Dirichlet series
| L(s) = 1 | − 9·3-s − 6·5-s + 11·7-s + 45·9-s − 11-s − 4·13-s + 54·15-s − 9·17-s + 8·19-s − 99·21-s + 7·23-s − 5·25-s − 165·27-s − 9·29-s + 25·31-s + 9·33-s − 66·35-s − 6·37-s + 36·39-s − 4·41-s + 24·43-s − 270·45-s + 16·47-s + 31·49-s + 81·51-s − 26·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 5.19·3-s − 2.68·5-s + 4.15·7-s + 15·9-s − 0.301·11-s − 1.10·13-s + 13.9·15-s − 2.18·17-s + 1.83·19-s − 21.6·21-s + 1.45·23-s − 25-s − 31.7·27-s − 1.67·29-s + 4.49·31-s + 1.56·33-s − 11.1·35-s − 0.986·37-s + 5.76·39-s − 0.624·41-s + 3.65·43-s − 40.2·45-s + 2.33·47-s + 31/7·49-s + 11.3·51-s − 3.57·53-s + 0.809·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{9} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{36} \cdot 3^{9} \cdot 167^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.80348\times 10^{16}\) |
| Root analytic conductor: | \(8.00050\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{36} \cdot 3^{9} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.5655770449\) |
| \(L(\frac12)\) | \(\approx\) | \(0.5655770449\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{9} \) | |
| 167 | \( ( 1 - T )^{9} \) | |
| good | 5 | \( 1 + 6 T + 41 T^{2} + 173 T^{3} + 719 T^{4} + 478 p T^{5} + 7554 T^{6} + 20743 T^{7} + 10721 p T^{8} + 123718 T^{9} + 10721 p^{2} T^{10} + 20743 p^{2} T^{11} + 7554 p^{3} T^{12} + 478 p^{5} T^{13} + 719 p^{5} T^{14} + 173 p^{6} T^{15} + 41 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 11 T + 90 T^{2} - 528 T^{3} + 2641 T^{4} - 11154 T^{5} + 42250 T^{6} - 141980 T^{7} + 62250 p T^{8} - 1204094 T^{9} + 62250 p^{2} T^{10} - 141980 p^{2} T^{11} + 42250 p^{3} T^{12} - 11154 p^{4} T^{13} + 2641 p^{5} T^{14} - 528 p^{6} T^{15} + 90 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + T + 52 T^{2} + 70 T^{3} + 1332 T^{4} + 2129 T^{5} + 23490 T^{6} + 38142 T^{7} + 322805 T^{8} + 481364 T^{9} + 322805 p T^{10} + 38142 p^{2} T^{11} + 23490 p^{3} T^{12} + 2129 p^{4} T^{13} + 1332 p^{5} T^{14} + 70 p^{6} T^{15} + 52 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 4 T + 74 T^{2} + 264 T^{3} + 2783 T^{4} + 8611 T^{5} + 67581 T^{6} + 182788 T^{7} + 1176589 T^{8} + 2768138 T^{9} + 1176589 p T^{10} + 182788 p^{2} T^{11} + 67581 p^{3} T^{12} + 8611 p^{4} T^{13} + 2783 p^{5} T^{14} + 264 p^{6} T^{15} + 74 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 9 T + 105 T^{2} + 636 T^{3} + 4431 T^{4} + 20141 T^{5} + 107753 T^{6} + 398006 T^{7} + 1917132 T^{8} + 6639284 T^{9} + 1917132 p T^{10} + 398006 p^{2} T^{11} + 107753 p^{3} T^{12} + 20141 p^{4} T^{13} + 4431 p^{5} T^{14} + 636 p^{6} T^{15} + 105 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 8 T + 74 T^{2} - 18 p T^{3} + 1683 T^{4} - 3555 T^{5} + 13241 T^{6} + 21770 T^{7} + 26115 T^{8} + 779550 T^{9} + 26115 p T^{10} + 21770 p^{2} T^{11} + 13241 p^{3} T^{12} - 3555 p^{4} T^{13} + 1683 p^{5} T^{14} - 18 p^{7} T^{15} + 74 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 7 T + 122 T^{2} - 616 T^{3} + 6996 T^{4} - 30587 T^{5} + 281820 T^{6} - 1091516 T^{7} + 8438553 T^{8} - 28681556 T^{9} + 8438553 p T^{10} - 1091516 p^{2} T^{11} + 281820 p^{3} T^{12} - 30587 p^{4} T^{13} + 6996 p^{5} T^{14} - 616 p^{6} T^{15} + 122 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 9 T + 160 T^{2} + 954 T^{3} + 10872 T^{4} + 50409 T^{5} + 484084 T^{6} + 1856662 T^{7} + 16472499 T^{8} + 56162028 T^{9} + 16472499 p T^{10} + 