Dirichlet series
| L(s) = 1 | − 3·5-s − 3·7-s − 2·8-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 + 6·40-s − 12·41-s + 9·43-s + 18·45-s − 9·47-s − 15·49-s − 12·53-s + 36·55-s + 6·56-s + 12·59-s − 36·61-s + 18·63-s + 64-s − 36·65-s + 36·67-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 0.707·8-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 + 0.948·40-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 + 0.801·56-s + 1.56·59-s − 4.60·61-s + 2.26·63-s + 1/8·64-s − 4.46·65-s + 4.39·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.13095\times 10^{-5}\) |
| Root analytic conductor: | \(0.656652\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.09293197381\) |
| \(L(\frac12)\) | \(\approx\) | \(0.09293197381\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 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 | \( 1 - 39 T + 993 T^{2} - 19028 T^{3} + 304221 T^{4} - 4077399 T^{5} + 47593567 T^{6} - 4909788 p T^{7} + 4125278700 T^{8} - 29493453512 T^{9} + 169040152338 T^{10} - 681702895626 T^{11} + 3451348757128 T^{12} - 681702895626 p T^{13} + 169040152338 p^{2} T^{14} - 29493453512 p^{3} T^{15} + 4125278700 p^{4} T^{16} - 4909788 p^{6} T^{17} + 47593567 p^{6} T^{18} - 4077399 p^{7} T^{19} + 304221 p^{8} T^{20} - 19028 p^{9} T^{21} + 993 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.95663554087565013889154636311, −5.88484242553135047079676747998, −5.86887439080793451778033588558, −5.34023075222685362304283042385, −5.24154657136006027338953748870, −5.22042601629672777934718492782, −5.21576089903790852035786638133, −4.92801450494612008646801785527, −4.90213228101025682137794310419, −4.86079399754260743684800646591, −4.83378555322537888982443692768, −4.33265229555500093110670406066, −4.07282333162853230174924346812, −3.94215172792586259298413553251, −3.84118534107726967099610747639, −3.63529219061052785628640177139, −3.45391049160437654506309570118, −3.07807768430759731796366311785, −3.04501983622561113941906123750, −2.95074388022841310194526929879, −2.91630507369118712360623845274, −2.83059906697665274024251745673, −2.39363991817582668294811212740, −2.07713370685981269596023571491, −1.47095584505163990617267129332, 1.47095584505163990617267129332, 2.07713370685981269596023571491, 2.39363991817582668294811212740, 2.83059906697665274024251745673, 2.91630507369118712360623845274, 2.95074388022841310194526929879, 3.04501983622561113941906123750, 3.07807768430759731796366311785, 3.45391049160437654506309570118, 3.63529219061052785628640177139, 3.84118534107726967099610747639, 3.94215172792586259298413553251, 4.07282333162853230174924346812, 4.33265229555500093110670406066, 4.83378555322537888982443692768, 4.86079399754260743684800646591, 4.90213228101025682137794310419, 4.92801450494612008646801785527, 5.21576089903790852035786638133, 5.22042601629672777934718492782, 5.24154657136006027338953748870, 5.34023075222685362304283042385, 5.86887439080793451778033588558, 5.88484242553135047079676747998, 5.95663554087565013889154636311
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.