Dirichlet series
| L(s) = 1 | − 3·2-s + 3-s + 8·4-s + 8·5-s − 3·6-s + 11·7-s − 23·8-s + 9-s − 24·10-s − 6·11-s + 8·12-s + 8·13-s − 33·14-s + 8·15-s + 51·16-s − 4·17-s − 3·18-s − 20·19-s + 64·20-s + 11·21-s + 18·22-s − 6·23-s − 23·24-s + 28·25-s − 24·26-s − 4·27-s + 88·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 4·4-s + 3.57·5-s − 1.22·6-s + 4.15·7-s − 8.13·8-s + 1/3·9-s − 7.58·10-s − 1.80·11-s + 2.30·12-s + 2.21·13-s − 8.81·14-s + 2.06·15-s + 51/4·16-s − 0.970·17-s − 0.707·18-s − 4.58·19-s + 14.3·20-s + 2.40·21-s + 3.83·22-s − 1.25·23-s − 4.69·24-s + 28/5·25-s − 4.70·26-s − 0.769·27-s + 16.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 5^{16} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.13957\times 10^{10}\) |
| Root analytic conductor: | \(2.16130\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 5^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.318369845\) |
| \(L(\frac12)\) | \(\approx\) | \(4.318369845\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 - T + 5 T^{3} - 8 T^{4} + 5 p T^{5} - p^{2} T^{6} - 5 p^{2} T^{7} + 17 p^{2} T^{8} - 5 p^{3} T^{9} - p^{4} T^{10} + 5 p^{4} T^{11} - 8 p^{4} T^{12} + 5 p^{5} T^{13} - p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 - T + T^{2} )^{8} \) | |
| 13 | \( ( 1 - T + T^{2} )^{8} \) | |
| good | 2 | \( 1 + 3 T + T^{2} + p T^{3} + p^{4} T^{4} + 5 p T^{5} + 17 T^{6} + 45 p T^{7} + 19 p T^{8} + 7 T^{9} + 283 T^{10} + 129 T^{11} + 69 T^{12} + 115 p^{3} T^{13} + 223 T^{14} + 95 T^{15} + 2431 T^{16} + 95 p T^{17} + 223 p^{2} T^{18} + 115 p^{6} T^{19} + 69 p^{4} T^{20} + 129 p^{5} T^{21} + 283 p^{6} T^{22} + 7 p^{7} T^{23} + 19 p^{9} T^{24} + 45 p^{10} T^{25} + 17 p^{10} T^{26} + 5 p^{12} T^{27} + p^{16} T^{28} + p^{14} T^{29} + p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 - 11 T + 41 T^{2} - 2 p T^{3} - 358 T^{4} + 1409 T^{5} - 3506 T^{6} + 580 p T^{7} + 21755 T^{8} - 114550 T^{9} + 159339 T^{10} + 193360 T^{11} - 1037527 T^{12} + 3015669 T^{13} - 5911715 T^{14} - 14130744 T^{15} + 94338616 T^{16} - 14130744 p T^{17} - 5911715 p^{2} T^{18} + 3015669 p^{3} T^{19} - 1037527 p^{4} T^{20} + 193360 p^{5} T^{21} + 159339 p^{6} T^{22} - 114550 p^{7} T^{23} + 21755 p^{8} T^{24} + 580 p^{10} T^{25} - 3506 p^{10} T^{26} + 1409 p^{11} T^{27} - 358 p^{12} T^{28} - 2 p^{14} T^{29} + 41 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 11 | \( 1 + 6 T - 14 T^{2} - 124 T^{3} - 65 T^{4} - 200 T^{5} - 230 p T^{6} + 10203 T^{7} + 98771 T^{8} + 203740 T^{9} - 547586 T^{10} - 252000 p T^{11} - 1350780 T^{12} - 14315965 T^{13} - 8413765 p T^{14} + 211509026 T^{15} + 2118658459 T^{16} + 211509026 p T^{17} - 8413765 p^{3} T^{18} - 14315965 p^{3} T^{19} - 1350780 p^{4} T^{20} - 252000 p^{6} T^{21} - 547586 p^{6} T^{22} + 203740 p^{7} T^{23} + 98771 p^{8} T^{24} + 10203 p^{9} T^{25} - 230 p^{11} T^{26} - 200 p^{11} T^{27} - 65 p^{12} T^{28} - 124 p^{13} T^{29} - 14 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 