Dirichlet series
| L(s) = 1 | − 10·3-s + 10·5-s + 7-s + 55·9-s + 7·11-s − 13-s − 100·15-s + 3·17-s + 4·19-s − 10·21-s − 13·23-s + 55·25-s − 220·27-s + 18·29-s − 9·31-s − 70·33-s + 10·35-s − 3·37-s + 10·39-s + 19·41-s + 5·43-s + 550·45-s − 8·47-s − 13·49-s − 30·51-s + 17·53-s + 70·55-s + ⋯ |
| L(s) = 1 | − 5.77·3-s + 4.47·5-s + 0.377·7-s + 55/3·9-s + 2.11·11-s − 0.277·13-s − 25.8·15-s + 0.727·17-s + 0.917·19-s − 2.18·21-s − 2.71·23-s + 11·25-s − 42.3·27-s + 3.34·29-s − 1.61·31-s − 12.1·33-s + 1.69·35-s − 0.493·37-s + 1.60·39-s + 2.96·41-s + 0.762·43-s + 81.9·45-s − 1.16·47-s − 1.85·49-s − 4.20·51-s + 2.33·53-s + 9.43·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 67^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18940\times 10^{18}\) |
| Root analytic conductor: | \(8.01247\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{30} \cdot 3^{10} \cdot 5^{10} \cdot 67^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(34.30689894\) |
| \(L(\frac12)\) | \(\approx\) | \(34.30689894\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T )^{10} \) | |
| 5 | \( ( 1 - T )^{10} \) | |
| 67 | \( ( 1 + T )^{10} \) | |
| good | 7 | \( 1 - T + 2 p T^{2} - 15 T^{3} + 186 T^{4} - 178 T^{5} + 1738 T^{6} - 1814 T^{7} + 16309 T^{8} - 16570 T^{9} + 112816 T^{10} - 16570 p T^{11} + 16309 p^{2} T^{12} - 1814 p^{3} T^{13} + 1738 p^{4} T^{14} - 178 p^{5} T^{15} + 186 p^{6} T^{16} - 15 p^{7} T^{17} + 2 p^{9} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 - 7 T + 64 T^{2} - 309 T^{3} + 1796 T^{4} - 6664 T^{5} + 29884 T^{6} - 89964 T^{7} + 358275 T^{8} - 957150 T^{9} + 3846360 T^{10} - 957150 p T^{11} + 358275 p^{2} T^{12} - 89964 p^{3} T^{13} + 29884 p^{4} T^{14} - 6664 p^{5} T^{15} + 1796 p^{6} T^{16} - 309 p^{7} T^{17} + 64 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + T + 51 T^{2} + 136 T^{3} + 1524 T^{4} + 5223 T^{5} + 35402 T^{6} + 122549 T^{7} + 655739 T^{8} + 2048919 T^{9} + 9683014 T^{10} + 2048919 p T^{11} + 655739 p^{2} T^{12} + 122549 p^{3} T^{13} + 35402 p^{4} T^{14} + 5223 p^{5} T^{15} + 1524 p^{6} T^{16} + 136 p^{7} T^{17} + 51 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 - 3 T + 84 T^{2} - 212 T^{3} + 3409 T^{4} - 7566 T^{5} + 92710 T^{6} - 188934 T^{7} + 1981278 T^{8} - 3801053 T^{9} + 2122284 p T^{10} - 3801053 p T^{11} + 1981278 p^{2} T^{12} - 188934 p^{3} T^{13} + 92710 p^{4} T^{14} - 7566 p^{5} T^{15} + 3409 p^{6} T^{16} - 212 p^{7} T^{17} + 84 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 4 T + 4 p T^{2} - 326 T^{3} + 3457 T^{4} - 15427 T^{5} + 111344 T^{6} - 500837 T^{7} + 150738 p T^{8} - 12256256 T^{9} + 59630952 T^{10} - 12256256 p T^{11} + 150738 p^{3} T^{12} - 500837 p^{3} T^{13} + 111344 p^{4} T^{14} - 15427 p^{5} T^{15} + 3457 p^{6} T^{16} - 326 p^{7} T^{17} + 4 p^{9} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 13 T + 8 p T^{2} + 1501 T^{3} + 12034 T^{4} + 69255 T^{5} + 382474 T^{6} + 1604521 T^{7} + 6619325 T^{8} + 22623090 T^{9} + 102582588 T^{10} + 22623090 p T^{11} + 6619325 p^{2} T^{12} + 1604521 p^{3} T^{13} + 382474 p^{4} T^{14} + 69255 p^{5} T^{15} + 12034 p^{6} T^{16} + 1501 p^{7} T^{17} + 8 p^{9} T^{18} + 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 - 18 T + 264 T^{2} - 2860 T^{3} + 27707 T^{4} - 230507 T^{5} + 1770874 T^{6} - 12268667 T^{7} + 79767356 T^{8} - 475065536 T^{9} + 2666465244 T^{10} - 475065536 p T^{11} + 79767356 p^{2} T^{12} - 12268667 p^{3} T^{13} + 1770874 p^{4} T^{14} - 230507 p^{5} T^{15} + 27707 p^{6} T^{16} - 2860 p^{7} T^{17} + 264 