Dirichlet series
| L(s) = 1 | − 3-s + 8·5-s − 4·7-s − 10·9-s + 6·11-s + 13·13-s − 8·15-s + 10·17-s − 19-s + 4·21-s + 3·23-s + 11·25-s + 8·27-s + 29·29-s − 3·31-s − 6·33-s − 32·35-s + 41·37-s − 13·39-s + 20·41-s − 43-s − 80·45-s + 5·47-s − 24·49-s − 10·51-s + 39·53-s + 48·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3.57·5-s − 1.51·7-s − 3.33·9-s + 1.80·11-s + 3.60·13-s − 2.06·15-s + 2.42·17-s − 0.229·19-s + 0.872·21-s + 0.625·23-s + 11/5·25-s + 1.53·27-s + 5.38·29-s − 0.538·31-s − 1.04·33-s − 5.40·35-s + 6.74·37-s − 2.08·39-s + 3.12·41-s − 0.152·43-s − 11.9·45-s + 0.729·47-s − 3.42·49-s − 1.40·51-s + 5.35·53-s + 6.47·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 167^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.17271\times 10^{12}\) |
| Root analytic conductor: | \(3.26619\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 167^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(57.21189998\) |
| \(L(\frac12)\) | \(\approx\) | \(57.21189998\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 167 | \( ( 1 + T )^{12} \) | |
| good | 3 | \( 1 + T + 11 T^{2} + 13 T^{3} + 68 T^{4} + 10 p^{2} T^{5} + 326 T^{6} + 160 p T^{7} + 146 p^{2} T^{8} + 2062 T^{9} + 4559 T^{10} + 2404 p T^{11} + 14242 T^{12} + 2404 p^{2} T^{13} + 4559 p^{2} T^{14} + 2062 p^{3} T^{15} + 146 p^{6} T^{16} + 160 p^{6} T^{17} + 326 p^{6} T^{18} + 10 p^{9} T^{19} + 68 p^{8} T^{20} + 13 p^{9} T^{21} + 11 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 8 T + 53 T^{2} - 248 T^{3} + 1047 T^{4} - 3688 T^{5} + 12156 T^{6} - 35512 T^{7} + 99309 T^{8} - 254088 T^{9} + 634687 T^{10} - 1479832 T^{11} + 3411222 T^{12} - 1479832 p T^{13} + 634687 p^{2} T^{14} - 254088 p^{3} T^{15} + 99309 p^{4} T^{16} - 35512 p^{5} T^{17} + 12156 p^{6} T^{18} - 3688 p^{7} T^{19} + 1047 p^{8} T^{20} - 248 p^{9} T^{21} + 53 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 4 T + 40 T^{2} + 136 T^{3} + 17 p^{2} T^{4} + 2529 T^{5} + 12154 T^{6} + 33386 T^{7} + 137681 T^{8} + 347839 T^{9} + 1274382 T^{10} + 2958957 T^{11} + 1396582 p T^{12} + 2958957 p T^{13} + 1274382 p^{2} T^{14} + 347839 p^{3} T^{15} + 137681 p^{4} T^{16} + 33386 p^{5} T^{17} + 12154 p^{6} T^{18} + 2529 p^{7} T^{19} + 17 p^{10} T^{20} + 136 p^{9} T^{21} + 40 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T + 78 T^{2} - 368 T^{3} + 2890 T^{4} - 11755 T^{5} + 71010 T^{6} - 258818 T^{7} + 1309339 T^{8} - 4348186 T^{9} + 19257904 T^{10} - 58566749 T^{11} + 232768628 T^{12} - 58566749 p T^{13} + 19257904 p^{2} T^{14} - 4348186 p^{3} T^{15} + 1309339 p^{4} T^{16} - 258818 p^{5} T^{17} + 71010 p^{6} T^{18} - 11755 p^{7} T^{19} + 2890 p^{8} T^{20} - 368 p^{9} T^{21} + 78 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - p T + 166 T^{2} - 1386 T^{3} + 843 p T^{4} - 70182 T^{5} + 426682 T^{6} - 2255577 T^{7} + 875071 p T^{8} - 51440742 T^{9} + 222705776 T^{10} - 876391344 T^{11} + 3307744314 T^{12} - 876391344 p T^{13} + 222705776 p^{2} T^{14} - 51440742 p^{3} T^{15} + 875071 p^{5} T^{16} - 2255577 p^{5} T^{17} + 426682 p^{6} T^{18} - 70182 p^{7} T^{19} + 843 p^{9} T^{20} - 1386 p^{9} T^{21} + 166 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 10 T + 165 T^{2} - 1192 T^{3} + 11741 T^{4} - 68630 T^{5} + 517025 T^{6} - 2584468 T^{7} + 16278147 T^{8} - 71577210 T^{9} + 392465306 T^{10} - 1537406410 T^{11} + 7474717374 T^{12} - 1537406410 p T^{13} + 392465306 p^{2} T^{14} - 71577210 p^{3} T^{15} + 16278147 p^{4} T^{16} - 2584468 p^{5} T^{17} + 