^{13} + 20798308 p^{16} T^{14} + 2854635 p^{18} T^{15} + 30080 p^{21} T^{16} + 135 p^{24} T^{17} + p^{27} T^{18} \) | |
| 19 | \( 1 + 102 T + 25732 T^{2} + 6930 p^{2} T^{3} + 302940346 T^{4} + 21989381006 T^{5} + 1832413082724 T^{6} + 79088274881942 T^{7} + 4825131079375389 T^{8} + 179453305948683688 T^{9} + 4825131079375389 p^{3} T^{10} + 79088274881942 p^{6} T^{11} + 1832413082724 p^{9} T^{12} + 21989381006 p^{12} T^{13} + 302940346 p^{15} T^{14} + 6930 p^{20} T^{15} + 25732 p^{21} T^{16} + 102 p^{24} T^{17} + p^{27} T^{18} \) | |
| 29 | \( 1 - 280 T + 114249 T^{2} - 26146034 T^{3} + 6260917959 T^{4} - 1182137559562 T^{5} + 222056519145515 T^{6} - 35794364871880542 T^{7} + 6099340855312710708 T^{8} - \)\(90\!\cdots\!04\)\( T^{9} + 6099340855312710708 p^{3} T^{10} - 35794364871880542 p^{6} T^{11} + 222056519145515 p^{9} T^{12} - 1182137559562 p^{12} T^{13} + 6260917959 p^{15} T^{14} - 26146034 p^{18} T^{15} + 114249 p^{21} T^{16} - 280 p^{24} T^{17} + p^{27} T^{18} \) | |
| 31 | \( 1 + 168 T + 100010 T^{2} + 5204672 T^{3} + 4819367979 T^{4} + 109871016988 T^{5} + 227880204896062 T^{6} + 4765975750096380 T^{7} + 8403068948268895673 T^{8} + \)\(12\!\cdots\!84\)\( T^{9} + 8403068948268895673 p^{3} T^{10} + 4765975750096380 p^{6} T^{11} + 227880204896062 p^{9} T^{12} + 109871016988 p^{12} T^{13} + 4819367979 p^{15} T^{14} + 5204672 p^{18} T^{15} + 100010 p^{21} T^{16} + 168 p^{24} T^{17} + p^{27} T^{18} \) | |
| 37 | \( 1 - 153 T + 284647 T^{2} - 35923820 T^{3} + 40854618186 T^{4} - 4569797686820 T^{5} + 3882334686608142 T^{6} - 384537171897705972 T^{7} + \)\(26\!\cdots\!60\)\( T^{8} - \)\(22\!\cdots\!34\)\( T^{9} + \)\(26\!\cdots\!60\)\( p^{3} T^{10} - 384537171897705972 p^{6} T^{11} + 3882334686608142 p^{9} T^{12} - 4569797686820 p^{12} T^{13} + 40854618186 p^{15} T^{14} - 35923820 p^{18} T^{15} + 284647 p^{21} T^{16} - 153 p^{24} T^{17} + p^{27} T^{18} \) | |
| 41 | \( 1 + 502 T + 259736 T^{2} + 86501476 T^{3} + 33384577449 T^{4} + 272686384346 p T^{5} + 3939370910437684 T^{6} + 1120397752521027278 T^{7} + \)\(32\!\cdots\!85\)\( T^{8} + \)\(79\!\cdots\!54\)\( T^{9} + \)\(32\!\cdots\!85\)\( p^{3} T^{10} + 1120397752521027278 p^{6} T^{11} + 3939370910437684 p^{9} T^{12} + 272686384346 p^{13} T^{13} + 33384577449 p^{15} T^{14} + 86501476 p^{18} T^{15} + 259736 p^{21} T^{16} + 502 p^{24} T^{17} + p^{27} T^{18} \) | |
| 43 | \( 1 + 110 T + 504179 T^{2} + 40035800 T^{3} + 117226619596 T^{4} + 5801519934808 T^{5} + 16957798929694244 T^{6} + 452267799166326632 T^{7} + \)\(17\!\cdots\!98\)\( T^{8} + \)\(30\!\cdots\!36\)\( T^{9} + \)\(17\!\cdots\!98\)\( p^{3} T^{10} + 452267799166326632 p^{6} T^{11} + 16957798929694244 p^{9} T^{12} + 5801519934808 p^{12} T^{13} + 117226619596 p^{15} T^{14} + 40035800 p^{18} T^{15} + 504179 p^{21} T^{16} + 110 p^{24} T^{17} + p^{27} T^{18} \) | |
| 47 | \( 1 - 153 T + 335978 T^{2} - 76301983 T^{3} + 63815892639 T^{4} - 18498478505870 T^{5} + 8684045307666267 T^{6} - 2980110571567440849 T^{7} + \)\(96\!\cdots\!05\)\( T^{8} - \)\(36\!\cdots\!94\)\( T^{9} + \)\(96\!\cdots\!05\)\( p^{3} T^{10} - 2980110571567440849 p^{6} T^{11} + 8684045307666267 p^{9} T^{12} - 18498478505870 p^{12} T^{13} + 63815892639 p^{15} T^{14} - 76301983 p^{18} T^{15} + 335978 p^{21} T^{16} - 153 p^{24} T^{17} + p^{27} T^{18} \) | |
