Dirichlet series
| L(s) = 1 | + 3·3-s − 6·4-s − 9·5-s − 4·8-s − 6·9-s − 18·12-s − 3·13-s − 27·15-s + 12·16-s − 9·17-s + 54·20-s − 12·24-s + 45·25-s − 28·27-s + 15·29-s + 30·31-s + 27·32-s + 36·36-s + 30·37-s − 9·39-s + 36·40-s + 18·41-s − 6·43-s + 54·45-s + 21·47-s + 36·48-s − 30·49-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3·4-s − 4.02·5-s − 1.41·8-s − 2·9-s − 5.19·12-s − 0.832·13-s − 6.97·15-s + 3·16-s − 2.18·17-s + 12.0·20-s − 2.44·24-s + 9·25-s − 5.38·27-s + 2.78·29-s + 5.38·31-s + 4.77·32-s + 6·36-s + 4.93·37-s − 1.44·39-s + 5.69·40-s + 2.81·41-s − 0.914·43-s + 8.04·45-s + 3.06·47-s + 5.19·48-s − 4.28·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(5^{9} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.68403\times 10^{10}\) |
| Root analytic conductor: | \(3.79644\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 5^{9} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.936964697\) |
| \(L(\frac12)\) | \(\approx\) | \(1.936964697\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{9} \) |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + 3 p T^{2} + p^{2} T^{3} + 3 p^{3} T^{4} + 21 T^{5} + 9 p^{3} T^{6} + 75 T^{7} + 171 T^{8} + 177 T^{9} + 171 p T^{10} + 75 p^{2} T^{11} + 9 p^{6} T^{12} + 21 p^{4} T^{13} + 3 p^{8} T^{14} + p^{8} T^{15} + 3 p^{8} T^{16} + p^{9} T^{18} \) |
| 3 | \( 1 - p T + 5 p T^{2} - 35 T^{3} + 37 p T^{4} - 79 p T^{5} + 193 p T^{6} - 373 p T^{7} + 749 p T^{8} - 3871 T^{9} + 749 p^{2} T^{10} - 373 p^{3} T^{11} + 193 p^{4} T^{12} - 79 p^{5} T^{13} + 37 p^{6} T^{14} - 35 p^{6} T^{15} + 5 p^{8} T^{16} - p^{9} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 + 30 T^{2} + 10 T^{3} + 474 T^{4} + 192 T^{5} + 5389 T^{6} + 285 p T^{7} + 6801 p T^{8} + 319 p^{2} T^{9} + 6801 p^{2} T^{10} + 285 p^{3} T^{11} + 5389 p^{3} T^{12} + 192 p^{4} T^{13} + 474 p^{5} T^{14} + 10 p^{6} T^{15} + 30 p^{7} T^{16} + p^{9} T^{18} \) | |
| 11 | \( 1 + 63 T^{2} - 40 T^{3} + 1866 T^{4} - 2319 T^{5} + 3211 p T^{6} - 58584 T^{7} + 492000 T^{8} - 834149 T^{9} + 492000 p T^{10} - 58584 p^{2} T^{11} + 3211 p^{4} T^{12} - 2319 p^{4} T^{13} + 1866 p^{5} T^{14} - 40 p^{6} T^{15} + 63 p^{7} T^{16} + p^{9} T^{18} \) | |
| 13 | \( 1 + 3 T + 60 T^{2} + 108 T^{3} + 1821 T^{4} + 2454 T^{5} + 40032 T^{6} + 44769 T^{7} + 669741 T^{8} + 647665 T^{9} + 669741 p T^{10} + 44769 p^{2} T^{11} + 40032 p^{3} T^{12} + 2454 p^{4} T^{13} + 1821 p^{5} T^{14} + 108 p^{6} T^{15} + 60 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 9 T + 141 T^{2} + 968 T^{3} + 8505 T^{4} + 46953 T^{5} + 300109 T^{6} + 1382454 T^{7} + 7113252 T^{8} + 27872793 T^{9} + 7113252 p T^{10} + 1382454 p^{2} T^{11} + 300109 p^{3} T^{12} + 46953 p^{4} T^{13} + 8505 p^{5} T^{14} + 968 p^{6} T^{15} + 141 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 93 T^{2} + 170 T^{3} + 4152 T^{4} + 16017 T^{5} + 5556 p T^{6} + 721299 T^{7} + 3260967 T^{8} + 20307351 T^{9} + 3260967 p T^{10} + 721299 p^{2} T^{11} + 5556 p^{4} T^{12} + 16017 p^{4} T^{13} + 4152 p^{5} T^{14} + 170 p^{6} T^{15} + 93 p^{7} T^{16} + p^{9} T^{18} \) | |
