Dirichlet series
| L(s) = 1 | − 3-s − 9·5-s − 7-s − 7·9-s + 7·11-s + 9·15-s + 13·17-s + 4·19-s + 21-s + 12·23-s + 45·25-s + 7·27-s + 16·29-s − 13·31-s − 7·33-s + 9·35-s − 37-s + 6·41-s + 43-s + 63·45-s + 2·47-s − 21·49-s − 13·51-s + 30·53-s − 63·55-s − 4·57-s − 15·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4.02·5-s − 0.377·7-s − 7/3·9-s + 2.11·11-s + 2.32·15-s + 3.15·17-s + 0.917·19-s + 0.218·21-s + 2.50·23-s + 9·25-s + 1.34·27-s + 2.97·29-s − 2.33·31-s − 1.21·33-s + 1.52·35-s − 0.164·37-s + 0.937·41-s + 0.152·43-s + 9.39·45-s + 0.291·47-s − 3·49-s − 1.82·51-s + 4.12·53-s − 8.49·55-s − 0.529·57-s − 1.95·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 5^{9} \cdot 13^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.59878\times 10^{12}\) |
| Root analytic conductor: | \(5.19513\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{18} \cdot 5^{9} \cdot 13^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.63004448\) |
| \(L(\frac12)\) | \(\approx\) | \(10.63004448\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T )^{9} \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + 8 T^{2} + 8 T^{3} + 31 T^{4} + 17 p T^{5} + 38 p T^{6} + 247 T^{7} + 388 T^{8} + 833 T^{9} + 388 p T^{10} + 247 p^{2} T^{11} + 38 p^{4} T^{12} + 17 p^{5} T^{13} + 31 p^{5} T^{14} + 8 p^{6} T^{15} + 8 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + T + 22 T^{2} - 4 T^{3} + 39 p T^{4} - 197 T^{5} + 2876 T^{6} - 2173 T^{7} + 25850 T^{8} - 15541 T^{9} + 25850 p T^{10} - 2173 p^{2} T^{11} + 2876 p^{3} T^{12} - 197 p^{4} T^{13} + 39 p^{6} T^{14} - 4 p^{6} T^{15} + 22 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 7 T + 41 T^{2} - 13 p T^{3} + 582 T^{4} - 2490 T^{5} + 11926 T^{6} - 43145 T^{7} + 145500 T^{8} - 428286 T^{9} + 145500 p T^{10} - 43145 p^{2} T^{11} + 11926 p^{3} T^{12} - 2490 p^{4} T^{13} + 582 p^{5} T^{14} - 13 p^{7} T^{15} + 41 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 13 T + 127 T^{2} - 891 T^{3} + 5818 T^{4} - 33238 T^{5} + 180078 T^{6} - 879477 T^{7} + 4060136 T^{8} - 17187546 T^{9} + 4060136 p T^{10} - 879477 p^{2} T^{11} + 180078 p^{3} T^{12} - 33238 p^{4} T^{13} + 5818 p^{5} T^{14} - 891 p^{6} T^{15} + 127 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 4 T + 84 T^{2} - 355 T^{3} + 3541 T^{4} - 15002 T^{5} + 110061 T^{6} - 429333 T^{7} + 2741855 T^{8} - 9310468 T^{9} + 2741855 p T^{10} - 429333 p^{2} T^{11} + 110061 p^{3} T^{12} - 15002 p^{4} T^{13} + 3541 p^{5} T^{14} - 355 p^{6} T^{15} + 84 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 12 T + 173 T^{2} - 1335 T^{3} + 10807 T^{4} - 61633 T^{5} + 359296 T^{6} - 1685957 T^{7} + 8393430 T^{8} - 38069843 T^{9} + 8393430 p T^{10} - 1685957 p^{2} T^{11} + 359296 p^{3} T^{12} - 61633 p^{4} T^{13} + 10807 p^{5} T^{14} - 1335 p^{6} T^{15} + 173 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 16 T + 219 T^{2} - 2133 T^{3} + 18623 T^{4} - 134367 T^{5} + 906672 T^{6} - 5432131 T^{7} + 31678126 T^{8} - 170565345 T^{9} + 31678126 p T^{10} - 5432131 p^{2} T^{11} + 906672 p^{3} T^{12} - 134367 p^{4} T^{13} + 18623 p^{5} T^{14} - 2133 p^{6} T^{15} + 219 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 13 T + 167 T^{2} + 1499 T^{3} + 13436 T^{4} + 95962 T^{5} + 698532 T^{6} + 4410373 T^{7} + 