Dirichlet series
| L(s) = 1 | + 12·3-s − 8·4-s − 12·5-s + 4·7-s + 78·9-s + 24·11-s − 96·12-s − 8·13-s − 144·15-s + 28·16-s + 28·17-s + 16·19-s + 96·20-s + 48·21-s + 78·25-s + 364·27-s − 32·28-s − 16·29-s + 4·32-s + 288·33-s − 48·35-s − 624·36-s + 20·37-s − 96·39-s − 4·41-s − 12·43-s − 192·44-s + ⋯ |
| L(s) = 1 | + 6.92·3-s − 4·4-s − 5.36·5-s + 1.51·7-s + 26·9-s + 7.23·11-s − 27.7·12-s − 2.21·13-s − 37.1·15-s + 7·16-s + 6.79·17-s + 3.67·19-s + 21.4·20-s + 10.4·21-s + 78/5·25-s + 70.0·27-s − 6.04·28-s − 2.97·29-s + 0.707·32-s + 50.1·33-s − 8.11·35-s − 104·36-s + 3.28·37-s − 15.3·39-s − 0.624·41-s − 1.82·43-s − 28.9·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 23^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 23^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 5^{12} \cdot 23^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.18686\times 10^{21}\) |
| Root analytic conductor: | \(7.95998\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 5^{12} \cdot 23^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8091.337401\) |
| \(L(\frac12)\) | \(\approx\) | \(8091.337401\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{12} \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + p^{3} T^{2} + 9 p^{2} T^{4} - p^{2} T^{5} + 59 p T^{6} - 3 p^{3} T^{7} + 77 p^{2} T^{8} - 21 p^{2} T^{9} + 175 p^{2} T^{10} - 27 p^{3} T^{11} + 727 p T^{12} - 27 p^{4} T^{13} + 175 p^{4} T^{14} - 21 p^{5} T^{15} + 77 p^{6} T^{16} - 3 p^{8} T^{17} + 59 p^{7} T^{18} - p^{9} T^{19} + 9 p^{10} T^{20} + p^{13} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 4 T + 38 T^{2} - 16 p T^{3} + 696 T^{4} - 1816 T^{5} + 9278 T^{6} - 22852 T^{7} + 100139 T^{8} - 233024 T^{9} + 894372 T^{10} - 277192 p T^{11} + 963360 p T^{12} - 277192 p^{2} T^{13} + 894372 p^{2} T^{14} - 233024 p^{3} T^{15} + 100139 p^{4} T^{16} - 22852 p^{5} T^{17} + 9278 p^{6} T^{18} - 1816 p^{7} T^{19} + 696 p^{8} T^{20} - 16 p^{10} T^{21} + 38 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 24 T + 356 T^{2} - 3816 T^{3} + 32819 T^{4} - 235744 T^{5} + 1462918 T^{6} - 7990044 T^{7} + 39073442 T^{8} - 172977808 T^{9} + 700443208 T^{10} - 2610388444 T^{11} + 8998383433 T^{12} - 2610388444 p T^{13} + 700443208 p^{2} T^{14} - 172977808 p^{3} T^{15} + 39073442 p^{4} T^{16} - 7990044 p^{5} T^{17} + 1462918 p^{6} T^{18} - 235744 p^{7} T^{19} + 32819 p^{8} T^{20} - 3816 p^{9} T^{21} + 356 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 8 T + 118 T^{2} + 740 T^{3} + 6114 T^{4} + 31276 T^{5} + 188720 T^{6} + 811232 T^{7} + 4001837 T^{8} + 14944084 T^{9} + 65098594 T^{10} + 220784964 T^{11} + 897906366 T^{12} + 220784964 p T^{13} + 65098594 p^{2} T^{14} + 14944084 p^{3} T^{15} + 4001837 p^{4} T^{16} + 811232 p^{5} T^{17} + 188720 p^{6} T^{18} + 31276 p^{7} T^{19} + 6114 p^{8} T^{20} + 740 p^{9} T^{21} + 118 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 28 T + 492 T^{2} - 6264 T^{3} + 64462 T^{4} - 32792 p T^{5} + 4198182 T^{6} - 28029484 T^{7} + 168765651 T^{8} - 924753636 T^{9} + 4651800142 T^{10} - 21563018788 T^{11} + 92447671354 T^{12} - 21563018788 p T^{13} + 4651800142 p^{2} T^{14} - 924753636 p^{3} T^{15} + 168765651 p^{4} T^{16} - 28029484 p^{5} T^{17} + 4198182 p^{6} T^{18} - 32792 p^{8} T^{19} + 64462 p^{8} T^{20} - 6264 p^{9} T^{21} + 492 