Dirichlet series
| L(s) = 1 | + 12·3-s + 4·5-s + 11·7-s + 78·9-s + 11-s + 8·13-s + 48·15-s + 3·17-s + 12·19-s + 132·21-s + 7·23-s − 13·25-s + 364·27-s + 5·29-s + 33·31-s + 12·33-s + 44·35-s + 8·37-s + 96·39-s − 6·41-s + 16·43-s + 312·45-s + 18·47-s + 31·49-s + 36·51-s + 20·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 6.92·3-s + 1.78·5-s + 4.15·7-s + 26·9-s + 0.301·11-s + 2.21·13-s + 12.3·15-s + 0.727·17-s + 2.75·19-s + 28.8·21-s + 1.45·23-s − 2.59·25-s + 70.0·27-s + 0.928·29-s + 5.92·31-s + 2.08·33-s + 7.43·35-s + 1.31·37-s + 15.3·39-s − 0.937·41-s + 2.43·43-s + 46.5·45-s + 2.62·47-s + 31/7·49-s + 5.04·51-s + 2.74·53-s + 0.539·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 3^{12} \cdot 167^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.15466\times 10^{18}\) |
| Root analytic conductor: | \(5.65721\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 3^{12} \cdot 167^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(47203.15431\) |
| \(L(\frac12)\) | \(\approx\) | \(47203.15431\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T )^{12} \) | |
| 167 | \( ( 1 - T )^{12} \) | |
| good | 5 | \( 1 - 4 T + 29 T^{2} - 89 T^{3} + 409 T^{4} - 1058 T^{5} + 3913 T^{6} - 1821 p T^{7} + 29926 T^{8} - 64597 T^{9} + 38604 p T^{10} - 77299 p T^{11} + 1053908 T^{12} - 77299 p^{2} T^{13} + 38604 p^{3} T^{14} - 64597 p^{3} T^{15} + 29926 p^{4} T^{16} - 1821 p^{6} T^{17} + 3913 p^{6} T^{18} - 1058 p^{7} T^{19} + 409 p^{8} T^{20} - 89 p^{9} T^{21} + 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 11 T + 90 T^{2} - 550 T^{3} + 2935 T^{4} - 1936 p T^{5} + 57133 T^{6} - 218439 T^{7} + 778294 T^{8} - 2566764 T^{9} + 7963941 T^{10} - 23067122 T^{11} + 63084108 T^{12} - 23067122 p T^{13} + 7963941 p^{2} T^{14} - 2566764 p^{3} T^{15} + 778294 p^{4} T^{16} - 218439 p^{5} T^{17} + 57133 p^{6} T^{18} - 1936 p^{8} T^{19} + 2935 p^{8} T^{20} - 550 p^{9} T^{21} + 90 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - T + 57 T^{2} + 9 T^{3} + 1303 T^{4} + 2130 T^{5} + 15693 T^{6} + 4980 p T^{7} + 142415 T^{8} + 536457 T^{9} + 1867410 T^{10} + 1451533 T^{11} + 25055730 T^{12} + 1451533 p T^{13} + 1867410 p^{2} T^{14} + 536457 p^{3} T^{15} + 142415 p^{4} T^{16} + 4980 p^{6} T^{17} + 15693 p^{6} T^{18} + 2130 p^{7} T^{19} + 1303 p^{8} T^{20} + 9 p^{9} T^{21} + 57 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 8 T + 103 T^{2} - 48 p T^{3} + 4818 T^{4} - 23799 T^{5} + 141509 T^{6} - 600471 T^{7} + 3036479 T^{8} - 11489293 T^{9} + 51865524 T^{10} - 178767981 T^{11} + 735638636 T^{12} - 178767981 p T^{13} + 51865524 p^{2} T^{14} - 11489293 p^{3} T^{15} + 3036479 p^{4} T^{16} - 600471 p^{5} T^{17} + 141509 p^{6} T^{18} - 23799 p^{7} T^{19} + 4818 p^{8} T^{20} - 48 p^{10} T^{21} + 103 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 3 T + 118 T^{2} - 387 T^{3} + 7251 T^{4} - 23602 T^{5} + 300910 T^{6} - 932520 T^{7} + 9256187 T^{8} - 26738797 T^{9} + 221263212 T^{10} - 584079475 T^{11} + 4205285970 T^{12} - 584079475 p T^{13} + 221263212 p^{2} T^{14} - 26738797 p^{3} T^{15} + 9256187 p^{4} T^{16} - 932520 p^{5} T^{17} + 300910 p^{6} T^{18} - 23602 p^{7} T^{19} + 7251 p^{8} T^{20} - 387 p^{9} T^{21} + 118 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 12 T + 199 T^{2} - 1674 T^{3} + 16320 T^{4} - 107611 T^{5} + 41355 p T^{6} - 4319201 T^{7} + 26066607 T^{8} - 124863285 T^{9} + 660397808 