| 29 | \( 1 - 15 T + 207 T^{2} - 1961 T^{3} + 17040 T^{4} - 122643 T^{5} + 832285 T^{6} - 4988805 T^{7} + 29400456 T^{8} - 158202393 T^{9} + 29400456 p T^{10} - 4988805 p^{2} T^{11} + 832285 p^{3} T^{12} - 122643 p^{4} T^{13} + 17040 p^{5} T^{14} - 1961 p^{6} T^{15} + 207 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 30 T + 564 T^{2} - 7480 T^{3} + 79374 T^{4} - 695745 T^{5} + 5298840 T^{6} - 35767185 T^{7} + 221521584 T^{8} - 1271602303 T^{9} + 221521584 p T^{10} - 35767185 p^{2} T^{11} + 5298840 p^{3} T^{12} - 695745 p^{4} T^{13} + 79374 p^{5} T^{14} - 7480 p^{6} T^{15} + 564 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 30 T + 600 T^{2} - 8522 T^{3} + 98484 T^{4} - 943845 T^{5} + 7904815 T^{6} - 1585047 p T^{7} + 399084690 T^{8} - 2506518869 T^{9} + 399084690 p T^{10} - 1585047 p^{3} T^{11} + 7904815 p^{3} T^{12} - 943845 p^{4} T^{13} + 98484 p^{5} T^{14} - 8522 p^{6} T^{15} + 600 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 18 T + 351 T^{2} - 3497 T^{3} + 36516 T^{4} - 213015 T^{5} + 33967 p T^{6} - 2472450 T^{7} + 9125322 T^{8} + 132745475 T^{9} + 9125322 p T^{10} - 2472450 p^{2} T^{11} + 33967 p^{4} T^{12} - 213015 p^{4} T^{13} + 36516 p^{5} T^{14} - 3497 p^{6} T^{15} + 351 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 6 T + 288 T^{2} + 1474 T^{3} + 38502 T^{4} + 166296 T^{5} + 3190276 T^{6} + 11690208 T^{7} + 185153736 T^{8} + 583746205 T^{9} + 185153736 p T^{10} + 11690208 p^{2} T^{11} + 3190276 p^{3} T^{12} + 166296 p^{4} T^{13} + 38502 p^{5} T^{14} + 1474 p^{6} T^{15} + 288 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 21 T + 363 T^{2} - 4203 T^{3} + 46251 T^{4} - 413523 T^{5} + 77413 p T^{6} - 27504027 T^{7} + 209788005 T^{8} - 1420501763 T^{9} + 209788005 p T^{10} - 27504027 p^{2} T^{11} + 77413 p^{4} T^{12} - 413523 p^{4} T^{13} + 46251 p^{5} T^{14} - 4203 p^{6} T^{15} + 363 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 9 T + 330 T^{2} + 2810 T^{3} + 55461 T^{4} + 421788 T^{5} + 5919750 T^{6} + 39597783 T^{7} + 438400545 T^{8} + 47510049 p T^{9} + 438400545 p T^{10} + 39597783 p^{2} T^{11} + 5919750 p^{3} T^{12} + 421788 p^{4} T^{13} + 55461 p^{5} T^{14} + 2810 p^{6} T^{15} + 330 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 27 T + 669 T^{2} - 11256 T^{3} + 171393 T^{4} - 2134647 T^{5} + 24312614 T^{6} - 239475900 T^{7} + 36822012 p T^{8} - 17366476051 T^{9} + 36822012 p^{2} T^{10} - 239475900 p^{2} T^{11} + 24312614 p^{3} T^{12} - 2134647 p^{4} T^{13} + 171393 p^{5} T^{14} - 11256 p^{6} T^{15} + 669 p^{7} T^{16} - 27 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 12 T + 330 T^{2} - 3534 T^{3} + 55302 T^{4} - 532932 T^{5} + 6274883 T^{6} - 52798755 T^{7} + 514842033 T^{8} - 3745507321 T^{9} + 514842033 p T^{10} - 52798755 p^{2} T^{11} + 6274883 p^{3} T^{12} - 532932 p^{4} T^{13} + 55302 p^{5} T^{14} - 3534 p^{6} T^{15} + 330 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 36 T + 786 T^{2} - 12125 T^{3} + 153165 T^{4} - 1670532 T^{5} + 16876290 T^{6} - 160030224 T^{7} + 1448180877 T^{8} - 12228577499 T^{9} + 1448180877 p T^{10} - 160030224 p^{2} T^{11} + 16876290 p^{3} T^{12} - 1670532 p^{4} T^{13} + 153165 p^{5} T^{14} - 12125 p^{6} T^{15} + 786 p^{7} T^{16} - 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 6 T + 462 T^{2} + 3172 T^{3} + 103323 T^{4} + 716955 T^{5} + 14792891 T^{6} + 94240119 T^{7} + 1475179173 T^{8} + 8119496483 T^{9} + 1475179173 p T^{10} + 94240119 p^{2} T^{11} + 14792891 p^{3} T^{12} + 716955 p^{4} T^{13} + 103323 p^{5} T^{14} + 3172 p^{6} T^{15} + 462 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 9 T + 390 T^{2} + 3040 T^{3} + 71094 T^{4} + 465699 T^{5} + 8187870 T^{6} + 45826335 