1856662 p^{2} T^{11} + 484084 p^{3} T^{12} + 50409 p^{4} T^{13} + 10872 p^{5} T^{14} + 954 p^{6} T^{15} + 160 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 25 T + 431 T^{2} - 5443 T^{3} + 57907 T^{4} - 525407 T^{5} + 4233640 T^{6} - 30306973 T^{7} + 196210743 T^{8} - 1145270768 T^{9} + 196210743 p T^{10} - 30306973 p^{2} T^{11} + 4233640 p^{3} T^{12} - 525407 p^{4} T^{13} + 57907 p^{5} T^{14} - 5443 p^{6} T^{15} + 431 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 6 T + 165 T^{2} + 967 T^{3} + 12241 T^{4} + 69142 T^{5} + 530842 T^{6} + 3130351 T^{7} + 16937275 T^{8} + 117767388 T^{9} + 16937275 p T^{10} + 3130351 p^{2} T^{11} + 530842 p^{3} T^{12} + 69142 p^{4} T^{13} + 12241 p^{5} T^{14} + 967 p^{6} T^{15} + 165 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 4 T + 257 T^{2} + 1351 T^{3} + 31614 T^{4} + 184863 T^{5} + 2521094 T^{6} + 14274085 T^{7} + 142940352 T^{8} + 712583638 T^{9} + 142940352 p T^{10} + 14274085 p^{2} T^{11} + 2521094 p^{3} T^{12} + 184863 p^{4} T^{13} + 31614 p^{5} T^{14} + 1351 p^{6} T^{15} + 257 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 24 T + 515 T^{2} - 7423 T^{3} + 96980 T^{4} - 1029695 T^{5} + 10046614 T^{6} - 84278929 T^{7} + 654510506 T^{8} - 4455064042 T^{9} + 654510506 p T^{10} - 84278929 p^{2} T^{11} + 10046614 p^{3} T^{12} - 1029695 p^{4} T^{13} + 96980 p^{5} T^{14} - 7423 p^{6} T^{15} + 515 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 16 T + 443 T^{2} - 5414 T^{3} + 84955 T^{4} - 827727 T^{5} + 9385058 T^{6} - 74534126 T^{7} + 662991021 T^{8} - 4317365814 T^{9} + 662991021 p T^{10} - 74534126 p^{2} T^{11} + 9385058 p^{3} T^{12} - 827727 p^{4} T^{13} + 84955 p^{5} T^{14} - 5414 p^{6} T^{15} + 443 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 26 T + 527 T^{2} + 7026 T^{3} + 77435 T^{4} + 638817 T^{5} + 4321678 T^{6} + 20643772 T^{7} + 78929197 T^{8} + 281350616 T^{9} + 78929197 p T^{10} + 20643772 p^{2} T^{11} + 4321678 p^{3} T^{12} + 638817 p^{4} T^{13} + 77435 p^{5} T^{14} + 7026 p^{6} T^{15} + 527 p^{7} T^{16} + 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 4 T + 325 T^{2} - 1542 T^{3} + 55375 T^{4} - 257845 T^{5} + 6265316 T^{6} - 26719662 T^{7} + 503343089 T^{8} - 1894242454 T^{9} + 503343089 p T^{10} - 26719662 p^{2} T^{11} + 6265316 p^{3} T^{12} - 257845 p^{4} T^{13} + 55375 p^{5} T^{14} - 1542 p^{6} T^{15} + 325 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 20 T + 602 T^{2} + 9248 T^{3} + 155243 T^{4} + 1900323 T^{5} + 22802367 T^{6} + 226604280 T^{7} + 2116420563 T^{8} + 17147925490 T^{9} + 2116420563 p T^{10} + 226604280 p^{2} T^{11} + 22802367 p^{3} T^{12} + 1900323 p^{4} T^{13} + 155243 p^{5} T^{14} + 9248 p^{6} T^{15} + 602 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 25 T + 568 T^{2} - 9209 T^{3} + 132959 T^{4} - 1619633 T^{5} + 18110518 T^{6} - 180254541 T^{7} + 1671954472 T^{8} - 14144938314 T^{9} + 1671954472 p T^{10} - 180254541 p^{2} T^{11} + 18110518 p^{3} T^{12} - 1619633 p^{4} T^{13} + 132959 p^{5} T^{14} - 9209 p^{6} T^{15} + 568 p^{7} T^{16} - 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 15 T + 449 T^{2} - 5499 T^{3} + 91049 T^{4} - 954797 T^{5} + 11431765 T^{6} - 106092249 T^{7} + 1035484196 T^{8} - 8617953600 T^{9} + 1035484196 p T^{10} - 106092249 p^{2} T^{11} + 11431765 p^{3} T^{12} - 954797 p^{4} T^{13} + 91049 p^{5} T^{14} - 5499 p^{6} T^{15} + 449 