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 9 T + 202 T^{2} + 1765 T^{3} + 21670 T^{4} + 166886 T^{5} + 1517680 T^{6} + 10138642 T^{7} + 74818857 T^{8} + 433054664 T^{9} + 2697545004 T^{10} + 433054664 p T^{11} + 74818857 p^{2} T^{12} + 10138642 p^{3} T^{13} + 1517680 p^{4} T^{14} + 166886 p^{5} T^{15} + 21670 p^{6} T^{16} + 1765 p^{7} T^{17} + 202 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 3 T + 250 T^{2} + 599 T^{3} + 29046 T^{4} + 54315 T^{5} + 2121624 T^{6} + 3095913 T^{7} + 111934021 T^{8} + 133624050 T^{9} + 4618254532 T^{10} + 133624050 p T^{11} + 111934021 p^{2} T^{12} + 3095913 p^{3} T^{13} + 2121624 p^{4} T^{14} + 54315 p^{5} T^{15} + 29046 p^{6} T^{16} + 599 p^{7} T^{17} + 250 p^{8} T^{18} + 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 19 T + 368 T^{2} - 4349 T^{3} + 49444 T^{4} - 430932 T^{5} + 3641500 T^{6} - 25712320 T^{7} + 182349347 T^{8} - 1150469860 T^{9} + 7673956504 T^{10} - 1150469860 p T^{11} + 182349347 p^{2} T^{12} - 25712320 p^{3} T^{13} + 3641500 p^{4} T^{14} - 430932 p^{5} T^{15} + 49444 p^{6} T^{16} - 4349 p^{7} T^{17} + 368 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 5 T + 182 T^{2} - 1001 T^{3} + 18658 T^{4} - 98736 T^{5} + 1369886 T^{6} - 6786236 T^{7} + 78837917 T^{8} - 366300732 T^{9} + 3705588920 T^{10} - 366300732 p T^{11} + 78837917 p^{2} T^{12} - 6786236 p^{3} T^{13} + 1369886 p^{4} T^{14} - 98736 p^{5} T^{15} + 18658 p^{6} T^{16} - 1001 p^{7} T^{17} + 182 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 8 T + 226 T^{2} + 1120 T^{3} + 473 p T^{4} + 63839 T^{5} + 1431748 T^{6} + 1464381 T^{7} + 70406744 T^{8} - 43073226 T^{9} + 3160585108 T^{10} - 43073226 p T^{11} + 70406744 p^{2} T^{12} + 1464381 p^{3} T^{13} + 1431748 p^{4} T^{14} + 63839 p^{5} T^{15} + 473 p^{7} T^{16} + 1120 p^{7} T^{17} + 226 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 17 T + 401 T^{2} - 5026 T^{3} + 74864 T^{4} - 775069 T^{5} + 8898650 T^{6} - 78246343 T^{7} + 742696671 T^{8} - 5636416925 T^{9} + 45646286698 T^{10} - 5636416925 p T^{11} + 742696671 p^{2} T^{12} - 78246343 p^{3} T^{13} + 8898650 p^{4} T^{14} - 775069 p^{5} T^{15} + 74864 p^{6} T^{16} - 5026 p^{7} T^{17} + 401 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 24 T + 546 T^{2} - 7898 T^{3} + 109061 T^{4} - 1166873 T^{5} + 12201806 T^{6} - 106524167 T^{7} + 943343442 T^{8} - 7297192556 T^{9} + 59355862400 T^{10} - 7297192556 p T^{11} + 943343442 p^{2} T^{12} - 106524167 p^{3} T^{13} + 12201806 p^{4} T^{14} - 1166873 p^{5} T^{15} + 109061 p^{6} T^{16} - 7898 p^{7} T^{17} + 546 p^{8} T^{18} - 24 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 21 T + 479 T^{2} - 6584 T^{3} + 90338 T^{4} - 1006219 T^{5} + 10801144 T^{6} - 105689765 T^{7} + 958038205 T^{8} - 8323898991 T^{9} + 65797631858 T^{10} - 8323898991 p T^{11} + 958038205 p^{2} T^{12} - 105689765 p^{3} T^{13} + 10801144 p^{4} T^{14} - 1006219 p^{5} T^{15} + 90338 p^{6} T^{16} - 6584 p^{7} T^{17} + 479 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 2 T + 512 T^{2} - 661 T^{3} + 126024 T^{4} - 101473 T^{5} + 19761222 T^{6} - 9921601 T^{7} + 2191280375 T^{8} - 756366019 T^{9} + 179727526516 T^{10} - 756366019 p T^{11} + 2191280375 p^{2} T^{12} - 9921601 p^{3} T^{13} + 19761222 p^{4} T^{14} - 101473 p^{5} T^{15} + 126024 p^{6} T^{16} - 661 p^{7} T^{17} + 512 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 25 T + 740 T^{2} - 12028 T^{3} + 215101 T^{4} - 2731424 T^{5} + 37378650 T^{6} - 395371462 T^{7} + 4440713018 T^{8} - 39905879573 T^{9} + 379065324404 