27899758 T^{8} + 154411698 T^{9} + 27899758 p T^{10} + 4410373 p^{2} T^{11} + 698532 p^{3} T^{12} + 95962 p^{4} T^{13} + 13436 p^{5} T^{14} + 1499 p^{6} T^{15} + 167 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + T + 164 T^{2} - 25 T^{3} + 13345 T^{4} - 19962 T^{5} + 748055 T^{6} - 1820111 T^{7} + 33023343 T^{8} - 86331982 T^{9} + 33023343 p T^{10} - 1820111 p^{2} T^{11} + 748055 p^{3} T^{12} - 19962 p^{4} T^{13} + 13345 p^{5} T^{14} - 25 p^{6} T^{15} + 164 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 6 T + 105 T^{2} - 657 T^{3} + 8491 T^{4} - 56617 T^{5} + 512164 T^{6} - 3237269 T^{7} + 24968146 T^{8} - 155625947 T^{9} + 24968146 p T^{10} - 3237269 p^{2} T^{11} + 512164 p^{3} T^{12} - 56617 p^{4} T^{13} + 8491 p^{5} T^{14} - 657 p^{6} T^{15} + 105 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - T + 144 T^{2} + 36 T^{3} + 14051 T^{4} + 4699 T^{5} + 997254 T^{6} + 464435 T^{7} + 54046602 T^{8} + 25642497 T^{9} + 54046602 p T^{10} + 464435 p^{2} T^{11} + 997254 p^{3} T^{12} + 4699 p^{4} T^{13} + 14051 p^{5} T^{14} + 36 p^{6} T^{15} + 144 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 2 T + 157 T^{2} - 27 T^{3} + 15847 T^{4} - 2755 T^{5} + 1203316 T^{6} + 22677 T^{7} + 70019046 T^{8} + 2721691 T^{9} + 70019046 p T^{10} + 22677 p^{2} T^{11} + 1203316 p^{3} T^{12} - 2755 p^{4} T^{13} + 15847 p^{5} T^{14} - 27 p^{6} T^{15} + 157 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 30 T + 764 T^{2} - 13045 T^{3} + 197201 T^{4} - 2405102 T^{5} + 26608679 T^{6} - 251697171 T^{7} + 2186503079 T^{8} - 16578690952 T^{9} + 2186503079 p T^{10} - 251697171 p^{2} T^{11} + 26608679 p^{3} T^{12} - 2405102 p^{4} T^{13} + 197201 p^{5} T^{14} - 13045 p^{6} T^{15} + 764 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 15 T + 410 T^{2} + 4305 T^{3} + 65047 T^{4} + 487110 T^{5} + 5390959 T^{6} + 29366503 T^{7} + 302432813 T^{8} + 1478780406 T^{9} + 302432813 p T^{10} + 29366503 p^{2} T^{11} + 5390959 p^{3} T^{12} + 487110 p^{4} T^{13} + 65047 p^{5} T^{14} + 4305 p^{6} T^{15} + 410 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 21 T + 450 T^{2} - 6360 T^{3} + 87635 T^{4} - 952857 T^{5} + 10212800 T^{6} - 92578873 T^{7} + 832837852 T^{8} - 6507880653 T^{9} + 832837852 p T^{10} - 92578873 p^{2} T^{11} + 10212800 p^{3} T^{12} - 952857 p^{4} T^{13} + 87635 p^{5} T^{14} - 6360 p^{6} T^{15} + 450 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 7 T + 271 T^{2} - 862 T^{3} + 31617 T^{4} - 22471 T^{5} + 2747942 T^{6} + 340663 T^{7} + 220931030 T^{8} + 6142161 T^{9} + 220931030 p T^{10} + 340663 p^{2} T^{11} + 2747942 p^{3} T^{12} - 22471 p^{4} T^{13} + 31617 p^{5} T^{14} - 862 p^{6} T^{15} + 271 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 7 T + 329 T^{2} - 1655 T^{3} + 57246 T^{4} - 227682 T^{5} + 6713342 T^{6} - 20970873 T^{7} + 596832992 T^{8} - 1621578350 T^{9} + 596832992 p T^{10} - 20970873 p^{2} T^{11} + 6713342 p^{3} T^{12} - 227682 p^{4} T^{13} + 57246 p^{5} T^{14} - 1655 p^{6} T^{15} + 329 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 28 T + 593 T^{2} - 9272 T^{3} + 133028 T^{4} - 1626944 T^{5} + 18735348 T^{6} - 192384968 T^{7} + 1868388734 T^{8} - 16364819272 T^{9} + 1868388734 p T^{10} - 192384968 p^{2} T^{11} + 18735348 p^{3} T^{12} - 1626944 