Dirichlet series
| L(s) = 1 | − 3·5-s + 3·7-s − 6·9-s + 12·11-s + 12·13-s − 6·17-s + 9·19-s − 30·23-s + 15·29-s − 9·35-s − 15·37-s − 12·41-s − 9·43-s + 18·45-s + 9·47-s − 15·49-s − 12·53-s − 36·55-s − 12·59-s − 36·61-s − 18·63-s − 36·65-s − 36·67-s − 12·71-s − 21·73-s + 36·77-s − 39·79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.13·7-s − 2·9-s + 3.61·11-s + 3.32·13-s − 1.45·17-s + 2.06·19-s − 6.25·23-s + 2.78·29-s − 1.52·35-s − 2.46·37-s − 1.87·41-s − 1.37·43-s + 2.68·45-s + 1.31·47-s − 2.14·49-s − 1.64·53-s − 4.85·55-s − 1.56·59-s − 4.60·61-s − 2.26·63-s − 4.46·65-s − 4.39·67-s − 1.42·71-s − 2.45·73-s + 4.10·77-s − 4.38·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.83877\times 10^{6}\) |
| Root analytic conductor: | \(1.85729\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4982714700\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4982714700\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 2 p T^{2} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + 19 p T^{6} + 2 p^{3} T^{7} + 2 p^{4} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) | |
| good | 5 | \( 1 + 3 T + 9 T^{2} + 9 T^{3} - 18 T^{4} - 42 T^{5} - 142 T^{6} + 9 T^{7} - 171 T^{8} - 702 T^{9} - 603 T^{10} - 1791 T^{11} + 8049 T^{12} - 1791 p T^{13} - 603 p^{2} T^{14} - 702 p^{3} T^{15} - 171 p^{4} T^{16} + 9 p^{5} T^{17} - 142 p^{6} T^{18} - 42 p^{7} T^{19} - 18 p^{8} T^{20} + 9 p^{9} T^{21} + 9 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 3 T + 24 T^{2} - 46 T^{3} + 321 T^{4} - 561 T^{5} + 472 p T^{6} - 4068 T^{7} + 3681 p T^{8} - 26386 T^{9} + 193845 T^{10} - 149838 T^{11} + 1294315 T^{12} - 149838 p T^{13} + 193845 p^{2} T^{14} - 26386 p^{3} T^{15} + 3681 p^{5} T^{16} - 4068 p^{5} T^{17} + 472 p^{7} T^{18} - 561 p^{7} T^{19} + 321 p^{8} T^{20} - 46 p^{9} T^{21} + 24 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 12 T + 90 T^{2} - 504 T^{3} + 2628 T^{4} - 12720 T^{5} + 57872 T^{6} - 239643 T^{7} + 87030 p T^{8} - 3660336 T^{9} + 13502205 T^{10} - 46654983 T^{11} + 157089783 T^{12} - 46654983 p T^{13} + 13502205 p^{2} T^{14} - 3660336 p^{3} T^{15} + 87030 p^{5} T^{16} - 239643 p^{5} T^{17} + 57872 p^{6} T^{18} - 12720 p^{7} T^{19} + 2628 p^{8} T^{20} - 504 p^{9} T^{21} + 90 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 12 T + 48 T^{2} - 74 T^{3} + 246 T^{4} - 1974 T^{5} + 8974 T^{6} - 33102 T^{7} + 94338 T^{8} - 225668 T^{9} + 896478 T^{10} - 3069618 T^{11} + 8300191 T^{12} - 3069618 p T^{13} + 896478 p^{2} T^{14} - 225668 p^{3} T^{15} + 94338 p^{4} T^{16} - 33102 p^{5} T^{17} + 8974 p^{6} T^{18} - 1974 p^{7} T^{19} + 246 p^{8} T^{20} - 74 p^{9} T^{21} + 48 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 6 T - 48 T^{2} - 252 T^{3} + 1722 T^{4} + 5946 T^{5} - 48874 T^{6} - 104832 T^{7} + 1071234 T^{8} + 1189944 T^{9} - 21101022 T^{10} - 6521976 T^{11} + 379698279 T^{12} - 6521976 p T^{13} - 21101022 p^{2} T^{14} + 1189944 p^{3} T^{15} + 1071234 p^{4} T^{16} - 104832 p^{5} T^{17} - 48874 p^{6} T^{18} + 5946 p^{7} T^{19} + 1722 p^{8} T^{20} - 252 p^{9} T^{21} - 48 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 9 T - 36 T^{2} + 419 T^{3} + 1611 T^{4} - 13806 T^{5} - 55610 T^{6} + 315819 