T^{10} - 39905879573 p T^{11} + 4440713018 p^{2} T^{12} - 395371462 p^{3} T^{13} + 37378650 p^{4} T^{14} - 2731424 p^{5} T^{15} + 215101 p^{6} T^{16} - 12028 p^{7} T^{17} + 740 p^{8} T^{18} - 25 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + T + 304 T^{2} - 19 T^{3} + 38768 T^{4} - 23842 T^{5} + 3323248 T^{6} + 1354642 T^{7} + 3804961 p T^{8} + 668164272 T^{9} + 26718419968 T^{10} + 668164272 p T^{11} + 3804961 p^{3} T^{12} + 1354642 p^{3} T^{13} + 3323248 p^{4} T^{14} - 23842 p^{5} T^{15} + 38768 p^{6} T^{16} - 19 p^{7} T^{17} + 304 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 6 T + 426 T^{2} + 3029 T^{3} + 94146 T^{4} + 738611 T^{5} + 14493228 T^{6} + 113357291 T^{7} + 1715204773 T^{8} + 12429006967 T^{9} + 160199677444 T^{10} + 12429006967 p T^{11} + 1715204773 p^{2} T^{12} + 113357291 p^{3} T^{13} + 14493228 p^{4} T^{14} + 738611 p^{5} T^{15} + 94146 p^{6} T^{16} + 3029 p^{7} T^{17} + 426 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 23 T + 920 T^{2} - 16422 T^{3} + 372263 T^{4} - 5371640 T^{5} + 88403594 T^{6} - 1055029014 T^{7} + 13741282336 T^{8} - 137002507093 T^{9} + 1466050969100 T^{10} - 137002507093 p T^{11} + 13741282336 p^{2} T^{12} - 1055029014 p^{3} T^{13} + 88403594 p^{4} T^{14} - 5371640 p^{5} T^{15} + 372263 p^{6} T^{16} - 16422 p^{7} T^{17} + 920 p^{8} T^{18} - 23 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 21 T + 673 T^{2} - 11470 T^{3} + 226572 T^{4} - 3176553 T^{5} + 48430586 T^{6} - 580095447 T^{7} + 7325043379 T^{8} - 75878830653 T^{9} + 821190737994 T^{10} - 75878830653 p T^{11} + 7325043379 p^{2} T^{12} - 580095447 p^{3} T^{13} + 48430586 p^{4} T^{14} - 3176553 p^{5} T^{15} + 226572 p^{6} T^{16} - 11470 p^{7} T^{17} + 673 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.59674609909080939603289186691, −2.54968595860358400937462276480, −2.44802889631829826118380434329, −2.42862462765773761362471434267, −2.36256790764638946339117234767, −1.87764061454748796186101457992, −1.84162736173756998742496848594, −1.81466316889829589176589664966, −1.74631430945324304866133911628, −1.71746180709347044760806123094, −1.70723279140214842566028830479, −1.70022159744480135731215201140, −1.60823695964659162710421281379, −1.40065334455660600779467251923, −1.39968246569389582995154719181, −1.03061172498476604525246130979, −0.938477129454797742795552299237, −0.886032085659990063915600896541, −0.826838683630003734832576291685, −0.818402482245154783180317386507, −0.70259551870566741499735481325, −0.59643916512523722783983310093, −0.49371402532621099290462706028, −0.43939665758109311471516992525, −0.19603975036898664007149337332, 0.19603975036898664007149337332, 0.43939665758109311471516992525, 0.49371402532621099290462706028, 0.59643916512523722783983310093, 0.70259551870566741499735481325, 0.818402482245154783180317386507, 0.826838683630003734832576291685, 0.886032085659990063915600896541, 0.938477129454797742795552299237, 1.03061172498476604525246130979, 1.39968246569389582995154719181, 1.40065334455660600779467251923, 1.60823695964659162710421281379, 1.70022159744480135731215201140, 1.70723279140214842566028830479, 1.71746180709347044760806123094, 1.74631430945324304866133911628, 1.81466316889829589176589664966, 1.84162736173756998742496848594, 1.87764061454748796186101457992, 2.36256790764638946339117234767, 2.42862462765773761362471434267, 2.44802889631829826118380434329, 2.54968595860358400937462276480, 2.59674609909080939603289186691
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.