T^{10} - 2851154001 T^{11} + 13691500992 T^{12} - 2851154001 p T^{13} + 660397808 p^{2} T^{14} - 124863285 p^{3} T^{15} + 26066607 p^{4} T^{16} - 4319201 p^{5} T^{17} + 41355 p^{7} T^{18} - 107611 p^{7} T^{19} + 16320 p^{8} T^{20} - 1674 p^{9} T^{21} + 199 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 7 T + 167 T^{2} - 941 T^{3} + 13183 T^{4} - 2628 p T^{5} + 655771 T^{6} - 2465474 T^{7} + 23482855 T^{8} - 73464215 T^{9} + 666819294 T^{10} - 1819767099 T^{11} + 16262816722 T^{12} - 1819767099 p T^{13} + 666819294 p^{2} T^{14} - 73464215 p^{3} T^{15} + 23482855 p^{4} T^{16} - 2465474 p^{5} T^{17} + 655771 p^{6} T^{18} - 2628 p^{8} T^{19} + 13183 p^{8} T^{20} - 941 p^{9} T^{21} + 167 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 5 T + 265 T^{2} - 939 T^{3} + 31695 T^{4} - 75812 T^{5} + 2339341 T^{6} - 3519780 T^{7} + 122806087 T^{8} - 110222539 T^{9} + 4959840978 T^{10} - 2882442317 T^{11} + 160066031378 T^{12} - 2882442317 p T^{13} + 4959840978 p^{2} T^{14} - 110222539 p^{3} T^{15} + 122806087 p^{4} T^{16} - 3519780 p^{5} T^{17} + 2339341 p^{6} T^{18} - 75812 p^{7} T^{19} + 31695 p^{8} T^{20} - 939 p^{9} T^{21} + 265 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 33 T + 715 T^{2} - 11453 T^{3} + 151689 T^{4} - 1717753 T^{5} + 17166335 T^{6} - 153572178 T^{7} + 40185534 p T^{8} - 9228392127 T^{9} + 2025747070 p T^{10} - 393831808170 T^{11} + 2281105342288 T^{12} - 393831808170 p T^{13} + 2025747070 p^{3} T^{14} - 9228392127 p^{3} T^{15} + 40185534 p^{5} T^{16} - 153572178 p^{5} T^{17} + 17166335 p^{6} T^{18} - 1717753 p^{7} T^{19} + 151689 p^{8} T^{20} - 11453 p^{9} T^{21} + 715 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 8 T + 199 T^{2} - 657 T^{3} + 15497 T^{4} - 11784 T^{5} + 1024693 T^{6} - 275281 T^{7} + 62717478 T^{8} + 2913275 T^{9} + 2925421190 T^{10} + 1550447139 T^{11} + 114197307556 T^{12} + 1550447139 p T^{13} + 2925421190 p^{2} T^{14} + 2913275 p^{3} T^{15} + 62717478 p^{4} T^{16} - 275281 p^{5} T^{17} + 1024693 p^{6} T^{18} - 11784 p^{7} T^{19} + 15497 p^{8} T^{20} - 657 p^{9} T^{21} + 199 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 6 T + 6 p T^{2} + 1573 T^{3} + 32094 T^{4} + 205504 T^{5} + 2951176 T^{6} + 17984407 T^{7} + 208488291 T^{8} + 1180521459 T^{9} + 11680066058 T^{10} + 1483371931 p T^{11} + 529803935660 T^{12} + 1483371931 p^{2} T^{13} + 11680066058 p^{2} T^{14} + 1180521459 p^{3} T^{15} + 208488291 p^{4} T^{16} + 17984407 p^{5} T^{17} + 2951176 p^{6} T^{18} + 205504 p^{7} T^{19} + 32094 p^{8} T^{20} + 1573 p^{9} T^{21} + 6 p^{11} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 16 T + 352 T^{2} - 4313 T^{3} + 57980 T^{4} - 594402 T^{5} + 6230176 T^{6} - 55553277 T^{7} + 491590559 T^{8} - 3876909777 T^{9} + 29940548104 T^{10} - 210459992337 T^{11} + 1441935098296 T^{12} - 210459992337 p T^{13} + 29940548104 p^{2} T^{14} - 3876909777 p^{3} T^{15} + 491590559 p^{4} T^{16} - 55553277 p^{5} T^{17} + 6230176 p^{6} T^{18} - 594402 p^{7} T^{19} + 57980 p^{8} T^{20} - 4313 p^{9} T^{21} + 352 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 18 T + 425 T^{2} - 5660 T^{3} + 83071 T^{4} - 912185 T^{5} + 10346123 T^{6} - 97439810 T^{7} + 923239432 T^{8} - 7625094882 T^{9} + 62597370528 T^{10} - 457914714495 T^{11} + 3314420997080 T^{12} - 457914714495 p T^{13} + 62597370528 p^{2} T^{14} - 7625094882 p^{3} T^{15} + 923239432 p^{4} T^{16} - 