Dirichlet series
| L(s) = 1 | − 3-s + 6·5-s − 3·7-s − 4·9-s − 9·11-s + 13-s − 6·15-s + 7·17-s + 9·19-s + 3·21-s − 13·23-s + 11·25-s + 7·27-s + 9·29-s + 4·31-s + 9·33-s − 18·35-s + 24·37-s − 39-s − 6·41-s + 14·43-s − 24·45-s − 24·47-s − 17·49-s − 7·51-s + 19·53-s − 54·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s − 1.13·7-s − 4/3·9-s − 2.71·11-s + 0.277·13-s − 1.54·15-s + 1.69·17-s + 2.06·19-s + 0.654·21-s − 2.71·23-s + 11/5·25-s + 1.34·27-s + 1.67·29-s + 0.718·31-s + 1.56·33-s − 3.04·35-s + 3.94·37-s − 0.160·39-s − 0.937·41-s + 2.13·43-s − 3.57·45-s − 3.50·47-s − 2.42·49-s − 0.980·51-s + 2.60·53-s − 7.28·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 11^{9} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 11^{9} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{36} \cdot 11^{9} \cdot 19^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.90065\times 10^{12}\) |
| Root analytic conductor: | \(5.16739\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{36} \cdot 11^{9} \cdot 19^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(28.80291104\) |
| \(L(\frac12)\) | \(\approx\) | \(28.80291104\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( ( 1 + T )^{9} \) | |
| 19 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 + T + 5 T^{2} + 2 T^{3} + 14 T^{4} - 8 T^{5} + 49 T^{6} + 5 T^{7} + 223 T^{8} + 88 T^{9} + 223 p T^{10} + 5 p^{2} T^{11} + 49 p^{3} T^{12} - 8 p^{4} T^{13} + 14 p^{5} T^{14} + 2 p^{6} T^{15} + 5 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - 6 T + p^{2} T^{2} - 78 T^{3} + 216 T^{4} - 84 p T^{5} + 601 T^{6} - 196 T^{7} - 1419 T^{8} + 5704 T^{9} - 1419 p T^{10} - 196 p^{2} T^{11} + 601 p^{3} T^{12} - 84 p^{5} T^{13} + 216 p^{5} T^{14} - 78 p^{6} T^{15} + p^{9} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 + 3 T + 26 T^{2} + 59 T^{3} + 384 T^{4} + 773 T^{5} + 4280 T^{6} + 7795 T^{7} + 37661 T^{8} + 62036 T^{9} + 37661 p T^{10} + 7795 p^{2} T^{11} + 4280 p^{3} T^{12} + 773 p^{4} T^{13} + 384 p^{5} T^{14} + 59 p^{6} T^{15} + 26 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - T + 44 T^{2} - p T^{3} + 1192 T^{4} + 41 T^{5} + 23950 T^{6} + 6411 T^{7} + 376013 T^{8} + 125236 T^{9} + 376013 p T^{10} + 6411 p^{2} T^{11} + 23950 p^{3} T^{12} + 41 p^{4} T^{13} + 1192 p^{5} T^{14} - p^{7} T^{15} + 44 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 7 T + 57 T^{2} - 326 T^{3} + 132 p T^{4} - 11632 T^{5} + 62836 T^{6} - 16378 p T^{7} + 1356254 T^{8} - 5529170 T^{9} + 1356254 p T^{10} - 16378 p^{3} T^{11} + 62836 p^{3} T^{12} - 11632 p^{4} T^{13} + 132 p^{6} T^{14} - 326 p^{6} T^{15} + 57 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 13 T + 154 T^{2} + 1307 T^{3} + 10527 T^{4} + 71230 T^{5} + 454563 T^{6} + 2573093 T^{7} + 13990133 T^{8} + 68512650 T^{9} + 13990133 p T^{10} + 2573093 p^{2} T^{11} + 454563 p^{3} T^{12} + 71230 p^{4} T^{13} + 10527 p^{5} T^{14} + 1307 p^{6} T^{15} + 154 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 9 T + 106 T^{2} - 403 T^{3} + 4186 T^{4} - 17463 T^{5} + 221316 T^{6} - 790199 T^{7} + 6538287 T^{8} - 15807868 T^{9} + 6538287 p T^{10} - 790199 p^{2} T^{11} + 221316 p^{3} T^{12} - 17463 p^{4} T^{13} + 4186 p^{5} T^{14} - 403 p^{6} T^{15} + 106 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 4 T + 99 T^{2} + 96 T^{3} + 