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 10 T + 276 T^{2} + 3380 T^{3} + 53683 T^{4} + 589381 T^{5} + 6867531 T^{6} + 71234924 T^{7} + 666366469 T^{8} + 5992553762 T^{9} + 666366469 p T^{10} + 71234924 p^{2} T^{11} + 6867531 p^{3} T^{12} + 589381 p^{4} T^{13} + 53683 p^{5} T^{14} + 3380 p^{6} T^{15} + 276 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 34 T + 941 T^{2} - 18473 T^{3} + 313218 T^{4} - 4495933 T^{5} + 57476662 T^{6} - 655683215 T^{7} + 6748904942 T^{8} - 63046688450 T^{9} + 6748904942 p T^{10} - 655683215 p^{2} T^{11} + 57476662 p^{3} T^{12} - 4495933 p^{4} T^{13} + 313218 p^{5} T^{14} - 18473 p^{6} T^{15} + 941 p^{7} T^{16} - 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 4 T + 451 T^{2} - 512 T^{3} + 86177 T^{4} + 2635 p T^{5} + 9317210 T^{6} + 68434988 T^{7} + 730048475 T^{8} + 8141053038 T^{9} + 730048475 p T^{10} + 68434988 p^{2} T^{11} + 9317210 p^{3} T^{12} + 2635 p^{5} T^{13} + 86177 p^{5} T^{14} - 512 p^{6} T^{15} + 451 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 13 T + 480 T^{2} - 6469 T^{3} + 126713 T^{4} - 1525667 T^{5} + 22163950 T^{6} - 229481013 T^{7} + 2736499528 T^{8} - 24097791692 T^{9} + 2736499528 p T^{10} - 229481013 p^{2} T^{11} + 22163950 p^{3} T^{12} - 1525667 p^{4} T^{13} + 126713 p^{5} T^{14} - 6469 p^{6} T^{15} + 480 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 4 T + 306 T^{2} + 941 T^{3} + 54811 T^{4} + 87444 T^{5} + 7554000 T^{6} + 3292913 T^{7} + 811591682 T^{8} - 7315132 T^{9} + 811591682 p T^{10} + 3292913 p^{2} T^{11} + 7554000 p^{3} T^{12} + 87444 p^{4} T^{13} + 54811 p^{5} T^{14} + 941 p^{6} T^{15} + 306 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.75438836864797443148529596704, −2.72430836331412888427472473847, −2.58326646069481317137128628291, −2.53431060417193783515299842258, −2.44703262183904292016352675582, −2.44260823666896551179121071802, −2.37716355209986162408921028392, −1.97358215802997982273294077535, −1.81068800750492243838849535684, −1.74740656008343010575332156701, −1.68243162750770505510340117892, −1.64393695518311994111807661368, −1.64034711528165687739693352508, −1.59381477912860457584154167299, −1.48599722487777907647564550268, −1.47833025938940282960356339080, −0.972681135288972757009574280332, −0.836329489761701077791381687715, −0.792429339941357491478249837240, −0.68748175377681242193768948320, −0.59261146243357259774131289172, −0.55464746413340524569983828518, −0.50927216099690422977278035870, −0.34829004676635298897594167364, −0.087809471718346019428789993430, 0.087809471718346019428789993430, 0.34829004676635298897594167364, 0.50927216099690422977278035870, 0.55464746413340524569983828518, 0.59261146243357259774131289172, 0.68748175377681242193768948320, 0.792429339941357491478249837240, 0.836329489761701077791381687715, 0.972681135288972757009574280332, 1.47833025938940282960356339080, 1.48599722487777907647564550268, 1.59381477912860457584154167299, 1.64034711528165687739693352508, 1.64393695518311994111807661368, 1.68243162750770505510340117892, 1.74740656008343010575332156701, 1.81068800750492243838849535684, 1.97358215802997982273294077535, 2.37716355209986162408921028392, 2.44260823666896551179121071802, 2.44703262183904292016352675582, 2.53431060417193783515299842258, 2.58326646069481317137128628291, 2.72430836331412888427472473847, 2.75438836864797443148529596704
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.