Dirichlet series
| L(s) = 1 | + 9·2-s + 45·4-s + 9·5-s + 165·8-s − 3·9-s + 81·10-s + 12·11-s + 9·13-s + 495·16-s + 6·17-s − 27·18-s + 405·20-s + 108·22-s + 18·23-s + 45·25-s + 81·26-s − 6·27-s − 6·31-s + 1.28e3·32-s + 54·34-s − 135·36-s + 6·37-s + 1.48e3·40-s + 18·43-s + 540·44-s − 27·45-s + 162·46-s + ⋯ |
| L(s) = 1 | + 6.36·2-s + 45/2·4-s + 4.02·5-s + 58.3·8-s − 9-s + 25.6·10-s + 3.61·11-s + 2.49·13-s + 123.·16-s + 1.45·17-s − 6.36·18-s + 90.5·20-s + 23.0·22-s + 3.75·23-s + 9·25-s + 15.8·26-s − 1.15·27-s − 1.07·31-s + 227.·32-s + 9.26·34-s − 22.5·36-s + 0.986·37-s + 234.·40-s + 2.74·43-s + 81.4·44-s − 4.02·45-s + 23.8·46-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 5^{9} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.37422\times 10^{13}\) |
| Root analytic conductor: | \(5.36898\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 5^{9} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(51913.62847\) |
| \(L(\frac12)\) | \(\approx\) | \(51913.62847\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{9} \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} + 2 p T^{3} + p T^{4} + 10 p T^{5} + 22 T^{6} + 14 p T^{7} + 43 p T^{8} + 44 T^{9} + 43 p^{2} T^{10} + 14 p^{3} T^{11} + 22 p^{3} T^{12} + 10 p^{5} T^{13} + p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} + p^{9} T^{18} \) |
| 7 | \( 1 + 12 T^{2} + 5 p T^{3} + 69 T^{4} + 234 T^{5} + 933 T^{6} + 489 T^{7} + 4299 T^{8} + 7788 T^{9} + 4299 p T^{10} + 489 p^{2} T^{11} + 933 p^{3} T^{12} + 234 p^{4} T^{13} + 69 p^{5} T^{14} + 5 p^{7} T^{15} + 12 p^{7} T^{16} + p^{9} T^{18} \) | |
| 11 | \( 1 - 12 T + 90 T^{2} - 509 T^{3} + 222 p T^{4} - 10710 T^{5} + 43570 T^{6} - 164238 T^{7} + 580644 T^{8} - 1953719 T^{9} + 580644 p T^{10} - 164238 p^{2} T^{11} + 43570 p^{3} T^{12} - 10710 p^{4} T^{13} + 222 p^{6} T^{14} - 509 p^{6} T^{15} + 90 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 9 T + 90 T^{2} - 521 T^{3} + 3141 T^{4} - 14112 T^{5} + 65423 T^{6} - 249615 T^{7} + 997173 T^{8} - 3481766 T^{9} + 997173 p T^{10} - 249615 p^{2} T^{11} + 65423 p^{3} T^{12} - 14112 p^{4} T^{13} + 3141 p^{5} T^{14} - 521 p^{6} T^{15} + 90 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 6 T + 78 T^{2} - 332 T^{3} + 2382 T^{4} - 5286 T^{5} + 1639 p T^{6} + 50292 T^{7} - 81060 T^{8} + 2490928 T^{9} - 81060 p T^{10} + 50292 p^{2} T^{11} + 1639 p^{4} T^{12} - 5286 p^{4} T^{13} + 2382 p^{5} T^{14} - 332 p^{6} T^{15} + 78 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 18 T + 288 T^{2} - 3141 T^{3} + 30537 T^{4} - 243450 T^{5} + 1753161 T^{6} - 10942983 T^{7} + 117567 p^{2} T^{8} - 312176952 T^{9} + 117567 p^{3} T^{10} - 10942983 p^{2} T^{11} + 1753161 p^{3} T^{12} - 243450 p^{4} T^{13} + 30537 p^{5} T^{14} - 3141 p^{6} T^{15} + 288 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 147 T^{2} - 128 T^{3} + 10878 T^{4} - 13248 T^{5} + 551122 T^{6} - 665856 T^{7} + 20874456 T^{8} - 22570304 T^{9} + 20874456 p T^{10} - 665856 p^{2} T^{11} + 551122 p^{3} T^{12} - 13248 p^{4} T^{13} + 10878 p^{5} T^{14} - 128 p^{6} T^{15} + 147 p^{7} T^{16} + p^{9} T^{18} \) | |