517025 p^{6} T^{18} - 68630 p^{7} T^{19} + 11741 p^{8} T^{20} - 1192 p^{9} T^{21} + 165 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + T + 143 T^{2} + 81 T^{3} + 10035 T^{4} + 1401 T^{5} + 458986 T^{6} - 103028 T^{7} + 15376081 T^{8} - 7311047 T^{9} + 21136557 p T^{10} - 231239358 T^{11} + 8453969526 T^{12} - 231239358 p T^{13} + 21136557 p^{3} T^{14} - 7311047 p^{3} T^{15} + 15376081 p^{4} T^{16} - 103028 p^{5} T^{17} + 458986 p^{6} T^{18} + 1401 p^{7} T^{19} + 10035 p^{8} T^{20} + 81 p^{9} T^{21} + 143 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 3 T + 156 T^{2} - 544 T^{3} + 12859 T^{4} - 46134 T^{5} + 722980 T^{6} - 2512287 T^{7} + 30281675 T^{8} - 98657940 T^{9} + 984601984 T^{10} - 2937072550 T^{11} + 25361223538 T^{12} - 2937072550 p T^{13} + 984601984 p^{2} T^{14} - 98657940 p^{3} T^{15} + 30281675 p^{4} T^{16} - 2512287 p^{5} T^{17} + 722980 p^{6} T^{18} - 46134 p^{7} T^{19} + 12859 p^{8} T^{20} - 544 p^{9} T^{21} + 156 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - p T + 548 T^{2} - 7552 T^{3} + 87152 T^{4} - 29776 p T^{5} + 7675970 T^{6} - 61582806 T^{7} + 455093348 T^{8} - 3101918876 T^{9} + 19739486944 T^{10} - 4037683581 p T^{11} + 652152687522 T^{12} - 4037683581 p^{2} T^{13} + 19739486944 p^{2} T^{14} - 3101918876 p^{3} T^{15} + 455093348 p^{4} T^{16} - 61582806 p^{5} T^{17} + 7675970 p^{6} T^{18} - 29776 p^{8} T^{19} + 87152 p^{8} T^{20} - 7552 p^{9} T^{21} + 548 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 3 T + 184 T^{2} + 16 p T^{3} + 17088 T^{4} + 38208 T^{5} + 1061590 T^{6} + 1787402 T^{7} + 49574412 T^{8} + 57911964 T^{9} + 1884191468 T^{10} + 1580225069 T^{11} + 61992076274 T^{12} + 1580225069 p T^{13} + 1884191468 p^{2} T^{14} + 57911964 p^{3} T^{15} + 49574412 p^{4} T^{16} + 1787402 p^{5} T^{17} + 1061590 p^{6} T^{18} + 38208 p^{7} T^{19} + 17088 p^{8} T^{20} + 16 p^{10} T^{21} + 184 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 41 T + 1033 T^{2} - 507 p T^{3} + 275331 T^{4} - 3406272 T^{5} + 36890664 T^{6} - 356083210 T^{7} + 3115607565 T^{8} - 24911189221 T^{9} + 183516644463 T^{10} - 1249548077529 T^{11} + 7892527209182 T^{12} - 1249548077529 p T^{13} + 183516644463 p^{2} T^{14} - 24911189221 p^{3} T^{15} + 3115607565 p^{4} T^{16} - 356083210 p^{5} T^{17} + 36890664 p^{6} T^{18} - 3406272 p^{7} T^{19} + 275331 p^{8} T^{20} - 507 p^{10} T^{21} + 1033 p^{10} T^{22} - 41 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 20 T + 462 T^{2} - 6324 T^{3} + 87975 T^{4} - 937816 T^{5} + 9887402 T^{6} - 87412528 T^{7} + 762361555 T^{8} - 5818085772 T^{9} + 44034155752 T^{10} - 298034708708 T^{11} + 2010901084426 T^{12} - 298034708708 p T^{13} + 44034155752 p^{2} T^{14} - 5818085772 p^{3} T^{15} + 762361555 p^{4} T^{16} - 87412528 p^{5} T^{17} + 9887402 p^{6} T^{18} - 937816 p^{7} T^{19} + 87975 p^{8} T^{20} - 6324 p^{9} T^{21} + 462 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + T + 322 T^{2} + 210 T^{3} + 52081 T^{4} + 20518 T^{5} + 5574006 T^{6} + 1198663 T^{7} + 438696187 T^{8} + 47048986 T^{9} + 26694033368 T^{10} + 1531431188 T^{11} + 1285689958822 T^{12} + 1531431188 p T^{13} + 26694033368 p^{2} T^{14} + 47048986 p^{3} T^{15} + 438696187 p^{4} T^{16} + 1198663 p^{5} T^{17} + 5574006 p^{6} T^{18} + 20518 p^{7} T^{19} + 52081 p^{8} T^{20} + 210 p^{9} T^{21} + 322 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 5 T + 272 T^{2} - 1154 T^{3} + 38220 T^{4} - 142892 T^{5} + 3793456 T^{6} - 12969688 T^{7} + 