T^{7} + 1580067 T^{8} - 4822864 T^{9} - 38744127 T^{10} + 34656777 T^{11} + 805313071 T^{12} + 34656777 p T^{13} - 38744127 p^{2} T^{14} - 4822864 p^{3} T^{15} + 1580067 p^{4} T^{16} + 315819 p^{5} T^{17} - 55610 p^{6} T^{18} - 13806 p^{7} T^{19} + 1611 p^{8} T^{20} + 419 p^{9} T^{21} - 36 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 30 T + 18 p T^{2} + 3222 T^{3} + 12348 T^{4} - 19608 T^{5} - 550618 T^{6} - 3222558 T^{7} - 7146234 T^{8} + 22769856 T^{9} + 232883658 T^{10} + 864182466 T^{11} + 2884098855 T^{12} + 864182466 p T^{13} + 232883658 p^{2} T^{14} + 22769856 p^{3} T^{15} - 7146234 p^{4} T^{16} - 3222558 p^{5} T^{17} - 550618 p^{6} T^{18} - 19608 p^{7} T^{19} + 12348 p^{8} T^{20} + 3222 p^{9} T^{21} + 18 p^{11} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 15 T + 81 T^{2} + 45 T^{3} - 4518 T^{4} + 31692 T^{5} - 51964 T^{6} - 564705 T^{7} + 4930839 T^{8} - 14676930 T^{9} - 17187291 T^{10} + 338065677 T^{11} - 2175351603 T^{12} + 338065677 p T^{13} - 17187291 p^{2} T^{14} - 14676930 p^{3} T^{15} + 4930839 p^{4} T^{16} - 564705 p^{5} T^{17} - 51964 p^{6} T^{18} + 31692 p^{7} T^{19} - 4518 p^{8} T^{20} + 45 p^{9} T^{21} + 81 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 81 T^{2} - 49 T^{3} + 4023 T^{4} + 2565 T^{5} + 163531 T^{6} + 464805 T^{7} + 4701186 T^{8} + 27482744 T^{9} + 112711716 T^{10} + 1250347077 T^{11} + 3229099084 T^{12} + 1250347077 p T^{13} + 112711716 p^{2} T^{14} + 27482744 p^{3} T^{15} + 4701186 p^{4} T^{16} + 464805 p^{5} T^{17} + 163531 p^{6} T^{18} + 2565 p^{7} T^{19} + 4023 p^{8} T^{20} - 49 p^{9} T^{21} + 81 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 + 15 T + 3 T^{2} - 578 T^{3} + 2505 T^{4} + 24945 T^{5} - 192167 T^{6} - 791163 T^{7} + 6816672 T^{8} + 1096072 T^{9} - 278194485 T^{10} + 271278831 T^{11} + 11925628483 T^{12} + 271278831 p T^{13} - 278194485 p^{2} T^{14} + 1096072 p^{3} T^{15} + 6816672 p^{4} T^{16} - 791163 p^{5} T^{17} - 192167 p^{6} T^{18} + 24945 p^{7} T^{19} + 2505 p^{8} T^{20} - 578 p^{9} T^{21} + 3 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 12 T + 117 T^{2} + 630 T^{3} + 5031 T^{4} + 23943 T^{5} + 171524 T^{6} + 408330 T^{7} + 3551247 T^{8} - 22823100 T^{9} - 180203112 T^{10} - 60994557 p T^{11} - 9745515051 T^{12} - 60994557 p^{2} T^{13} - 180203112 p^{2} T^{14} - 22823100 p^{3} T^{15} + 3551247 p^{4} T^{16} + 408330 p^{5} T^{17} + 171524 p^{6} T^{18} + 23943 p^{7} T^{19} + 5031 p^{8} T^{20} + 630 p^{9} T^{21} + 117 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 9 T + 36 T^{2} + 248 T^{3} + 1629 T^{4} + 13671 T^{5} + 99658 T^{6} + 1291248 T^{7} + 7996347 T^{8} + 44034284 T^{9} + 342067257 T^{10} + 2064609882 T^{11} + 16750938331 T^{12} + 2064609882 p T^{13} + 342067257 p^{2} T^{14} + 44034284 p^{3} T^{15} + 7996347 p^{4} T^{16} + 1291248 p^{5} T^{17} + 99658 p^{6} T^{18} + 13671 p^{7} T^{19} + 1629 p^{8} T^{20} + 248 p^{9} T^{21} + 36 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 9 T + 99 T^{2} - 981 T^{3} + 6354 T^{4} - 60768 T^{5} + 331868 T^{6} - 2211183 T^{7} + 13193361 T^{8} - 82909494 T^{9} + 638454717 T^{10} - 3119341005 T^{11} + 27438934035 T^{12} - 3119341005 p T^{13} + 638454717 p^{2} T^{14} - 