Dirichlet series
| L(s) = 1 | + 9·2-s + 3·3-s + 45·4-s − 9·5-s + 27·6-s + 3·7-s + 165·8-s − 2·9-s − 81·10-s + 6·11-s + 135·12-s + 9·13-s + 27·14-s − 27·15-s + 495·16-s + 3·17-s − 18·18-s + 6·19-s − 405·20-s + 9·21-s + 54·22-s + 14·23-s + 495·24-s + 45·25-s + 81·26-s − 15·27-s + 135·28-s + ⋯ |
| L(s) = 1 | + 6.36·2-s + 1.73·3-s + 45/2·4-s − 4.02·5-s + 11.0·6-s + 1.13·7-s + 58.3·8-s − 2/3·9-s − 25.6·10-s + 1.80·11-s + 38.9·12-s + 2.49·13-s + 7.21·14-s − 6.97·15-s + 123.·16-s + 0.727·17-s − 4.24·18-s + 1.37·19-s − 90.5·20-s + 1.96·21-s + 11.5·22-s + 2.91·23-s + 101.·24-s + 9·25-s + 15.8·26-s − 2.88·27-s + 25.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.70032\times 10^{13}\) |
| Root analytic conductor: | \(5.67271\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(14892.40989\) |
| \(L(\frac12)\) | \(\approx\) | \(14892.40989\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{9} \) |
| 5 | \( ( 1 + T )^{9} \) | |
| 13 | \( ( 1 - T )^{9} \) | |
| 31 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 - p T + 11 T^{2} - 8 p T^{3} + 2 p^{3} T^{4} - 94 T^{5} + 53 p T^{6} - 10 p^{3} T^{7} + 419 T^{8} - 754 T^{9} + 419 p T^{10} - 10 p^{5} T^{11} + 53 p^{4} T^{12} - 94 p^{4} T^{13} + 2 p^{8} T^{14} - 8 p^{7} T^{15} + 11 p^{7} T^{16} - p^{9} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 3 T + 32 T^{2} - 74 T^{3} + 431 T^{4} - 965 T^{5} + 4043 T^{6} - 10820 T^{7} + 34464 T^{8} - 93260 T^{9} + 34464 p T^{10} - 10820 p^{2} T^{11} + 4043 p^{3} T^{12} - 965 p^{4} T^{13} + 431 p^{5} T^{14} - 74 p^{6} T^{15} + 32 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 6 T + 68 T^{2} - 326 T^{3} + 2203 T^{4} - 9120 T^{5} + 46440 T^{6} - 167070 T^{7} + 693932 T^{8} - 2158676 T^{9} + 693932 p T^{10} - 167070 p^{2} T^{11} + 46440 p^{3} T^{12} - 9120 p^{4} T^{13} + 2203 p^{5} T^{14} - 326 p^{6} T^{15} + 68 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 3 T + 3 p T^{2} - 253 T^{3} + 1848 T^{4} - 9348 T^{5} + 50379 T^{6} - 241809 T^{7} + 1098114 T^{8} - 4571770 T^{9} + 1098114 p T^{10} - 241809 p^{2} T^{11} + 50379 p^{3} T^{12} - 9348 p^{4} T^{13} + 1848 p^{5} T^{14} - 253 p^{6} T^{15} + 3 p^{8} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 6 T + 104 T^{2} - 24 p T^{3} + 4337 T^{4} - 14771 T^{5} + 102015 T^{6} - 295199 T^{7} + 1796016 T^{8} - 5248880 T^{9} + 1796016 p T^{10} - 295199 p^{2} T^{11} + 102015 p^{3} T^{12} - 14771 p^{4} T^{13} + 4337 p^{5} T^{14} - 24 p^{7} T^{15} + 104 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 14 T + 223 T^{2} - 2115 T^{3} + 20201 T^{4} - 147170 T^{5} + 1051026 T^{6} - 6195899 T^{7} + 35535718 T^{8} - 172919562 T^{9} + 35535718 p T^{10} - 6195899 p^{2} T^{11} + 1051026 p^{3} T^{12} - 147170 p^{4} T^{13} + 20201 p^{5} T^{14} - 2115 p^{6} T^{15} + 223 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 17 T + 299 T^{2} - 3403 T^{3} + 35914 T^{4} - 310736 T^{5} + 2437809 T^{6} - 16900493 T^{7} + 105701360 T^{8} - 599567110 T^{9} + 105701360 p T^{10} - 16900493 p^{2} T^{11} + 2437809 p^{3} T^{12} - 310736 p^{4} T^{13} + 35914 p^{5} T^{14} - 3403 p^{6} T^{15} + 299 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 