p^{4} T^{13} + 133028 p^{5} T^{14} - 9272 p^{6} T^{15} + 593 p^{7} T^{16} - 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 31 T + 757 T^{2} - 12457 T^{3} + 188314 T^{4} - 2369594 T^{5} + 28747466 T^{6} - 304054231 T^{7} + 3100351316 T^{8} - 27989535950 T^{9} + 3100351316 p T^{10} - 304054231 p^{2} T^{11} + 28747466 p^{3} T^{12} - 2369594 p^{4} T^{13} + 188314 p^{5} T^{14} - 12457 p^{6} T^{15} + 757 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 45 T + 1348 T^{2} + 29160 T^{3} + 518079 T^{4} + 7760711 T^{5} + 102182454 T^{6} + 1194959175 T^{7} + 12622802288 T^{8} + 120577066249 T^{9} + 12622802288 p T^{10} + 1194959175 p^{2} T^{11} + 102182454 p^{3} T^{12} + 7760711 p^{4} T^{13} + 518079 p^{5} T^{14} + 29160 p^{6} T^{15} + 1348 p^{7} T^{16} + 45 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 41 T + 1239 T^{2} - 26092 T^{3} + 471499 T^{4} - 7079921 T^{5} + 96248956 T^{6} - 1149115833 T^{7} + 12643775152 T^{8} - 124080739275 T^{9} + 12643775152 p T^{10} - 1149115833 p^{2} T^{11} + 96248956 p^{3} T^{12} - 7079921 p^{4} T^{13} + 471499 p^{5} T^{14} - 26092 p^{6} T^{15} + 1239 p^{7} T^{16} - 41 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 8 T + 472 T^{2} + 4349 T^{3} + 122457 T^{4} + 1088022 T^{5} + 21739787 T^{6} + 174776115 T^{7} + 2806532707 T^{8} + 19946280100 T^{9} + 2806532707 p T^{10} + 174776115 p^{2} T^{11} + 21739787 p^{3} T^{12} + 1088022 p^{4} T^{13} + 122457 p^{5} T^{14} + 4349 p^{6} T^{15} + 472 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.17304319054807237154532330479, −3.16965716936598643866810293799, −3.13302460850048888387647011014, −3.10167753715177111999272914968, −2.98812201062305296275578416509, −2.98711107324965190837821119660, −2.88564202127660553285656139232, −2.43002034324168670057367512959, −2.41866964867671312173733544877, −2.21576788347672628107816617377, −2.20436535097459117143031595584, −2.10618544760363159502973440591, −2.03835543815424482368952275828, −1.80535886259663062629920809832, −1.51271341052675582796799305511, −1.49609864538553886379069701153, −1.13738128751829717391961567626, −1.09579032743028686844083808447, −0.978912672935551598642057857915, −0.911472599080910970591483310459, −0.75692166907515915478312774237, −0.65170403552692261229580577739, −0.50233529056238550359141325513, −0.42499448230794507774849499281, −0.34428503770653198672297448576, 0.34428503770653198672297448576, 0.42499448230794507774849499281, 0.50233529056238550359141325513, 0.65170403552692261229580577739, 0.75692166907515915478312774237, 0.911472599080910970591483310459, 0.978912672935551598642057857915, 1.09579032743028686844083808447, 1.13738128751829717391961567626, 1.49609864538553886379069701153, 1.51271341052675582796799305511, 1.80535886259663062629920809832, 2.03835543815424482368952275828, 2.10618544760363159502973440591, 2.20436535097459117143031595584, 2.21576788347672628107816617377, 2.41866964867671312173733544877, 2.43002034324168670057367512959, 2.88564202127660553285656139232, 2.98711107324965190837821119660, 2.98812201062305296275578416509, 3.10167753715177111999272914968, 3.13302460850048888387647011014, 3.16965716936598643866810293799, 3.17304319054807237154532330479
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.