2858 T^{4} + 792 p T^{5} + 103825 T^{6} + 735348 T^{7} + 7077441 T^{8} + 12004768 T^{9} + 7077441 p T^{10} + 735348 p^{2} T^{11} + 103825 p^{3} T^{12} + 792 p^{5} T^{13} + 2858 p^{5} T^{14} + 96 p^{6} T^{15} + 99 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 24 T + 490 T^{2} - 6728 T^{3} + 82415 T^{4} - 817440 T^{5} + 7413025 T^{6} - 57697080 T^{7} + 415583025 T^{8} - 2622869520 T^{9} + 415583025 p T^{10} - 57697080 p^{2} T^{11} + 7413025 p^{3} T^{12} - 817440 p^{4} T^{13} + 82415 p^{5} T^{14} - 6728 p^{6} T^{15} + 490 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 6 T + 68 T^{2} - 74 T^{3} + 2012 T^{4} - 1402 T^{5} + 231846 T^{6} + 363818 T^{7} + 8138505 T^{8} - 14118616 T^{9} + 8138505 p T^{10} + 363818 p^{2} T^{11} + 231846 p^{3} T^{12} - 1402 p^{4} T^{13} + 2012 p^{5} T^{14} - 74 p^{6} T^{15} + 68 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 14 T + 254 T^{2} - 2678 T^{3} + 31624 T^{4} - 276660 T^{5} + 2556338 T^{6} - 19184970 T^{7} + 147986311 T^{8} - 958482140 T^{9} + 147986311 p T^{10} - 19184970 p^{2} T^{11} + 2556338 p^{3} T^{12} - 276660 p^{4} T^{13} + 31624 p^{5} T^{14} - 2678 p^{6} T^{15} + 254 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 24 T + 535 T^{2} + 8080 T^{3} + 110100 T^{4} + 1234624 T^{5} + 12541660 T^{6} + 111148912 T^{7} + 896801774 T^{8} + 6437770320 T^{9} + 896801774 p T^{10} + 111148912 p^{2} T^{11} + 12541660 p^{3} T^{12} + 1234624 p^{4} T^{13} + 110100 p^{5} T^{14} + 8080 p^{6} T^{15} + 535 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 19 T + 307 T^{2} - 3760 T^{3} + 37734 T^{4} - 324916 T^{5} + 2650242 T^{6} - 19093200 T^{7} + 140258456 T^{8} - 1018201490 T^{9} + 140258456 p T^{10} - 19093200 p^{2} T^{11} + 2650242 p^{3} T^{12} - 324916 p^{4} T^{13} + 37734 p^{5} T^{14} - 3760 p^{6} T^{15} + 307 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 19 T + 340 T^{2} - 3991 T^{3} + 45537 T^{4} - 381278 T^{5} + 3223785 T^{6} - 21415825 T^{7} + 162863719 T^{8} - 1062195390 T^{9} + 162863719 p T^{10} - 21415825 p^{2} T^{11} + 3223785 p^{3} T^{12} - 381278 p^{4} T^{13} + 45537 p^{5} T^{14} - 3991 p^{6} T^{15} + 340 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 28 T + 627 T^{2} - 9780 T^{3} + 136382 T^{4} - 1582344 T^{5} + 17024506 T^{6} - 161068332 T^{7} + 1436603472 T^{8} - 11517757880 T^{9} + 1436603472 p T^{10} - 161068332 p^{2} T^{11} + 17024506 p^{3} T^{12} - 1582344 p^{4} T^{13} + 136382 p^{5} T^{14} - 9780 p^{6} T^{15} + 627 p^{7} T^{16} - 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 5 T + 135 T^{2} + 552 T^{3} + 10926 T^{4} + 33734 T^{5} + 699561 T^{6} + 2163495 T^{7} + 41421273 T^{8} + 83338180 T^{9} + 41421273 p T^{10} + 2163495 p^{2} T^{11} + 699561 p^{3} T^{12} + 33734 p^{4} T^{13} + 10926 p^{5} T^{14} + 552 p^{6} T^{15} + 135 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 16 T + 421 T^{2} + 6108 T^{3} + 92398 T^{4} + 1151220 T^{5} + 13202389 T^{6} + 138201880 T^{7} + 1314896979 T^{8} + 11599635392 T^{9} + 1314896979 p T^{10} + 138201880 p^{2} T^{11} + 13202389 p^{3} T^{12} + 1151220 p^{4} T^{13} + 92398 p^{5} T^{14} + 6108 p^{6} T^{15} + 421 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 15 T + 617 T^{2} - 7400 T^{3} + 167340 T^{4} - 1658180 T^{5} + 26808140 T^{6} - 222759064 T^{7} + 2836798598 T^{8} - 19755406874 