3 T + 257 T^{2} + 717 T^{3} + 31295 T^{4} + 78646 T^{5} + 2369130 T^{6} + 5239252 T^{7} + 123049776 T^{8} + 233739262 T^{9} + 123049776 p T^{10} + 5239252 p^{2} T^{11} + 2369130 p^{3} T^{12} + 78646 p^{4} T^{13} + 31295 p^{5} T^{14} + 717 p^{6} T^{15} + 257 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 4 T + 272 T^{2} - 1139 T^{3} + 34513 T^{4} - 148165 T^{5} + 2730648 T^{6} - 11463853 T^{7} + 151601330 T^{8} - 14056774 p T^{9} + 151601330 p T^{10} - 11463853 p^{2} T^{11} + 2730648 p^{3} T^{12} - 148165 p^{4} T^{13} + 34513 p^{5} T^{14} - 1139 p^{6} T^{15} + 272 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 5 T + 142 T^{2} - 390 T^{3} + 9762 T^{4} - 15174 T^{5} + 519771 T^{6} - 670852 T^{7} + 25907114 T^{8} - 36982262 T^{9} + 25907114 p T^{10} - 670852 p^{2} T^{11} + 519771 p^{3} T^{12} - 15174 p^{4} T^{13} + 9762 p^{5} T^{14} - 390 p^{6} T^{15} + 142 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 25 T + 500 T^{2} - 6452 T^{3} + 69763 T^{4} - 559969 T^{5} + 3790341 T^{6} - 18604876 T^{7} + 83842004 T^{8} - 370711736 T^{9} + 83842004 p T^{10} - 18604876 p^{2} T^{11} + 3790341 p^{3} T^{12} - 559969 p^{4} T^{13} + 69763 p^{5} T^{14} - 6452 p^{6} T^{15} + 500 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 18 T + 474 T^{2} - 6353 T^{3} + 97653 T^{4} - 1037907 T^{5} + 11740872 T^{6} - 102091845 T^{7} + 918398760 T^{8} - 6598306202 T^{9} + 918398760 p T^{10} - 102091845 p^{2} T^{11} + 11740872 p^{3} T^{12} - 1037907 p^{4} T^{13} + 97653 p^{5} T^{14} - 6353 p^{6} T^{15} + 474 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 6 T + 371 T^{2} + 2729 T^{3} + 67363 T^{4} + 519084 T^{5} + 7953510 T^{6} + 56859033 T^{7} + 659469998 T^{8} + 4071061112 T^{9} + 659469998 p T^{10} + 56859033 p^{2} T^{11} + 7953510 p^{3} T^{12} + 519084 p^{4} T^{13} + 67363 p^{5} T^{14} + 2729 p^{6} T^{15} + 371 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 26 T + 744 T^{2} - 12696 T^{3} + 212216 T^{4} - 2699981 T^{5} + 32806722 T^{6} - 328137798 T^{7} + 3112858718 T^{8} - 24983064122 T^{9} + 3112858718 p T^{10} - 328137798 p^{2} T^{11} + 32806722 p^{3} T^{12} - 2699981 p^{4} T^{13} + 212216 p^{5} T^{14} - 12696 p^{6} T^{15} + 744 p^{7} T^{16} - 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 2 T + 525 T^{2} + 802 T^{3} + 125884 T^{4} + 147515 T^{5} + 18204491 T^{6} + 16752306 T^{7} + 1757570683 T^{8} + 1322234686 T^{9} + 1757570683 p T^{10} + 16752306 p^{2} T^{11} + 18204491 p^{3} T^{12} + 147515 p^{4} T^{13} + 125884 p^{5} T^{14} + 802 p^{6} T^{15} + 525 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 18 T + 654 T^{2} - 8684 T^{3} + 177246 T^{4} - 1852914 T^{5} + 27477057 T^{6} - 235330908 T^{7} + 2794952640 T^{8} - 20037290312 T^{9} + 2794952640 p T^{10} - 235330908 p^{2} T^{11} + 27477057 p^{3} T^{12} - 1852914 p^{4} T^{13} + 177246 p^{5} T^{14} - 8684 p^{6} T^{15} + 654 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 5 T + 302 T^{2} + 1298 T^{3} + 39267 T^{4} + 152319 T^{5} + 3003058 T^{6} + 11844364 T^{7} + 179892820 T^{8} + 833799268 T^{9} + 179892820 p T^{10} + 11844364 p^{2} T^{11} + 3003058 p^{3} T^{12} + 152319 p^{4} T^{13} + 39267 