| 53 | \( 1 + 273 T + 523869 T^{2} + 183155640 T^{3} + 153366088636 T^{4} + 54028899288828 T^{5} + 33096042278049580 T^{6} + 11130876193192091208 T^{7} + \)\(55\!\cdots\!74\)\( T^{8} + \)\(18\!\cdots\!62\)\( T^{9} + \)\(55\!\cdots\!74\)\( p^{3} T^{10} + 11130876193192091208 p^{6} T^{11} + 33096042278049580 p^{9} T^{12} + 54028899288828 p^{12} T^{13} + 153366088636 p^{15} T^{14} + 183155640 p^{18} T^{15} + 523869 p^{21} T^{16} + 273 p^{24} T^{17} + p^{27} T^{18} \) | |
| 59 | \( 1 + 827 T + 1499439 T^{2} + 1044839972 T^{3} + 1085104255880 T^{4} + 630183147315164 T^{5} + 479617458926198464 T^{6} + \)\(23\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!22\)\( T^{8} + \)\(58\!\cdots\!42\)\( T^{9} + \)\(14\!\cdots\!22\)\( p^{3} T^{10} + \)\(23\!\cdots\!00\)\( p^{6} T^{11} + 479617458926198464 p^{9} T^{12} + 630183147315164 p^{12} T^{13} + 1085104255880 p^{15} T^{14} + 1044839972 p^{18} T^{15} + 1499439 p^{21} T^{16} + 827 p^{24} T^{17} + p^{27} T^{18} \) | |
| 61 | \( 1 - 1976 T + 3075400 T^{2} - 3262602900 T^{3} + 3013287624910 T^{4} - 2271369268835106 T^{5} + 1556199414337381624 T^{6} - \)\(92\!\cdots\!56\)\( T^{7} + \)\(51\!\cdots\!33\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{9} + \)\(51\!\cdots\!33\)\( p^{3} T^{10} - \)\(92\!\cdots\!56\)\( p^{6} T^{11} + 1556199414337381624 p^{9} T^{12} - 2271369268835106 p^{12} T^{13} + 3013287624910 p^{15} T^{14} - 3262602900 p^{18} T^{15} + 3075400 p^{21} T^{16} - 1976 p^{24} T^{17} + p^{27} T^{18} \) | |
| 67 | \( 1 + 1613 T + 3056699 T^{2} + 3241275972 T^{3} + 3580107189908 T^{4} + 2848552051924996 T^{5} + 2323652137507265020 T^{6} + \)\(14\!\cdots\!76\)\( T^{7} + \)\(98\!\cdots\!58\)\( T^{8} + \)\(53\!\cdots\!02\)\( T^{9} + \)\(98\!\cdots\!58\)\( p^{3} T^{10} + \)\(14\!\cdots\!76\)\( p^{6} T^{11} + 2323652137507265020 p^{9} T^{12} + 2848552051924996 p^{12} T^{13} + 3580107189908 p^{15} T^{14} + 3241275972 p^{18} T^{15} + 3056699 p^{21} T^{16} + 1613 p^{24} T^{17} + p^{27} T^{18} \) | |
| 71 | \( 1 + 1370 T + 1599336 T^{2} + 1430564556 T^{3} + 1308255548585 T^{4} + 1014445974684680 T^{5} + 773855850389129196 T^{6} + \)\(53\!\cdots\!06\)\( T^{7} + \)\(35\!\cdots\!61\)\( T^{8} + \)\(21\!\cdots\!04\)\( T^{9} + \)\(35\!\cdots\!61\)\( p^{3} T^{10} + \)\(53\!\cdots\!06\)\( p^{6} T^{11} + 773855850389129196 p^{9} T^{12} + 1014445974684680 p^{12} T^{13} + 1308255548585 p^{15} T^{14} + 1430564556 p^{18} T^{15} + 1599336 p^{21} T^{16} + 1370 p^{24} T^{17} + p^{27} T^{18} \) | |
| 73 | \( 1 - 425 T + 1639776 T^{2} - 485309631 T^{3} + 1483427453093 T^{4} - 340476643673134 T^{5} + 915037395346314271 T^{6} - \)\(15\!\cdots\!17\)\( T^{7} + \)\(43\!\cdots\!23\)\( T^{8} - \)\(63\!\cdots\!26\)\( T^{9} + \)\(43\!\cdots\!23\)\( p^{3} T^{10} - \)\(15\!\cdots\!17\)\( p^{6} T^{11} + 915037395346314271 p^{9} T^{12} - 340476643673134 p^{12} T^{13} + 1483427453093 p^{15} T^{14} - 485309631 p^{18} T^{15} + 1639776 p^{21} T^{16} - 425 p^{24} T^{17} + p^{27} T^{18} \) | |
| 79 | \( 1 + 2624 T + 6338851 T^{2} + 9805353032 T^{3} + 14048858013568 T^{4} + 15833065033976144 T^{5} + 16753925827879254424 T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!62\)\( T^{8} + \)\(90\!\cdots\!12\)\( T^{9} + \)\(12\!\cdots\!62\)\( p^{3} T^{10} + \)\(14\!\cdots\!20\)\( p^{6} T^{11} + 16753925827879254424 p^{9} T^{12} + 15833065033976144 p^{12} T^{13} + 14048858013568 p^{15} T^{14} + 9805353032 p^{18} T^{15} + 6338851 p^{21} T^{16} + 2624 p^{24} T^{17} + p^{27} T^{18} \) | |