| 31 | \( 1 + 6 T + 195 T^{2} + 1104 T^{3} + 18828 T^{4} + 97104 T^{5} + 1162516 T^{6} + 5317056 T^{7} + 50056266 T^{8} + 197742964 T^{9} + 50056266 p T^{10} + 5317056 p^{2} T^{11} + 1162516 p^{3} T^{12} + 97104 p^{4} T^{13} + 18828 p^{5} T^{14} + 1104 p^{6} T^{15} + 195 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 6 T + 204 T^{2} - 1033 T^{3} + 18417 T^{4} - 77142 T^{5} + 995859 T^{6} - 3507723 T^{7} + 39994371 T^{8} - 130541864 T^{9} + 39994371 p T^{10} - 3507723 p^{2} T^{11} + 995859 p^{3} T^{12} - 77142 p^{4} T^{13} + 18417 p^{5} T^{14} - 1033 p^{6} T^{15} + 204 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 144 T^{2} + 293 T^{3} + 11916 T^{4} + 34740 T^{5} + 767066 T^{6} + 2269944 T^{7} + 39623592 T^{8} + 105977717 T^{9} + 39623592 p T^{10} + 2269944 p^{2} T^{11} + 767066 p^{3} T^{12} + 34740 p^{4} T^{13} + 11916 p^{5} T^{14} + 293 p^{6} T^{15} + 144 p^{7} T^{16} + p^{9} T^{18} \) | |
| 43 | \( 1 - 18 T + 390 T^{2} - 4272 T^{3} + 50730 T^{4} - 371802 T^{5} + 3062413 T^{6} - 15604020 T^{7} + 112246716 T^{8} - 523231720 T^{9} + 112246716 p T^{10} - 15604020 p^{2} T^{11} + 3062413 p^{3} T^{12} - 371802 p^{4} T^{13} + 50730 p^{5} T^{14} - 4272 p^{6} T^{15} + 390 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 3 T + 264 T^{2} + 355 T^{3} + 645 p T^{4} - 16836 T^{5} + 2065115 T^{6} - 5291979 T^{7} + 104321775 T^{8} - 378629702 T^{9} + 104321775 p T^{10} - 5291979 p^{2} T^{11} + 2065115 p^{3} T^{12} - 16836 p^{4} T^{13} + 645 p^{6} T^{14} + 355 p^{6} T^{15} + 264 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 384 T^{2} + 7 T^{3} + 69441 T^{4} + 870 T^{5} + 7733771 T^{6} + 38901 T^{7} + 583656519 T^{8} + 1164532 T^{9} + 583656519 p T^{10} + 38901 p^{2} T^{11} + 7733771 p^{3} T^{12} + 870 p^{4} T^{13} + 69441 p^{5} T^{14} + 7 p^{6} T^{15} + 384 p^{7} T^{16} + p^{9} T^{18} \) | |
| 59 | \( 1 - 21 T + 498 T^{2} - 5933 T^{3} + 77016 T^{4} - 576393 T^{5} + 5179286 T^{6} - 22424553 T^{7} + 189224706 T^{8} - 564693197 T^{9} + 189224706 p T^{10} - 22424553 p^{2} T^{11} + 5179286 p^{3} T^{12} - 576393 p^{4} T^{13} + 77016 p^{5} T^{14} - 5933 p^{6} T^{15} + 498 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 18 T + 273 T^{2} - 1792 T^{3} + 14676 T^{4} - 74736 T^{5} + 1199364 T^{6} - 7328400 T^{7} + 66610770 T^{8} - 232484892 T^{9} + 66610770 p T^{10} - 7328400 p^{2} T^{11} + 1199364 p^{3} T^{12} - 74736 p^{4} T^{13} + 14676 p^{5} T^{14} - 1792 p^{6} T^{15} + 273 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 399 T^{2} - 566 T^{3} + 75165 T^{4} - 194238 T^{5} + 9039552 T^{6} - 29576946 T^{7} + 788493777 T^{8} - 2559394164 T^{9} + 788493777 p T^{10} - 29576946 p^{2} T^{11} + 9039552 p^{3} T^{12} - 194238 p^{4} T^{13} + 75165 p^{5} T^{14} - 566 p^{6} T^{15} + 399 p^{7} T^{16} + p^{9} T^{18} \) | |
| 71 | \( 1 - 18 T + 447 T^{2} - 6320 T^{3} + 94848 T^{4} - 1103376 T^{5} + 12738824 T^{6} - 125422944 T^{7} + 1214299614 T^{8} - 10308038108 T^{9} + 1214299614 p T^{10} - 125422944 p^{2} T^{11} + 12738824 p^{3} T^{12} - 1103376 p^{4} T^{13} + 94848 p^{5} T^{14} - 6320 p^{6} T^{15} + 447 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 36 T + 1077 T^{2} + 21862 T^{3} + 386925 T^{4} + 5568882 T^{5} + 71958864 T^{6} + 800688714 T^{7} + 8125857021 T^{8} + 72511926572 T^{9} + 8125857021 p T^{10} + 800688714 