82909494 p^{3} T^{15} + 13193361 p^{4} T^{16} - 2211183 p^{5} T^{17} + 331868 p^{6} T^{18} - 60768 p^{7} T^{19} + 6354 p^{8} T^{20} - 981 p^{9} T^{21} + 99 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 6 T + 255 T^{2} + 1593 T^{3} + 29193 T^{4} + 168801 T^{5} + 1959550 T^{6} + 168801 p T^{7} + 29193 p^{2} T^{8} + 1593 p^{3} T^{9} + 255 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 12 T + 9 T^{2} - 288 T^{3} - 7416 T^{4} - 73437 T^{5} + 128189 T^{6} + 3672837 T^{7} + 36794322 T^{8} + 361771704 T^{9} - 137989746 T^{10} - 18392855166 T^{11} - 121796603070 T^{12} - 18392855166 p T^{13} - 137989746 p^{2} T^{14} + 361771704 p^{3} T^{15} + 36794322 p^{4} T^{16} + 3672837 p^{5} T^{17} + 128189 p^{6} T^{18} - 73437 p^{7} T^{19} - 7416 p^{8} T^{20} - 288 p^{9} T^{21} + 9 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 36 T + 531 T^{2} + 3559 T^{3} + 3123 T^{4} - 91953 T^{5} - 447281 T^{6} - 4553469 T^{7} - 84400488 T^{8} - 593953490 T^{9} - 1210754052 T^{10} - 5462757351 T^{11} - 104608387268 T^{12} - 5462757351 p T^{13} - 1210754052 p^{2} T^{14} - 593953490 p^{3} T^{15} - 84400488 p^{4} T^{16} - 4553469 p^{5} T^{17} - 447281 p^{6} T^{18} - 91953 p^{7} T^{19} + 3123 p^{8} T^{20} + 3559 p^{9} T^{21} + 531 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 36 T + 630 T^{2} + 7592 T^{3} + 71910 T^{4} + 581724 T^{5} + 4963294 T^{6} + 52102143 T^{7} + 557562474 T^{8} + 5169740204 T^{9} + 40448438361 T^{10} + 276054423399 T^{11} + 1991438619445 T^{12} + 276054423399 p T^{13} + 40448438361 p^{2} T^{14} + 5169740204 p^{3} T^{15} + 557562474 p^{4} T^{16} + 52102143 p^{5} T^{17} + 4963294 p^{6} T^{18} + 581724 p^{7} T^{19} + 71910 p^{8} T^{20} + 7592 p^{9} T^{21} + 630 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 12 T - 201 T^{2} - 2430 T^{3} + 25608 T^{4} + 247278 T^{5} - 2990308 T^{6} - 17740170 T^{7} + 310819581 T^{8} + 1005967566 T^{9} - 26322276303 T^{10} - 30298327380 T^{11} + 1913975417427 T^{12} - 30298327380 p T^{13} - 26322276303 p^{2} T^{14} + 1005967566 p^{3} T^{15} + 310819581 p^{4} T^{16} - 17740170 p^{5} T^{17} - 2990308 p^{6} T^{18} + 247278 p^{7} T^{19} + 25608 p^{8} T^{20} - 2430 p^{9} T^{21} - 201 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 21 T - 48 T^{2} - 3887 T^{3} - 4845 T^{4} + 448266 T^{5} + 1641574 T^{6} - 22871205 T^{7} - 33688827 T^{8} + 739848004 T^{9} - 9072236607 T^{10} + 8731491045 T^{11} + 1415682555505 T^{12} + 8731491045 p T^{13} - 9072236607 p^{2} T^{14} + 739848004 p^{3} T^{15} - 33688827 p^{4} T^{16} - 22871205 p^{5} T^{17} + 1641574 p^{6} T^{18} + 448266 p^{7} T^{19} - 4845 p^{8} T^{20} - 3887 p^{9} T^{21} - 48 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 39 T + 822 T^{2} + 11936 T^{3} + 137379 T^{4} + 1304607 T^{5} + 9780148 T^{6} + 42815682 T^{7} - 154130391 T^{8} - 6081344458 T^{9} - 88425769017 T^{10} - 1013814962454 T^{11} - 9774915393497 T^{12} - 1013814962454 p T^{13} - 88425769017 p^{2} T^{14} - 6081344458 p^{3} T^{15} - 154130391 p^{4} T^{16} + 42815682 p^{5} T^{17} + 9780148 p^{6} T^{18} + 1304607 p^{7} T^{19} + 137379 p^{8} T^{20} + 11936 p^{9} T^{21} + 822 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 18 T + 126 T^{2} - 