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 16 T + 232 T^{2} - 2184 T^{3} + 19145 T^{4} - 134488 T^{5} + 906100 T^{6} - 5278016 T^{7} + 30084510 T^{8} - 153456064 T^{9} + 774512416 T^{10} - 3552638440 T^{11} + 16206623305 T^{12} - 3552638440 p T^{13} + 774512416 p^{2} T^{14} - 153456064 p^{3} T^{15} + 30084510 p^{4} T^{16} - 5278016 p^{5} T^{17} + 906100 p^{6} T^{18} - 134488 p^{7} T^{19} + 19145 p^{8} T^{20} - 2184 p^{9} T^{21} + 232 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 16 T + 12 p T^{2} + 3896 T^{3} + 50562 T^{4} + 448768 T^{5} + 4398796 T^{6} + 32728536 T^{7} + 263235255 T^{8} + 1686487448 T^{9} + 11557677480 T^{10} + 64532141960 T^{11} + 383712889884 T^{12} + 64532141960 p T^{13} + 11557677480 p^{2} T^{14} + 1686487448 p^{3} T^{15} + 263235255 p^{4} T^{16} + 32728536 p^{5} T^{17} + 4398796 p^{6} T^{18} + 448768 p^{7} T^{19} + 50562 p^{8} T^{20} + 3896 p^{9} T^{21} + 12 p^{11} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 170 T^{2} - 96 T^{3} + 15181 T^{4} - 21296 T^{5} + 917126 T^{6} - 2167168 T^{7} + 42075362 T^{8} - 136497872 T^{9} + 1584755298 T^{10} - 5917944080 T^{11} + 52024116733 T^{12} - 5917944080 p T^{13} + 1584755298 p^{2} T^{14} - 136497872 p^{3} T^{15} + 42075362 p^{4} T^{16} - 2167168 p^{5} T^{17} + 917126 p^{6} T^{18} - 21296 p^{7} T^{19} + 15181 p^{8} T^{20} - 96 p^{9} T^{21} + 170 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 - 20 T + 486 T^{2} - 6596 T^{3} + 95578 T^{4} - 1006100 T^{5} + 10942044 T^{6} - 94983644 T^{7} + 843561249 T^{8} - 6247285172 T^{9} + 47230080702 T^{10} - 304030177748 T^{11} + 1997833613670 T^{12} - 304030177748 p T^{13} + 47230080702 p^{2} T^{14} - 6247285172 p^{3} T^{15} + 843561249 p^{4} T^{16} - 94983644 p^{5} T^{17} + 10942044 p^{6} T^{18} - 1006100 p^{7} T^{19} + 95578 p^{8} T^{20} - 6596 p^{9} T^{21} + 486 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 4 T + 244 T^{2} + 1328 T^{3} + 30309 T^{4} + 195112 T^{5} + 2593306 T^{6} + 17792708 T^{7} + 4185072 p T^{8} + 1164728408 T^{9} + 9239631824 T^{10} + 59224983280 T^{11} + 413961962145 T^{12} + 59224983280 p T^{13} + 9239631824 p^{2} T^{14} + 1164728408 p^{3} T^{15} + 4185072 p^{5} T^{16} + 17792708 p^{5} T^{17} + 2593306 p^{6} T^{18} + 195112 p^{7} T^{19} + 30309 p^{8} T^{20} + 1328 p^{9} T^{21} + 244 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 12 T + 280 T^{2} + 3144 T^{3} + 40224 T^{4} + 394196 T^{5} + 3897006 T^{6} + 32545572 T^{7} + 278567269 T^{8} + 2048589852 T^{9} + 15597613238 T^{10} + 105439800136 T^{11} + 725577533946 T^{12} + 105439800136 p T^{13} + 15597613238 p^{2} T^{14} + 2048589852 p^{3} T^{15} + 278567269 p^{4} T^{16} + 32545572 p^{5} T^{17} + 3897006 p^{6} T^{18} + 394196 p^{7} T^{19} + 40224 p^{8} T^{20} + 3144 p^{9} T^{21} + 280 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 4 T + 224 T^{2} - 468 T^{3} + 26594 T^{4} - 20844 T^{5} + 2235896 T^{6} + 668804 T^{7} + 149836767 T^{8} + 149538024 T^{9} + 8588634600 T^{10} + 11150196552 T^{11} + 429508552796 T^{12} + 11150196552 p T^{13} + 8588634600 p^{2} T^{14} + 149538024 p^{3} T^{15} + 149836767 p^{4} T^{16} + 668804 p^{5} T^{17} + 2235896 p^{6} T^{18} - 20844 p^{7} T^{19} + 26594 p^{8} T^{20} - 468 p^{9} T^{21} + 224 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 28 T + 626 T^{2} - 9620 T^{3} + 127372 T^{4} - 1376552 