p^{2} T^{11} + 71958864 p^{3} T^{12} + 5568882 p^{4} T^{13} + 386925 p^{5} T^{14} + 21862 p^{6} T^{15} + 1077 p^{7} T^{16} + 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 6 T + 243 T^{2} + 1776 T^{3} + 30384 T^{4} + 138504 T^{5} + 2603080 T^{6} + 2853264 T^{7} + 141395322 T^{8} + 11678692 T^{9} + 141395322 p T^{10} + 2853264 p^{2} T^{11} + 2603080 p^{3} T^{12} + 138504 p^{4} T^{13} + 30384 p^{5} T^{14} + 1776 p^{6} T^{15} + 243 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 6 T + 387 T^{2} + 1630 T^{3} + 73989 T^{4} + 237780 T^{5} + 9946760 T^{6} + 26530032 T^{7} + 1037545905 T^{8} + 2414804488 T^{9} + 1037545905 p T^{10} + 26530032 p^{2} T^{11} + 9946760 p^{3} T^{12} + 237780 p^{4} T^{13} + 73989 p^{5} T^{14} + 1630 p^{6} T^{15} + 387 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 18 T + 453 T^{2} + 4687 T^{3} + 63492 T^{4} + 341199 T^{5} + 2979988 T^{6} - 8179722 T^{7} - 62459847 T^{8} - 2524106903 T^{9} - 62459847 p T^{10} - 8179722 p^{2} T^{11} + 2979988 p^{3} T^{12} + 341199 p^{4} T^{13} + 63492 p^{5} T^{14} + 4687 p^{6} T^{15} + 453 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 18 T + 633 T^{2} - 9130 T^{3} + 185679 T^{4} - 2261148 T^{5} + 34695954 T^{6} - 364698168 T^{7} + 4607774769 T^{8} - 41679458144 T^{9} + 4607774769 p T^{10} - 364698168 p^{2} T^{11} + 34695954 p^{3} T^{12} - 2261148 p^{4} T^{13} + 185679 p^{5} T^{14} - 9130 p^{6} T^{15} + 633 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.16282398614624596116231376999, −3.05927841266333027790952051064, −3.04612731792552559395269908145, −3.03663339818442931735348867150, −3.01517380670781093362305858682, −3.00906318682171519996010797484, −2.84370999373780132232200609757, −2.47149651225059500438886496142, −2.46752472402498464388205243639, −2.36682438860616569266307672228, −2.22150618097912975422179483555, −2.00971495385049747604006197841, −2.00772137920953868035564072516, −1.94375734677858146174310703673, −1.78383145341501670933185078319, −1.72558907068914447466801760113, −1.60389811036118110937625059498, −1.52526398695793841790028635146, −1.39252294996221243510332203796, −1.17646163167329742606465488367, −0.978962325675130327566354794888, −0.904377273925308280941232307614, −0.884756369936596766114495567642, −0.77299859370099034304740552103, −0.67950312700842511767884619780, 0.67950312700842511767884619780, 0.77299859370099034304740552103, 0.884756369936596766114495567642, 0.904377273925308280941232307614, 0.978962325675130327566354794888, 1.17646163167329742606465488367, 1.39252294996221243510332203796, 1.52526398695793841790028635146, 1.60389811036118110937625059498, 1.72558907068914447466801760113, 1.78383145341501670933185078319, 1.94375734677858146174310703673, 2.00772137920953868035564072516, 2.00971495385049747604006197841, 2.22150618097912975422179483555, 2.36682438860616569266307672228, 2.46752472402498464388205243639, 2.47149651225059500438886496142, 2.84370999373780132232200609757, 3.00906318682171519996010797484, 3.01517380670781093362305858682, 3.03663339818442931735348867150, 3.04612731792552559395269908145, 3.05927841266333027790952051064, 3.16282398614624596116231376999
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.