T^{9} + 2836798598 p T^{10} - 222759064 p^{2} T^{11} + 26808140 p^{3} T^{12} - 1658180 p^{4} T^{13} + 167340 p^{5} T^{14} - 7400 p^{6} T^{15} + 617 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 2 T + 461 T^{2} + 544 T^{3} + 100994 T^{4} + 42712 T^{5} + 14132290 T^{6} - 1519264 T^{7} + 1439895484 T^{8} - 393744820 T^{9} + 1439895484 p T^{10} - 1519264 p^{2} T^{11} + 14132290 p^{3} T^{12} + 42712 p^{4} T^{13} + 100994 p^{5} T^{14} + 544 p^{6} T^{15} + 461 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 8 T + 464 T^{2} + 4064 T^{3} + 113776 T^{4} + 937586 T^{5} + 18395756 T^{6} + 135506248 T^{7} + 2096409587 T^{8} + 13449651548 T^{9} + 2096409587 p T^{10} + 135506248 p^{2} T^{11} + 18395756 p^{3} T^{12} + 937586 p^{4} T^{13} + 113776 p^{5} T^{14} + 4064 p^{6} T^{15} + 464 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 12 T + 458 T^{2} - 4672 T^{3} + 90511 T^{4} - 759712 T^{5} + 10724501 T^{6} - 72030720 T^{7} + 973260953 T^{8} - 5867220008 T^{9} + 973260953 p T^{10} - 72030720 p^{2} T^{11} + 10724501 p^{3} T^{12} - 759712 p^{4} T^{13} + 90511 p^{5} T^{14} - 4672 p^{6} T^{15} + 458 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 4 T + 230 T^{2} + 1920 T^{3} + 46455 T^{4} + 335040 T^{5} + 7014141 T^{6} + 46441536 T^{7} + 800240453 T^{8} + 5387468024 T^{9} + 800240453 p T^{10} + 46441536 p^{2} T^{11} + 7014141 p^{3} T^{12} + 335040 p^{4} T^{13} + 46455 p^{5} T^{14} + 1920 p^{6} T^{15} + 230 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 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
−3.08776385338385622200408068724, −3.08234979131084153970396068417, −2.98016536488151245915654880890, −2.89729565103851273198870731534, −2.87598102298463329051879212618, −2.80090459947056070764888378042, −2.68994442126110925483593537441, −2.57356105840017217536798557930, −2.25173688249219537138368524004, −2.18653798125076845500030349915, −2.10697740987068321748694180492, −2.10079225969882716257356502154, −2.02157741303540660339650693164, −1.96806810731284119319196704759, −1.67457827655053431155562075567, −1.66471697471263570970087506107, −1.56846475104487573032851499350, −1.13186090556216701910209475388, −1.02703220699927617873917194693, −0.910548424048769001036091130302, −0.883895073871911066502301519748, −0.54808372036542595179463430954, −0.48336551344959397750699491395, −0.48228884407548635810598435631, −0.37834104367879552518582411484, 0.37834104367879552518582411484, 0.48228884407548635810598435631, 0.48336551344959397750699491395, 0.54808372036542595179463430954, 0.883895073871911066502301519748, 0.910548424048769001036091130302, 1.02703220699927617873917194693, 1.13186090556216701910209475388, 1.56846475104487573032851499350, 1.66471697471263570970087506107, 1.67457827655053431155562075567, 1.96806810731284119319196704759, 2.02157741303540660339650693164, 2.10079225969882716257356502154, 2.10697740987068321748694180492, 2.18653798125076845500030349915, 2.25173688249219537138368524004, 2.57356105840017217536798557930, 2.68994442126110925483593537441, 2.80090459947056070764888378042, 2.87598102298463329051879212618, 2.89729565103851273198870731534, 2.98016536488151245915654880890, 3.08234979131084153970396068417, 3.08776385338385622200408068724
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.