2 T + 58 T^{2} + 165 T^{3} + 1668 T^{4} + 393 p T^{5} + 35766 T^{6} + 10139 p T^{7} + 655698 T^{8} + 10139 p^{2} T^{9} + 35766 p^{2} T^{10} + 393 p^{4} T^{11} + 1668 p^{4} T^{12} + 165 p^{5} T^{13} + 58 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 10 T + 115 T^{2} + 789 T^{3} + 5477 T^{4} + 29161 T^{5} + 155913 T^{6} + 705427 T^{7} + 3285644 T^{8} + 705427 p T^{9} + 155913 p^{2} T^{10} + 29161 p^{3} T^{11} + 5477 p^{4} T^{12} + 789 p^{5} T^{13} + 115 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 6 T - 3 p T^{2} + 102 T^{3} + 5728 T^{4} - 17889 T^{5} - 152619 T^{6} + 1474869 T^{7} + 1290028 T^{8} - 48151878 T^{9} + 5173155 p T^{10} + 915675852 T^{11} - 5234330630 T^{12} - 9247804869 T^{13} + 110813888712 T^{14} - 6511801374 T^{15} - 2475610635356 T^{16} - 6511801374 p T^{17} + 110813888712 p^{2} T^{18} - 9247804869 p^{3} T^{19} - 5234330630 p^{4} T^{20} + 915675852 p^{5} T^{21} + 5173155 p^{7} T^{22} - 48151878 p^{7} T^{23} + 1290028 p^{8} T^{24} + 1474869 p^{9} T^{25} - 152619 p^{10} T^{26} - 17889 p^{11} T^{27} + 5728 p^{12} T^{28} + 102 p^{13} T^{29} - 3 p^{15} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32} \) | |
| 29 | \( 1 + 14 T - 41 T^{2} - 1514 T^{3} - 2162 T^{4} + 76701 T^{5} + 222627 T^{6} - 2195775 T^{7} - 3434042 T^{8} + 58264782 T^{9} - 266623611 T^{10} - 2482782898 T^{11} + 16046080568 T^{12} + 94185515191 T^{13} - 400827827970 T^{14} - 1346822728290 T^{15} + 8830422122554 T^{16} - 1346822728290 p T^{17} - 400827827970 p^{2} T^{18} + 94185515191 p^{3} T^{19} + 16046080568 p^{4} T^{20} - 2482782898 p^{5} T^{21} - 266623611 p^{6} T^{22} + 58264782 p^{7} T^{23} - 3434042 p^{8} T^{24} - 2195775 p^{9} T^{25} + 222627 p^{10} T^{26} + 76701 p^{11} T^{27} - 2162 p^{12} T^{28} - 1514 p^{13} T^{29} - 41 p^{14} T^{30} + 14 p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 - p T + 437 T^{2} - 4684 T^{3} + 50094 T^{4} - 468734 T^{5} + 3707584 T^{6} - 29347535 T^{7} + 226953608 T^{8} - 1555429988 T^{9} + 10531314797 T^{10} - 72476887018 T^{11} + 452591828931 T^{12} - 2729822979532 T^{13} + 17003917808772 T^{14} - 98139156516864 T^{15} + 536034169527679 T^{16} - 98139156516864 p T^{17} + 17003917808772 p^{2} T^{18} - 2729822979532 p^{3} T^{19} + 452591828931 p^{4} T^{20} - 72476887018 p^{5} T^{21} + 10531314797 p^{6} T^{22} - 1555429988 p^{7} T^{23} + 226953608 p^{8} T^{24} - 29347535 p^{9} T^{25} + 3707584 p^{10} T^{26} - 468734 p^{11} T^{27} + 50094 p^{12} T^{28} - 4684 p^{13} T^{29} + 437 p^{14} T^{30} - p^{16} T^{31} + p^{16} T^{32} \) | |