p^{5} T^{14} + 1298 p^{6} T^{15} + 302 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 18 T + 582 T^{2} - 7034 T^{3} + 129423 T^{4} - 1151070 T^{5} + 16167662 T^{6} - 114934422 T^{7} + 1458169524 T^{8} - 9329847056 T^{9} + 1458169524 p T^{10} - 114934422 p^{2} T^{11} + 16167662 p^{3} T^{12} - 1151070 p^{4} T^{13} + 129423 p^{5} T^{14} - 7034 p^{6} T^{15} + 582 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 13 T + 244 T^{2} - 2436 T^{3} + 33629 T^{4} - 257179 T^{5} + 3216915 T^{6} - 19582906 T^{7} + 247031002 T^{8} - 1560041268 T^{9} + 247031002 p T^{10} - 19582906 p^{2} T^{11} + 3216915 p^{3} T^{12} - 257179 p^{4} T^{13} + 33629 p^{5} T^{14} - 2436 p^{6} T^{15} + 244 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 9 T + 177 T^{2} - 739 T^{3} + 23976 T^{4} - 149190 T^{5} + 2891847 T^{6} - 14344869 T^{7} + 313124952 T^{8} - 1863251872 T^{9} + 313124952 p T^{10} - 14344869 p^{2} T^{11} + 2891847 p^{3} T^{12} - 149190 p^{4} T^{13} + 23976 p^{5} T^{14} - 739 p^{6} T^{15} + 177 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 18 T + 821 T^{2} + 11434 T^{3} + 293816 T^{4} + 3336292 T^{5} + 62271944 T^{6} + 590155591 T^{7} + 8747268651 T^{8} + 69391285948 T^{9} + 8747268651 p T^{10} + 590155591 p^{2} T^{11} + 62271944 p^{3} T^{12} + 3336292 p^{4} T^{13} + 293816 p^{5} T^{14} + 11434 p^{6} T^{15} + 821 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.15031409429891803908589721388, −3.14889829546467743411843309907, −3.13355522174673446940392690492, −3.12618930187154019550215016805, −3.02364921360726517540221728496, −2.98204592521497483966454979936, −2.95077200223110750456487382870, −2.54096078120525341099629277275, −2.36468078777033185389415254977, −2.34323184196212689273806488374, −2.26671539999507629535918338125, −2.20528882255948944581608683650, −2.16090180720807710949361133117, −2.10134136522168052569135617865, −1.90406813982819245440662618765, −1.64051406078699272800613870248, −1.34970409942234394712252682866, −1.26734850387862052411016022387, −1.19280338956479130910494335836, −1.00206094482527716477187893061, −0.944224929106252213037306613977, −0.876445763163244354104558612460, −0.803818902919450142444794385399, −0.58358184417592675011091411348, −0.55010447011081702749889623739, 0.55010447011081702749889623739, 0.58358184417592675011091411348, 0.803818902919450142444794385399, 0.876445763163244354104558612460, 0.944224929106252213037306613977, 1.00206094482527716477187893061, 1.19280338956479130910494335836, 1.26734850387862052411016022387, 1.34970409942234394712252682866, 1.64051406078699272800613870248, 1.90406813982819245440662618765, 2.10134136522168052569135617865, 2.16090180720807710949361133117, 2.20528882255948944581608683650, 2.26671539999507629535918338125, 2.34323184196212689273806488374, 2.36468078777033185389415254977, 2.54096078120525341099629277275, 2.95077200223110750456487382870, 2.98204592521497483966454979936, 3.02364921360726517540221728496, 3.12618930187154019550215016805, 3.13355522174673446940392690492, 3.14889829546467743411843309907, 3.15031409429891803908589721388
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.