| 83 | \( 1 + 2505 T + 4841575 T^{2} + 5921154168 T^{3} + 6244394820536 T^{4} + 4660398571327884 T^{5} + 3051123301044991344 T^{6} + \)\(11\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!30\)\( T^{8} - \)\(98\!\cdots\!10\)\( T^{9} + \)\(33\!\cdots\!30\)\( p^{3} T^{10} + \)\(11\!\cdots\!16\)\( p^{6} T^{11} + 3051123301044991344 p^{9} T^{12} + 4660398571327884 p^{12} T^{13} + 6244394820536 p^{15} T^{14} + 5921154168 p^{18} T^{15} + 4841575 p^{21} T^{16} + 2505 p^{24} T^{17} + p^{27} T^{18} \) | |
| 89 | \( 1 - 1120 T + 2193301 T^{2} - 1094054248 T^{3} + 1813601496304 T^{4} - 668800925098032 T^{5} + 1531170734301446936 T^{6} - \)\(31\!\cdots\!88\)\( T^{7} + \)\(80\!\cdots\!26\)\( T^{8} + \)\(88\!\cdots\!36\)\( T^{9} + \)\(80\!\cdots\!26\)\( p^{3} T^{10} - \)\(31\!\cdots\!88\)\( p^{6} T^{11} + 1531170734301446936 p^{9} T^{12} - 668800925098032 p^{12} T^{13} + 1813601496304 p^{15} T^{14} - 1094054248 p^{18} T^{15} + 2193301 p^{21} T^{16} - 1120 p^{24} T^{17} + p^{27} T^{18} \) | |
| 97 | \( 1 - 2026 T + 5085342 T^{2} - 8558477894 T^{3} + 13362866750624 T^{4} - 18191982232836658 T^{5} + 23054644895043811006 T^{6} - \)\(26\!\cdots\!14\)\( T^{7} + \)\(28\!\cdots\!43\)\( T^{8} - \)\(27\!\cdots\!28\)\( T^{9} + \)\(28\!\cdots\!43\)\( p^{3} T^{10} - \)\(26\!\cdots\!14\)\( p^{6} T^{11} + 23054644895043811006 p^{9} T^{12} - 18191982232836658 p^{12} T^{13} + 13362866750624 p^{15} T^{14} - 8558477894 p^{18} T^{15} + 5085342 p^{21} T^{16} - 2026 p^{24} T^{17} + p^{27} T^{18} \) | |
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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
−3.67877085116585660949338010205, −3.42330541293075433978201348740, −3.30916125212103059318572507276, −3.29582016845075520543482973700, −3.28866244953617053060645911961, −3.24384851767838902696436061558, −3.22492054905952247804863664688, −2.81588136020400253106772714907, −2.70015069101442011699805302054, −2.61242292536146215975202255404, −2.55011976734063389878485165184, −2.52367835088160206855465445893, −2.43914518301184209029814190538, −2.43568952797812085267718207070, −2.36167399613267921961668463334, −2.12951192541883760771335697268, −1.75663338568116712674365700190, −1.46635957609314482884607517528, −1.38093492662889465375602379225, −1.31737291477286692634115403659, −1.23839204646157960764267333940, −1.21275604865329635971322013278, −0.993800067126839626635724396185, −0.857648842329206787333890148430, −0.73509230947028949756311715863, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.73509230947028949756311715863, 0.857648842329206787333890148430, 0.993800067126839626635724396185, 1.21275604865329635971322013278, 1.23839204646157960764267333940, 1.31737291477286692634115403659, 1.38093492662889465375602379225, 1.46635957609314482884607517528, 1.75663338568116712674365700190, 2.12951192541883760771335697268, 2.36167399613267921961668463334, 2.43568952797812085267718207070, 2.43914518301184209029814190538, 2.52367835088160206855465445893, 2.55011976734063389878485165184, 2.61242292536146215975202255404, 2.70015069101442011699805302054, 2.81588136020400253106772714907, 3.22492054905952247804863664688, 3.24384851767838902696436061558, 3.28866244953617053060645911961, 3.29582016845075520543482973700, 3.30916125212103059318572507276, 3.42330541293075433978201348740, 3.67877085116585660949338010205
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.