| 37 | \( ( 1 - T + 176 T^{2} - 470 T^{3} + 14414 T^{4} - 60011 T^{5} + 765222 T^{6} - 3845703 T^{7} + 31208770 T^{8} - 3845703 p T^{9} + 765222 p^{2} T^{10} - 60011 p^{3} T^{11} + 14414 p^{4} T^{12} - 470 p^{5} T^{13} + 176 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 12 T - 102 T^{2} + 1056 T^{3} + 11179 T^{4} - 54414 T^{5} - 823146 T^{6} + 1892481 T^{7} + 37341988 T^{8} - 47575014 T^{9} - 1425750228 T^{10} + 3721116918 T^{11} + 62671259911 T^{12} - 230279397597 T^{13} - 3466745446779 T^{14} + 4373363873607 T^{15} + 172080621913498 T^{16} + 4373363873607 p T^{17} - 3466745446779 p^{2} T^{18} - 230279397597 p^{3} T^{19} + 62671259911 p^{4} T^{20} + 3721116918 p^{5} T^{21} - 1425750228 p^{6} T^{22} - 47575014 p^{7} T^{23} + 37341988 p^{8} T^{24} + 1892481 p^{9} T^{25} - 823146 p^{10} T^{26} - 54414 p^{11} T^{27} + 11179 p^{12} T^{28} + 1056 p^{13} T^{29} - 102 p^{14} T^{30} - 12 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( 1 + 15 T + 48 T^{2} - 29 T^{3} - 1029 T^{4} - 31892 T^{5} - 181788 T^{6} - 217863 T^{7} + 644359 T^{8} + 72703066 T^{9} + 672872745 T^{10} + 24646614 p T^{11} - 1856544524 T^{12} + 13394264448 T^{13} - 289708608143 T^{14} - 2096033279687 T^{15} - 1219490051355 T^{16} - 2096033279687 p T^{17} - 289708608143 p^{2} T^{18} + 13394264448 p^{3} T^{19} - 1856544524 p^{4} T^{20} + 24646614 p^{6} T^{21} + 672872745 p^{6} T^{22} + 72703066 p^{7} T^{23} + 644359 p^{8} T^{24} - 217863 p^{9} T^{25} - 181788 p^{10} T^{26} - 31892 p^{11} T^{27} - 1029 p^{12} T^{28} - 29 p^{13} T^{29} + 48 p^{14} T^{30} + 15 p^{15} T^{31} + p^{16} T^{32} \) | |
| 47 | \( 1 - 18 T + 34 T^{2} + 244 T^{3} + 8128 T^{4} - 33190 T^{5} - 269692 T^{6} - 1471674 T^{7} + 4896491 T^{8} + 35843078 T^{9} + 1343456464 T^{10} + 700651134 T^{11} - 21915581340 T^{12} - 660627480728 T^{13} - 2860347638 T^{14} + 6373124983198 T^{15} + 194655320953312 T^{16} + 6373124983198 p T^{17} - 2860347638 p^{2} T^{18} - 660627480728 p^{3} T^{19} - 21915581340 p^{4} T^{20} + 700651134 p^{5} T^{21} + 1343456464 p^{6} T^{22} + 35843078 p^{7} T^{23} + 4896491 p^{8} T^{24} - 1471674 p^{9} T^{25} - 269692 p^{10} T^{26} - 33190 p^{11} T^{27} + 8128 p^{12} T^{28} + 244 p^{13} T^{29} + 34 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - 2 T + 142 T^{2} - 708 T^{3} + 12986 T^{4} - 68564 T^{5} + 1017309 T^{6} - 4483381 T^{7} + 61490950 T^{8} - 4483381 p T^{9} + 1017309 p^{2} T^{10} - 68564 p^{3} T^{11} + 12986 p^{4} T^{12} - 708 p^{5} T^{13} + 142 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( 1 + 24 T + 44 T^{2} - 1764 T^{3} + 13779 T^{4} + 258642 T^{5} - 26284 p T^{6} - 14506275 T^{7} + 171808019 T^{8} + 428729796 T^{9} - 12514336110 T^{10} + 6199179012 T^{11} + 628867090330 T^{12} - 926330255247 T^{13} - 18244364863045 T^{14} + 87398232691752 T^{15} + 1395587144711007 T^{16} + 87398232691752 p T^{17} - 18244364863045 p^{2} T^{18} - 926330255247 p^{3} T^{19} + 628867090330 p^{4} T^{20} + 6199179012 p^{5} T^{21} - 12514336110 p^{6} T^{22} + 428729796 p^{7} T^{23} + 171808019 p^{8} T^{24} - 14506275 p^{9} T^{25} - 26284 p^{11} T^{26} + 258642 p^{11} T^{27} + 13779 p^{12} T^{28} - 1764 p^{13} T^{29} + 44 p^{14} T^{30} + 24 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 9 T - 31 T^{2} + 186 T^{3} - 693 T^{4} + 27501 T^{5} + 343696 T^{6} - 4778382 T^{7} - 5468956 T^{8} - 35134779 T^{9} + 1246967670 T^{10} + 