97439810 p^{5} T^{17} + 10346123 p^{6} T^{18} - 912185 p^{7} T^{19} + 83071 p^{8} T^{20} - 5660 p^{9} T^{21} + 425 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 20 T + 549 T^{2} - 144 p T^{3} + 122313 T^{4} - 1328735 T^{5} + 16067905 T^{6} - 147185648 T^{7} + 1496194206 T^{8} - 12145564678 T^{9} + 109080630288 T^{10} - 798675767107 T^{11} + 6425851329756 T^{12} - 798675767107 p T^{13} + 109080630288 p^{2} T^{14} - 12145564678 p^{3} T^{15} + 1496194206 p^{4} T^{16} - 147185648 p^{5} T^{17} + 16067905 p^{6} T^{18} - 1328735 p^{7} T^{19} + 122313 p^{8} T^{20} - 144 p^{10} T^{21} + 549 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 4 T + 377 T^{2} - 1836 T^{3} + 73773 T^{4} - 392035 T^{5} + 9904875 T^{6} - 53653804 T^{7} + 1009471726 T^{8} - 5306097190 T^{9} + 81672478638 T^{10} - 400645649097 T^{11} + 5344564122780 T^{12} - 400645649097 p T^{13} + 81672478638 p^{2} T^{14} - 5306097190 p^{3} T^{15} + 1009471726 p^{4} T^{16} - 53653804 p^{5} T^{17} + 9904875 p^{6} T^{18} - 392035 p^{7} T^{19} + 73773 p^{8} T^{20} - 1836 p^{9} T^{21} + 377 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 10 T + 445 T^{2} - 4156 T^{3} + 94520 T^{4} - 789057 T^{5} + 12579213 T^{6} - 91911701 T^{7} + 1186381407 T^{8} - 7600064489 T^{9} + 87750053054 T^{10} - 509722415835 T^{11} + 5618056854064 T^{12} - 509722415835 p T^{13} + 87750053054 p^{2} T^{14} - 7600064489 p^{3} T^{15} + 1186381407 p^{4} T^{16} - 91911701 p^{5} T^{17} + 12579213 p^{6} T^{18} - 789057 p^{7} T^{19} + 94520 p^{8} T^{20} - 4156 p^{9} T^{21} + 445 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 9 T + 364 T^{2} - 2307 T^{3} + 64381 T^{4} - 335261 T^{5} + 8131027 T^{6} - 36716836 T^{7} + 807347350 T^{8} - 3186819725 T^{9} + 66922528995 T^{10} - 241334650304 T^{11} + 4818627301004 T^{12} - 241334650304 p T^{13} + 66922528995 p^{2} T^{14} - 3186819725 p^{3} T^{15} + 807347350 p^{4} T^{16} - 36716836 p^{5} T^{17} + 8131027 p^{6} T^{18} - 335261 p^{7} T^{19} + 64381 p^{8} T^{20} - 2307 p^{9} T^{21} + 364 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 11 T + 396 T^{2} - 4588 T^{3} + 97063 T^{4} - 1016101 T^{5} + 16323044 T^{6} - 156812678 T^{7} + 2072326023 T^{8} - 18020382775 T^{9} + 206089443520 T^{10} - 1618074602797 T^{11} + 16319516903874 T^{12} - 1618074602797 p T^{13} + 206089443520 p^{2} T^{14} - 18020382775 p^{3} T^{15} + 2072326023 p^{4} T^{16} - 156812678 p^{5} T^{17} + 16323044 p^{6} T^{18} - 1016101 p^{7} T^{19} + 97063 p^{8} T^{20} - 4588 p^{9} T^{21} + 396 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 22 T + 633 T^{2} - 9692 T^{3} + 162972 T^{4} - 1961473 T^{5} + 25060151 T^{6} - 256579429 T^{7} + 2800657223 T^{8} - 25744486769 T^{9} + 255423588848 T^{10} - 2167713986391 T^{11} + 20005755817240 T^{12} - 2167713986391 p T^{13} + 255423588848 p^{2} T^{14} - 25744486769 p^{3} T^{15} + 2800657223 p^{4} T^{16} - 256579429 p^{5} T^{17} + 25060151 p^{6} T^{18} - 1961473 p^{7} T^{19} + 162972 p^{8} T^{20} - 9692 p^{9} T^{21} + 633 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 56 T + 1860 T^{2} - 44299 T^{3} + 844526 T^{4} - 13509480 T^{5} + 189081904 T^{6} - 2376082387 T^{7} + 27488453199 T^{8} - 296941458635 T^{9} + 3028562934604 T^{10} - 29215410638157 T^{11} + 267020165736708 T^{12} - 29215410638157 p T^{13} + 3028562934604 p^{2} T^{14} - 296941458635 p^{3} T^{15} + 27488453199 p^{4} T^{16} - 2376082387 p^{5} T^{17} + 