1260 T^{3} - 41166 T^{4} - 525114 T^{5} - 2039866 T^{6} + 36031104 T^{7} + 749106684 T^{8} + 6189995052 T^{9} + 9354005928 T^{10} - 5973189228 p T^{11} - 6862378086069 T^{12} - 5973189228 p^{2} T^{13} + 9354005928 p^{2} T^{14} + 6189995052 p^{3} T^{15} + 749106684 p^{4} T^{16} + 36031104 p^{5} T^{17} - 2039866 p^{6} T^{18} - 525114 p^{7} T^{19} - 41166 p^{8} T^{20} - 1260 p^{9} T^{21} + 126 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 12 T - 120 T^{2} + 3294 T^{3} - 12273 T^{4} - 248142 T^{5} + 2973695 T^{6} - 4697397 T^{7} - 153066366 T^{8} + 1115642430 T^{9} + 2098738827 T^{10} - 25141746861 T^{11} - 47336761416 T^{12} - 25141746861 p T^{13} + 2098738827 p^{2} T^{14} + 1115642430 p^{3} T^{15} - 153066366 p^{4} T^{16} - 4697397 p^{5} T^{17} + 2973695 p^{6} T^{18} - 248142 p^{7} T^{19} - 12273 p^{8} T^{20} + 3294 p^{9} T^{21} - 120 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 39 T + 993 T^{2} - 19028 T^{3} + 304221 T^{4} - 4077399 T^{5} + 47593567 T^{6} - 4909788 p T^{7} + 4125278700 T^{8} - 29493453512 T^{9} + 169040152338 T^{10} - 681702895626 T^{11} + 3451348757128 T^{12} - 681702895626 p T^{13} + 169040152338 p^{2} T^{14} - 29493453512 p^{3} T^{15} + 4125278700 p^{4} T^{16} - 4909788 p^{6} T^{17} + 47593567 p^{6} T^{18} - 4077399 p^{7} T^{19} + 304221 p^{8} T^{20} - 19028 p^{9} T^{21} + 993 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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
−3.94066619675452973401831284416, −3.79145687531844217354721645613, −3.36292797177033876798745477579, −3.30341174872567418858409379222, −3.27619022313481557896254805857, −3.13338502996299308233611571447, −3.06290671158270886078487766402, −3.03437225664081770974242098121, −3.01213077860560511718502847606, −2.89219133658631772058151191254, −2.85999481685813448805564262019, −2.83351768009256821410317735415, −2.05137714833831937419036793744, −2.02712261330053105866968532049, −1.92691697761152914618400295227, −1.90485053444794798586844392883, −1.74342809477071322192724775066, −1.65871973172107526798145154584, −1.52839090323605257612022635418, −1.46905740307732193467926497504, −1.39461673962789376139434420042, −1.26959327938067227957567673431, −0.65312565889262156399565005773, −0.44114123928277800188285263797, −0.12634927150988609280376610089, 0.12634927150988609280376610089, 0.44114123928277800188285263797, 0.65312565889262156399565005773, 1.26959327938067227957567673431, 1.39461673962789376139434420042, 1.46905740307732193467926497504, 1.52839090323605257612022635418, 1.65871973172107526798145154584, 1.74342809477071322192724775066, 1.90485053444794798586844392883, 1.92691697761152914618400295227, 2.02712261330053105866968532049, 2.05137714833831937419036793744, 2.83351768009256821410317735415, 2.85999481685813448805564262019, 2.89219133658631772058151191254, 3.01213077860560511718502847606, 3.03437225664081770974242098121, 3.06290671158270886078487766402, 3.13338502996299308233611571447, 3.27619022313481557896254805857, 3.30341174872567418858409379222, 3.36292797177033876798745477579, 3.79145687531844217354721645613, 3.94066619675452973401831284416
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.