12986627037 T^{11} - 29245418747 T^{12} - 889594680129 T^{13} - 2760893275954 T^{14} + 26895560646 p T^{15} + 612724366845426 T^{16} + 26895560646 p^{2} T^{17} - 2760893275954 p^{2} T^{18} - 889594680129 p^{3} T^{19} - 29245418747 p^{4} T^{20} + 12986627037 p^{5} T^{21} + 1246967670 p^{6} T^{22} - 35134779 p^{7} T^{23} - 5468956 p^{8} T^{24} - 4778382 p^{9} T^{25} + 343696 p^{10} T^{26} + 27501 p^{11} T^{27} - 693 p^{12} T^{28} + 186 p^{13} T^{29} - 31 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( 1 - 18 T + 2 p T^{2} - 102 T^{3} - 9996 T^{4} + 118194 T^{5} - 317585 T^{6} - 4638330 T^{7} + 51137513 T^{8} + 52143060 T^{9} - 5135590461 T^{10} + 43299397893 T^{11} - 14059734959 T^{12} - 2604006818442 T^{13} + 19582400339609 T^{14} - 14089301494524 T^{15} - 490351124632464 T^{16} - 14089301494524 p T^{17} + 19582400339609 p^{2} T^{18} - 2604006818442 p^{3} T^{19} - 14059734959 p^{4} T^{20} + 43299397893 p^{5} T^{21} - 5135590461 p^{6} T^{22} + 52143060 p^{7} T^{23} + 51137513 p^{8} T^{24} - 4638330 p^{9} T^{25} - 317585 p^{10} T^{26} + 118194 p^{11} T^{27} - 9996 p^{12} T^{28} - 102 p^{13} T^{29} + 2 p^{15} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 10 T + 357 T^{2} - 3055 T^{3} + 59273 T^{4} - 450587 T^{5} + 6317853 T^{6} - 43591239 T^{7} + 504141882 T^{8} - 43591239 p T^{9} + 6317853 p^{2} T^{10} - 450587 p^{3} T^{11} + 59273 p^{4} T^{12} - 3055 p^{5} T^{13} + 357 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 6 T + 284 T^{2} - 2319 T^{3} + 42202 T^{4} - 421371 T^{5} + 4380068 T^{6} - 46287561 T^{7} + 355062718 T^{8} - 46287561 p T^{9} + 4380068 p^{2} T^{10} - 421371 p^{3} T^{11} + 42202 p^{4} T^{12} - 2319 p^{5} T^{13} + 284 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 3 T - 125 T^{2} + 1026 T^{3} - 7950 T^{4} - 45927 T^{5} + 1045475 T^{6} - 7713051 T^{7} + 83581976 T^{8} + 265731840 T^{9} - 5660048280 T^{10} + 52977317085 T^{11} - 633668155952 T^{12} - 927463229058 T^{13} + 29709313820026 T^{14} - 130961640862620 T^{15} + 1616370146632755 T^{16} - 130961640862620 p T^{17} + 29709313820026 p^{2} T^{18} - 927463229058 p^{3} T^{19} - 633668155952 p^{4} T^{20} + 52977317085 p^{5} T^{21} - 5660048280 p^{6} T^{22} + 265731840 p^{7} T^{23} + 83581976 p^{8} T^{24} - 7713051 p^{9} T^{25} + 1045475 p^{10} T^{26} - 45927 p^{11} T^{27} - 7950 p^{12} T^{28} + 1026 p^{13} T^{29} - 125 p^{14} T^{30} - 3 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 10 T - 449 T^{2} + 3952 T^{3} + 118708 T^{4} - 882093 T^{5} - 21915750 T^{6} + 131032245 T^{7} + 3133726150 T^{8} - 14218181058 T^{9} - 364804526130 T^{10} + 1151919629993 T^{11} + 36285893177681 T^{12} - 67291032805205 T^{13} - 3246246532371240 T^{14} + 1988274678598317 T^{15} + 274096334839639474 T^{16} + 1988274678598317 p T^{17} - 3246246532371240 p^{2} T^{18} - 67291032805205 p^{3} T^{19} + 36285893177681 p^{4} T^{20} + 1151919629993 p^{5} T^{21} - 364804526130 p^{6} T^{22} - 14218181058 p^{7} T^{23} + 3133726150 