189081904 p^{6} T^{18} - 13509480 p^{7} T^{19} + 844526 p^{8} T^{20} - 44299 p^{9} T^{21} + 1860 p^{10} T^{22} - 56 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 26 T + 973 T^{2} - 17670 T^{3} + 376595 T^{4} - 5194783 T^{5} + 80296787 T^{6} - 880200858 T^{7} + 10936562682 T^{8} - 99486639658 T^{9} + 1085115323418 T^{10} - 8825068327103 T^{11} + 92536860445288 T^{12} - 8825068327103 p T^{13} + 1085115323418 p^{2} T^{14} - 99486639658 p^{3} T^{15} + 10936562682 p^{4} T^{16} - 880200858 p^{5} T^{17} + 80296787 p^{6} T^{18} - 5194783 p^{7} T^{19} + 376595 p^{8} T^{20} - 17670 p^{9} T^{21} + 973 p^{10} T^{22} - 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 15 T + 264 T^{2} + 2377 T^{3} + 35623 T^{4} + 309093 T^{5} + 4329853 T^{6} + 33743512 T^{7} + 444605126 T^{8} + 3224059839 T^{9} + 42312991591 T^{10} + 313857383284 T^{11} + 4118090623756 T^{12} + 313857383284 p T^{13} + 42312991591 p^{2} T^{14} + 3224059839 p^{3} T^{15} + 444605126 p^{4} T^{16} + 33743512 p^{5} T^{17} + 4329853 p^{6} T^{18} + 309093 p^{7} T^{19} + 35623 p^{8} T^{20} + 2377 p^{9} T^{21} + 264 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 8 T + 622 T^{2} - 5261 T^{3} + 199675 T^{4} - 1747832 T^{5} + 43601123 T^{6} - 383408405 T^{7} + 7204409176 T^{8} - 61549406631 T^{9} + 947657007183 T^{10} - 7599297626119 T^{11} + 101521370408824 T^{12} - 7599297626119 p T^{13} + 947657007183 p^{2} T^{14} - 61549406631 p^{3} T^{15} + 7204409176 p^{4} T^{16} - 383408405 p^{5} T^{17} + 43601123 p^{6} T^{18} - 1747832 p^{7} T^{19} + 199675 p^{8} T^{20} - 5261 p^{9} T^{21} + 622 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.62067657964600691705809484343, −2.36345706960652270200607565194, −2.17123518985839021774917262741, −2.15571078608387514564322932728, −2.07753994388778658422917351544, −2.06685459868209504507168099010, −2.02677107435435936677294233700, −2.02180665267423109110864251405, −2.01699940121795650077591486045, −1.97854773439203712344921041907, −1.90968660246799491948094041128, −1.78361052107674969049509889923, −1.74283436535647703138444855752, −1.45674301598096392611951289615, −1.21133205348591590088771980689, −1.11343893244387602947438037759, −1.11101240598675156660724598223, −1.09130026746448953081455861403, −1.08246060973286994053641051517, −0.992704419129737678826076666887, −0.944889107847477811213670416716, −0.853361803869348927451257825855, −0.70454365069032092724161603499, −0.68603730339794973346128434820, −0.39380280619834365659702284329, 0.39380280619834365659702284329, 0.68603730339794973346128434820, 0.70454365069032092724161603499, 0.853361803869348927451257825855, 0.944889107847477811213670416716, 0.992704419129737678826076666887, 1.08246060973286994053641051517, 1.09130026746448953081455861403, 1.11101240598675156660724598223, 1.11343893244387602947438037759, 1.21133205348591590088771980689, 1.45674301598096392611951289615, 1.74283436535647703138444855752, 1.78361052107674969049509889923, 1.90968660246799491948094041128, 1.97854773439203712344921041907, 2.01699940121795650077591486045, 2.02180665267423109110864251405, 2.02677107435435936677294233700, 2.06685459868209504507168099010, 2.07753994388778658422917351544, 2.15571078608387514564322932728, 2.17123518985839021774917262741, 2.36345706960652270200607565194, 2.62067657964600691705809484343
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.