T^{7} + 706947231 T^{8} + 3580256437 T^{9} + 706947231 p T^{10} + 45826335 p^{2} T^{11} + 8187870 p^{3} T^{12} + 465699 p^{4} T^{13} + 71094 p^{5} T^{14} + 3040 p^{6} T^{15} + 390 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 45 T + 1338 T^{2} - 28725 T^{3} + 508509 T^{4} - 7567899 T^{5} + 98988848 T^{6} - 1142795487 T^{7} + 11884603458 T^{8} - 110936558185 T^{9} + 11884603458 p T^{10} - 1142795487 p^{2} T^{11} + 98988848 p^{3} T^{12} - 7567899 p^{4} T^{13} + 508509 p^{5} T^{14} - 28725 p^{6} T^{15} + 1338 p^{7} T^{16} - 45 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 576 T^{2} - 507 T^{3} + 151803 T^{4} - 247698 T^{5} + 24576246 T^{6} - 50790690 T^{7} + 2768966469 T^{8} - 5598273343 T^{9} + 2768966469 p T^{10} - 50790690 p^{2} T^{11} + 24576246 p^{3} T^{12} - 247698 p^{4} T^{13} + 151803 p^{5} T^{14} - 507 p^{6} T^{15} + 576 p^{7} T^{16} + p^{9} T^{18} \) | |
| 89 | \( 1 + 9 T + 327 T^{2} + 4225 T^{3} + 72243 T^{4} + 850371 T^{5} + 11822154 T^{6} + 119250132 T^{7} + 1360333200 T^{8} + 12582561855 T^{9} + 1360333200 p T^{10} + 119250132 p^{2} T^{11} + 11822154 p^{3} T^{12} + 850371 p^{4} T^{13} + 72243 p^{5} T^{14} + 4225 p^{6} T^{15} + 327 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 45 T + 1344 T^{2} - 28316 T^{3} + 489126 T^{4} - 7111941 T^{5} + 92246475 T^{6} - 1084413258 T^{7} + 11843233752 T^{8} - 120453864257 T^{9} + 11843233752 p T^{10} - 1084413258 p^{2} T^{11} + 92246475 p^{3} T^{12} - 7111941 p^{4} T^{13} + 489126 p^{5} T^{14} - 28316 p^{6} T^{15} + 1344 p^{7} T^{16} - 45 p^{8} T^{17} + p^{9} T^{18} \) | |
| 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
−3.58751361162960603954872186055, −3.57148284426678461526587329762, −3.39420361806411458175355893421, −3.32690586093229870034970102676, −3.05215522703167314616269811377, −2.99472504558902527579870838620, −2.86791642161866863468653794370, −2.80112984372276701231890967178, −2.68972792927502863913790261676, −2.58383508635090023943248208434, −2.57959371274466554328873777931, −2.43384355747603962787730236606, −2.25593230279398055999845908699, −2.24005988986061361287929703200, −2.18075112209619640807878174253, −2.07321737907890338102367460069, −1.49902817316852925318894985135, −1.03440030517709971195313413570, −1.00643514399306799469999581845, −0.875883791869178934438611308451, −0.841075889633285071126227289892, −0.51219577351622973307792080487, −0.49911373524436470861405824914, −0.42600190367418406222618356254, −0.37158441969459668925570043229, 0.37158441969459668925570043229, 0.42600190367418406222618356254, 0.49911373524436470861405824914, 0.51219577351622973307792080487, 0.841075889633285071126227289892, 0.875883791869178934438611308451, 1.00643514399306799469999581845, 1.03440030517709971195313413570, 1.49902817316852925318894985135, 2.07321737907890338102367460069, 2.18075112209619640807878174253, 2.24005988986061361287929703200, 2.25593230279398055999845908699, 2.43384355747603962787730236606, 2.57959371274466554328873777931, 2.58383508635090023943248208434, 2.68972792927502863913790261676, 2.80112984372276701231890967178, 2.86791642161866863468653794370, 2.99472504558902527579870838620, 3.05215522703167314616269811377, 3.32690586093229870034970102676, 3.39420361806411458175355893421, 3.57148284426678461526587329762, 3.58751361162960603954872186055
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.