p^{8} T^{24} + 131032245 p^{9} T^{25} - 21915750 p^{10} T^{26} - 882093 p^{11} T^{27} + 118708 p^{12} T^{28} + 3952 p^{13} T^{29} - 449 p^{14} T^{30} - 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 13 T + 484 T^{2} + 5856 T^{3} + 116018 T^{4} + 1250692 T^{5} + 17794923 T^{6} + 165272834 T^{7} + 1887496867 T^{8} + 165272834 p T^{9} + 17794923 p^{2} T^{10} + 1250692 p^{3} T^{11} + 116018 p^{4} T^{12} + 5856 p^{5} T^{13} + 484 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 34 T + 218 T^{2} + 6026 T^{3} - 94651 T^{4} - 241538 T^{5} + 12178531 T^{6} - 9346921 T^{7} - 1187995261 T^{8} + 2601484768 T^{9} + 111447158382 T^{10} - 615949330462 T^{11} - 4907227076284 T^{12} + 41757419702547 T^{13} + 109698726445420 T^{14} - 429315107898876 T^{15} - 15946862344505411 T^{16} - 429315107898876 p T^{17} + 109698726445420 p^{2} T^{18} + 41757419702547 p^{3} T^{19} - 4907227076284 p^{4} T^{20} - 615949330462 p^{5} T^{21} + 111447158382 p^{6} T^{22} + 2601484768 p^{7} T^{23} - 1187995261 p^{8} T^{24} - 9346921 p^{9} T^{25} + 12178531 p^{10} T^{26} - 241538 p^{11} T^{27} - 94651 p^{12} T^{28} + 6026 p^{13} T^{29} + 218 p^{14} T^{30} - 34 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.67141626228240150485758561019, −2.64488251671974312239635788018, −2.49286484786563013472845096939, −2.41690708252113713300327897863, −2.41285575460481785083753865394, −2.33309427357688076210928844420, −2.25645023095001832941158901293, −2.22773596093241142507562066238, −2.21421014198268510835319625559, −2.15926971738357951998811209019, −2.13506426372550140385484782914, −1.95659681196968179459465901619, −1.85450338786010052387081451929, −1.69305604000094333125267356719, −1.64884253743066931481331033992, −1.53038100398638208290171981775, −1.46609976303041572358017334039, −1.37595211080371477162542160750, −1.27020575592909544872021576099, −1.01480494893843027980633457920, −0.988359491574311493623613773316, −0.77459067520787707870265516409, −0.61248553333314483788196496609, −0.53712341617138170740605114861, −0.11133264065886383651619888157, 0.11133264065886383651619888157, 0.53712341617138170740605114861, 0.61248553333314483788196496609, 0.77459067520787707870265516409, 0.988359491574311493623613773316, 1.01480494893843027980633457920, 1.27020575592909544872021576099, 1.37595211080371477162542160750, 1.46609976303041572358017334039, 1.53038100398638208290171981775, 1.64884253743066931481331033992, 1.69305604000094333125267356719, 1.85450338786010052387081451929, 1.95659681196968179459465901619, 2.13506426372550140385484782914, 2.15926971738357951998811209019, 2.21421014198268510835319625559, 2.22773596093241142507562066238, 2.25645023095001832941158901293, 2.33309427357688076210928844420, 2.41285575460481785083753865394, 2.41690708252113713300327897863, 2.49286484786563013472845096939, 2.64488251671974312239635788018, 2.67141626228240150485758561019
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.