294553462 T^{8} - 924023442 T^{9} + 392317650 p T^{10} - 52757098657 T^{11} + 949938052894 T^{12} - 52757098657 p T^{13} + 392317650 p^{3} T^{14} - 924023442 p^{3} T^{15} + 294553462 p^{4} T^{16} - 12969688 p^{5} T^{17} + 3793456 p^{6} T^{18} - 142892 p^{7} T^{19} + 38220 p^{8} T^{20} - 1154 p^{9} T^{21} + 272 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 39 T + 1040 T^{2} - 20272 T^{3} + 328087 T^{4} - 4495172 T^{5} + 54394544 T^{6} - 587035227 T^{7} + 5781909051 T^{8} - 52318711658 T^{9} + 441970840512 T^{10} - 3498408843716 T^{11} + 26210432476442 T^{12} - 3498408843716 p T^{13} + 441970840512 p^{2} T^{14} - 52318711658 p^{3} T^{15} + 5781909051 p^{4} T^{16} - 587035227 p^{5} T^{17} + 54394544 p^{6} T^{18} - 4495172 p^{7} T^{19} + 328087 p^{8} T^{20} - 20272 p^{9} T^{21} + 1040 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 8 T + 360 T^{2} - 2574 T^{3} + 64925 T^{4} - 419628 T^{5} + 7794256 T^{6} - 46045554 T^{7} + 706940547 T^{8} - 3846500746 T^{9} + 52187047304 T^{10} - 263973152262 T^{11} + 3297051899198 T^{12} - 263973152262 p T^{13} + 52187047304 p^{2} T^{14} - 3846500746 p^{3} T^{15} + 706940547 p^{4} T^{16} - 46045554 p^{5} T^{17} + 7794256 p^{6} T^{18} - 419628 p^{7} T^{19} + 64925 p^{8} T^{20} - 2574 p^{9} T^{21} + 360 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 30 T + 843 T^{2} - 15790 T^{3} + 274782 T^{4} - 3898493 T^{5} + 51874541 T^{6} - 601780665 T^{7} + 6604262409 T^{8} - 65123828282 T^{9} + 611139914150 T^{10} - 5226719073036 T^{11} + 42670329733292 T^{12} - 5226719073036 p T^{13} + 611139914150 p^{2} T^{14} - 65123828282 p^{3} T^{15} + 6604262409 p^{4} T^{16} - 601780665 p^{5} T^{17} + 51874541 p^{6} T^{18} - 3898493 p^{7} T^{19} + 274782 p^{8} T^{20} - 15790 p^{9} T^{21} + 843 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 9 T + 601 T^{2} + 3775 T^{3} + 157069 T^{4} + 604972 T^{5} + 23942342 T^{6} + 34337308 T^{7} + 2438929425 T^{8} - 2442927869 T^{9} + 188022122801 T^{10} - 539005678755 T^{11} + 12811914209986 T^{12} - 539005678755 p T^{13} + 188022122801 p^{2} T^{14} - 2442927869 p^{3} T^{15} + 2438929425 p^{4} T^{16} + 34337308 p^{5} T^{17} + 23942342 p^{6} T^{18} + 604972 p^{7} T^{19} + 157069 p^{8} T^{20} + 3775 p^{9} T^{21} + 601 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 29 T + 933 T^{2} - 18131 T^{3} + 349216 T^{4} - 5237441 T^{5} + 75956737 T^{6} - 938754783 T^{7} + 11147389591 T^{8} - 117660839402 T^{9} + 1191836294858 T^{10} - 10951471090550 T^{11} + 96586453776592 T^{12} - 10951471090550 p T^{13} + 1191836294858 p^{2} T^{14} - 117660839402 p^{3} T^{15} + 11147389591 p^{4} T^{16} - 938754783 p^{5} T^{17} + 75956737 p^{6} T^{18} - 5237441 p^{7} T^{19} + 349216 p^{8} T^{20} - 18131 p^{9} T^{21} + 933 p^{10} T^{22} - 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 12 T + 635 T^{2} - 7394 T^{3} + 194695 T^{4} - 2147414 T^{5} + 38452515 T^{6} - 390830824 T^{7} + 5462001571 T^{8} - 50006017692 T^{9} + 586344593106 T^{10} - 4769156371240 T^{11} + 48581803711146 T^{12} - 4769156371240 p T^{13} + 586344593106 p^{2} T^{14} - 50006017692 p^{3} T^{15} + 5462001571 p^{4} T^{16} - 390830824 p^{5} T^{17} + 38452515 p^{6} T^{18} - 2147414 p^{7} T^{19} + 194695 p^{8} T^{20} - 7394 p^{9} T^{21} + 635 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 2 T + 478 T^{2} + 662 T^{3} + 116823 T^{4} + 163500 T^{5} + 19383770 T^{6} + 31924712 T^{7} + 2440536275 T^{8} + 4660284782 T^{9} + 248660799624 T^{10} + 498568629982 T^{11} + 