Dirichlet series
| L(s) = 1 | − 5·2-s − 8·3-s + 9·4-s + 40·6-s − 8·7-s − 5·8-s + 25·9-s − 72·12-s − 6·13-s + 40·14-s − 6·16-s − 18·17-s − 125·18-s − 4·19-s + 64·21-s − 16·23-s + 40·24-s + 30·26-s − 32·27-s − 72·28-s − 2·29-s − 6·31-s + 14·32-s + 90·34-s + 225·36-s + 9·37-s + 20·38-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 4.61·3-s + 9/2·4-s + 16.3·6-s − 3.02·7-s − 1.76·8-s + 25/3·9-s − 20.7·12-s − 1.66·13-s + 10.6·14-s − 3/2·16-s − 4.36·17-s − 29.4·18-s − 0.917·19-s + 13.9·21-s − 3.33·23-s + 8.16·24-s + 5.88·26-s − 6.15·27-s − 13.6·28-s − 0.371·29-s − 1.07·31-s + 2.47·32-s + 15.4·34-s + 75/2·36-s + 1.47·37-s + 3.24·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 37^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 37^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(5^{18} \cdot 37^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.54288\times 10^{7}\) |
| Root analytic conductor: | \(2.71774\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 5^{18} \cdot 37^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 5 | \( 1 \) |
| 37 | \( ( 1 - T )^{9} \) | |
| good | 2 | \( 1 + 5 T + p^{4} T^{2} + 5 p^{3} T^{3} + 87 T^{4} + 171 T^{5} + 39 p^{3} T^{6} + 131 p^{2} T^{7} + 205 p^{2} T^{8} + 299 p^{2} T^{9} + 205 p^{3} T^{10} + 131 p^{4} T^{11} + 39 p^{6} T^{12} + 171 p^{4} T^{13} + 87 p^{5} T^{14} + 5 p^{9} T^{15} + p^{11} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 3 | \( 1 + 8 T + 13 p T^{2} + 16 p^{2} T^{3} + 149 p T^{4} + 1216 T^{5} + 985 p T^{6} + 718 p^{2} T^{7} + 12820 T^{8} + 2578 p^{2} T^{9} + 12820 p T^{10} + 718 p^{4} T^{11} + 985 p^{4} T^{12} + 1216 p^{4} T^{13} + 149 p^{6} T^{14} + 16 p^{8} T^{15} + 13 p^{8} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 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 | \( 1 - 2 T + 435 T^{2} - 2304 T^{3} + 103150 T^{4} - 691808 T^{5} + 17711706 T^{6} - 115122720 T^{7} + 2305098788 T^{8} - 13060706492 T^{9} + 2305098788 p T^{10} - 115122720 p^{2} T^{11} + 17711706 p^{3} T^{12} - 691808 p^{4} T^{13} + 103150 p^{5} T^{14} - 2304 p^{6} T^{15} + 435 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−4.53438958677899348795034551263, −4.49037731227175557622300288200, −4.42388197595327796156176802897, −4.22546047504805985197839934358, −4.16313419195263585453444073597, −3.94484896575209794768631521442, −3.91918144551939433313075768993, −3.73788644761699208973938273183, −3.58309087408736951244109473350, −3.32637079438596387846153615633, −3.23409868340090748840923119783, −3.11315893653660012523093732876, −3.05501236325565606637552568786, −2.92778995494525692062711071566, −2.53165525453817353485493738071, −2.48067292487186262191356776144, −2.45286833704972981336879617767, −2.40496315569077127182706367555, −2.01874241393456093052090794966, −1.95018822238348956058779491146, −1.87327632126738729686697154647, −1.39557708438843205617424548482, −1.32398558895225253924310895061, −1.18597353867793193973866949463, −1.17304735551230339377282055759, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.17304735551230339377282055759, 1.18597353867793193973866949463, 1.32398558895225253924310895061, 1.39557708438843205617424548482, 1.87327632126738729686697154647, 1.95018822238348956058779491146, 2.01874241393456093052090794966, 2.40496315569077127182706367555, 2.45286833704972981336879617767, 2.48067292487186262191356776144, 2.53165525453817353485493738071, 2.92778995494525692062711071566, 3.05501236325565606637552568786, 3.11315893653660012523093732876, 3.23409868340090748840923119783, 3.32637079438596387846153615633, 3.58309087408736951244109473350, 3.73788644761699208973938273183, 3.91918144551939433313075768993, 3.94484896575209794768631521442, 4.16313419195263585453444073597, 4.22546047504805985197839934358, 4.42388197595327796156176802897, 4.49037731227175557622300288200, 4.53438958677899348795034551263
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.