T^{5} + 13183148 T^{6} - 107462020 T^{7} + 781015831 T^{8} - 4844946476 T^{9} + 27509709458 T^{10} - 143614048904 T^{11} + 925410596374 T^{12} - 143614048904 p T^{13} + 27509709458 p^{2} T^{14} - 4844946476 p^{3} T^{15} + 781015831 p^{4} T^{16} - 107462020 p^{5} T^{17} + 13183148 p^{6} T^{18} - 1376552 p^{7} T^{19} + 127372 p^{8} T^{20} - 9620 p^{9} T^{21} + 626 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 20 T + 550 T^{2} - 7648 T^{3} + 129132 T^{4} - 1435184 T^{5} + 18785578 T^{6} - 177979084 T^{7} + 1971634887 T^{8} - 16482778984 T^{9} + 160783162952 T^{10} - 20430217120 p T^{11} + 10534905919788 T^{12} - 20430217120 p^{2} T^{13} + 160783162952 p^{2} T^{14} - 16482778984 p^{3} T^{15} + 1971634887 p^{4} T^{16} - 177979084 p^{5} T^{17} + 18785578 p^{6} T^{18} - 1435184 p^{7} T^{19} + 129132 p^{8} T^{20} - 7648 p^{9} T^{21} + 550 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 32 T + 944 T^{2} - 18688 T^{3} + 339177 T^{4} - 5033056 T^{5} + 69288980 T^{6} - 832019264 T^{7} + 9357791310 T^{8} - 94455295712 T^{9} + 898446380840 T^{10} - 7770819960112 T^{11} + 63523577578601 T^{12} - 7770819960112 p T^{13} + 898446380840 p^{2} T^{14} - 94455295712 p^{3} T^{15} + 9357791310 p^{4} T^{16} - 832019264 p^{5} T^{17} + 69288980 p^{6} T^{18} - 5033056 p^{7} T^{19} + 339177 p^{8} T^{20} - 18688 p^{9} T^{21} + 944 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 4 T + 248 T^{2} + 372 T^{3} + 36834 T^{4} + 50700 T^{5} + 4268056 T^{6} + 4124636 T^{7} + 393824799 T^{8} + 375597416 T^{9} + 32261607600 T^{10} + 25974951496 T^{11} + 2253686426396 T^{12} + 25974951496 p T^{13} + 32261607600 p^{2} T^{14} + 375597416 p^{3} T^{15} + 393824799 p^{4} T^{16} + 4124636 p^{5} T^{17} + 4268056 p^{6} T^{18} + 50700 p^{7} T^{19} + 36834 p^{8} T^{20} + 372 p^{9} T^{21} + 248 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 8 T + 550 T^{2} + 3972 T^{3} + 148743 T^{4} + 991036 T^{5} + 26373520 T^{6} + 163401032 T^{7} + 3426104714 T^{8} + 19651036716 T^{9} + 343769262154 T^{10} + 1799983931068 T^{11} + 27320826927589 T^{12} + 1799983931068 p T^{13} + 343769262154 p^{2} T^{14} + 19651036716 p^{3} T^{15} + 3426104714 p^{4} T^{16} + 163401032 p^{5} T^{17} + 26373520 p^{6} T^{18} + 991036 p^{7} T^{19} + 148743 p^{8} T^{20} + 3972 p^{9} T^{21} + 550 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 4 T + 402 T^{2} - 672 T^{3} + 79396 T^{4} + 480 T^{5} + 10789442 T^{6} + 14306844 T^{7} + 1136669507 T^{8} + 2670690424 T^{9} + 100093241716 T^{10} + 285085123296 T^{11} + 7745903478008 T^{12} + 285085123296 p T^{13} + 100093241716 p^{2} T^{14} + 2670690424 p^{3} T^{15} + 1136669507 p^{4} T^{16} + 14306844 p^{5} T^{17} + 10789442 p^{6} T^{18} + 480 p^{7} T^{19} + 79396 p^{8} T^{20} - 672 p^{9} T^{21} + 402 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 40 T + 960 T^{2} - 14296 T^{3} + 148525 T^{4} - 938968 T^{5} + 3631680 T^{6} - 18105640 T^{7} + 596654598 T^{8} - 9835459624 T^{9} + 106916974656 T^{10} - 786673898968 T^{11} + 6336425458665 T^{12} - 786673898968 p T^{13} + 106916974656 p^{2} T^{14} - 9835459624 p^{3} T^{15} + 596654598 p^{4} T^{16} - 18105640 p^{5} T^{17} + 3631680 p^{6} T^{18} - 938968 p^{7} T^{19} + 148525 p^{8} T^{20} - 14296 p^{9} T^{21} + 960 p^{10} T^{22} - 40 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 100 T + 5378 T^{2} - 201812 T^{3} + 5862548 T^{4} - 139210968 T^{5} + 2792741524 T^{6} - 48361320028 T^{7} + 8838409533 p T^{8} - 9847242227612 T^{9} + 117795390011914 T^{10} - 1261484581862936 T^{11} + 12125626996077206 T^{12} - 1261484581862936 p T^{13} + 117795390011914 p^{2} T^{14} - 9847242227612 p^{3} T^{15} + 8838409533 p^{5} T^{16} - 48361320028 p^{5} T^{17} + 2792741524 p^{6} T^{18} - 139210968 p^{7} T^{19} + 5862548 p^{8} T^{20} - 201812 p^{9} T^{21} + 5378 p^{10} T^{22} - 100 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 80 T + 3794 T^{2} - 129536 T^{3} + 3517276 T^{4} - 79471552 T^{5} + 1541043218 T^{6} - 26126448592 T^{7} + 392606254839 T^{8} - 5276203649416 T^{9} + 63830965003804 T^{10} - 697951044291640 T^{11} + 6915033216311708 T^{12} - 697951044291640 p T^{13} + 63830965003804 p^{2} T^{14} - 5276203649416 p^{3} T^{15} + 392606254839 p^{4} T^{16} - 26126448592 p^{5} T^{17} + 1541043218 p^{6} T^{18} - 79471552 p^{7} T^{19} + 3517276 p^{8} T^{20} - 129536 p^{9} T^{21} + 3794 p^{10} T^{22} - 80 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 8 T + 524 T^{2} + 4876 T^{3} + 148100 T^{4} + 1455836 T^{5} + 30749058 T^{6} + 289202556 T^{7} + 5014298433 T^{8} + 43702439936 T^{9} + 653108320042 T^{10} + 5282660821196 T^{11} + 69582685585666 T^{12} + 5282660821196 p T^{13} + 653108320042 p^{2} T^{14} + 43702439936 p^{3} T^{15} + 5014298433 p^{4} T^{16} + 289202556 p^{5} T^{17} + 30749058 p^{6} T^{18} + 1455836 p^{7} T^{19} + 148100 p^{8} T^{20} + 4876 p^{9} T^{21} + 524 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.35617653216165769364689448985, −2.33165649081389832027643091207, −2.07292294414274574313468770839, −2.06872929804720306705648165638, −1.97445779308897079730633336811, −1.88976430728003688537469010581, −1.70635240712906758528553529979, −1.69023999226504611243210369991, −1.60711219525719513157315076323, −1.57410996292646603894813317595, −1.51826231015634110915916497545, −1.43590871297710868870237020220, −1.41616370440354208248751698663, −1.35286245899002217553537330239, −0.940974129781193425715853097682, −0.910234751289091881286827694303, −0.896879037654984767184035760836, −0.75472547024661415141438754308, −0.72922705898571647700632462992, −0.72010337191029619575912190255, −0.69163463875582923663965781924, −0.65438909267406323452492577316, −0.56596249880044130467028873079, −0.55441443630871217357999903986, −0.46290715810283136135853141244, 0.46290715810283136135853141244, 0.55441443630871217357999903986, 0.56596249880044130467028873079, 0.65438909267406323452492577316, 0.69163463875582923663965781924, 0.72010337191029619575912190255, 0.72922705898571647700632462992, 0.75472547024661415141438754308, 0.896879037654984767184035760836, 0.910234751289091881286827694303, 0.940974129781193425715853097682, 1.35286245899002217553537330239, 1.41616370440354208248751698663, 1.43590871297710868870237020220, 1.51826231015634110915916497545, 1.57410996292646603894813317595, 1.60711219525719513157315076323, 1.69023999226504611243210369991, 1.70635240712906758528553529979, 1.88976430728003688537469010581, 1.97445779308897079730633336811, 2.06872929804720306705648165638, 2.07292294414274574313468770839, 2.33165649081389832027643091207, 2.35617653216165769364689448985
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.