21281094687146 T^{12} + 498568629982 p T^{13} + 248660799624 p^{2} T^{14} + 4660284782 p^{3} T^{15} + 2440536275 p^{4} T^{16} + 31924712 p^{5} T^{17} + 19383770 p^{6} T^{18} + 163500 p^{7} T^{19} + 116823 p^{8} T^{20} + 662 p^{9} T^{21} + 478 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 5 T + 575 T^{2} + 3737 T^{3} + 167083 T^{4} + 1276028 T^{5} + 32443144 T^{6} + 271988078 T^{7} + 4692313117 T^{8} + 40595516297 T^{9} + 533788509385 T^{10} + 4470477865035 T^{11} + 49073769984782 T^{12} + 4470477865035 p T^{13} + 533788509385 p^{2} T^{14} + 40595516297 p^{3} T^{15} + 4692313117 p^{4} T^{16} + 271988078 p^{5} T^{17} + 32443144 p^{6} T^{18} + 1276028 p^{7} T^{19} + 167083 p^{8} T^{20} + 3737 p^{9} T^{21} + 575 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 7 T + 525 T^{2} - 2629 T^{3} + 131137 T^{4} - 350743 T^{5} + 21160946 T^{6} - 7265218 T^{7} + 2583323699 T^{8} + 4194750279 T^{9} + 267349298145 T^{10} + 731344991270 T^{11} + 24912791136118 T^{12} + 731344991270 p T^{13} + 267349298145 p^{2} T^{14} + 4194750279 p^{3} T^{15} + 2583323699 p^{4} T^{16} - 7265218 p^{5} T^{17} + 21160946 p^{6} T^{18} - 350743 p^{7} T^{19} + 131137 p^{8} T^{20} - 2629 p^{9} T^{21} + 525 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 14 T + 471 T^{2} + 4178 T^{3} + 85936 T^{4} + 546301 T^{5} + 9377607 T^{6} + 39835307 T^{7} + 587492441 T^{8} - 333339196 T^{9} + 5928057546 T^{10} - 362031342048 T^{11} - 1474830902652 T^{12} - 362031342048 p T^{13} + 5928057546 p^{2} T^{14} - 333339196 p^{3} T^{15} + 587492441 p^{4} T^{16} + 39835307 p^{5} T^{17} + 9377607 p^{6} T^{18} + 546301 p^{7} T^{19} + 85936 p^{8} T^{20} + 4178 p^{9} T^{21} + 471 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.19636612368354051107705773384, −2.76652621113286438428306661756, −2.63842526533332100693388387889, −2.57173354912922029586624860215, −2.56811936323561939009742809961, −2.52787812721698619186031186493, −2.50752871509487936923165976578, −2.46706728195288175370077846436, −2.28053199297104851177852492938, −2.27705267046501090372870094314, −2.22553645527067132989650449421, −2.08560275905975347737519616713, −1.90785374501856895093850225941, −1.57293867000652323425899183943, −1.51788712502748915716106788013, −1.42102125427932849813244011553, −1.35016191060245711884461783201, −1.26351967056821844930212407490, −1.02106848776333508023550376181, −0.957666243795140857488366111043, −0.76679874117256961077544409395, −0.76114014011926294330767719391, −0.71760205230954866711681060204, −0.68326119589542720648670334673, −0.25258999586443835393340449380, 0.25258999586443835393340449380, 0.68326119589542720648670334673, 0.71760205230954866711681060204, 0.76114014011926294330767719391, 0.76679874117256961077544409395, 0.957666243795140857488366111043, 1.02106848776333508023550376181, 1.26351967056821844930212407490, 1.35016191060245711884461783201, 1.42102125427932849813244011553, 1.51788712502748915716106788013, 1.57293867000652323425899183943, 1.90785374501856895093850225941, 2.08560275905975347737519616713, 2.22553645527067132989650449421, 2.27705267046501090372870094314, 2.28053199297104851177852492938, 2.46706728195288175370077846436, 2.50752871509487936923165976578, 2.52787812721698619186031186493, 2.56811936323561939009742809961, 2.57173354912922029586624860215, 2.63842526533332100693388387889, 2.76652621113286438428306661756, 3.19636612368354051107705773384
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.