Dirichlet series
| L(s) = 1 | − 6·5-s + 2·7-s + 12·11-s − 6·19-s − 12·23-s + 15·25-s − 6·31-s − 12·35-s − 2·37-s − 8·41-s + 36·43-s − 16·47-s + 13·49-s + 12·53-s − 72·55-s + 2·67-s + 6·73-s + 24·77-s + 14·79-s + 40·83-s − 20·89-s + 36·95-s − 16·101-s + 6·103-s − 84·107-s − 2·109-s + 72·115-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 0.755·7-s + 3.61·11-s − 1.37·19-s − 2.50·23-s + 3·25-s − 1.07·31-s − 2.02·35-s − 0.328·37-s − 1.24·41-s + 5.48·43-s − 2.33·47-s + 13/7·49-s + 1.64·53-s − 9.70·55-s + 0.244·67-s + 0.702·73-s + 2.73·77-s + 1.57·79-s + 4.39·83-s − 2.11·89-s + 3.69·95-s − 1.59·101-s + 0.591·103-s − 8.12·107-s − 0.191·109-s + 6.71·115-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.07589\times 10^{12}\) |
| Root analytic conductor: | \(3.17193\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03786588938\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03786588938\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( ( 1 + T + T^{2} )^{6} \) | |
| 7 | \( 1 - 2 T - 9 T^{2} + 58 T^{3} - 78 T^{4} - 298 T^{5} + 1341 T^{6} - 298 p T^{7} - 78 p^{2} T^{8} + 58 p^{3} T^{9} - 9 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 11 | \( 1 - 12 T + 104 T^{2} - 672 T^{3} + 3665 T^{4} - 1548 p T^{5} + 71556 T^{6} - 279468 T^{7} + 1050010 T^{8} - 3896592 T^{9} + 14213614 T^{10} - 50673252 T^{11} + 171753277 T^{12} - 50673252 p T^{13} + 14213614 p^{2} T^{14} - 3896592 p^{3} T^{15} + 1050010 p^{4} T^{16} - 279468 p^{5} T^{17} + 71556 p^{6} T^{18} - 1548 p^{8} T^{19} + 3665 p^{8} T^{20} - 672 p^{9} T^{21} + 104 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) |
| 13 | \( 1 - 54 T^{2} + 113 p T^{4} - 27038 T^{6} + 379302 T^{8} - 4418150 T^{10} + 52195829 T^{12} - 4418150 p^{2} T^{14} + 379302 p^{4} T^{16} - 27038 p^{6} T^{18} + 113 p^{9} T^{20} - 54 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 50 T^{2} + 216 T^{3} + 1439 T^{4} - 10536 T^{5} - 8286 T^{6} + 311700 T^{7} - 563096 T^{8} - 5178888 T^{9} + 25876100 T^{10} + 40629132 T^{11} - 545782199 T^{12} + 40629132 p T^{13} + 25876100 p^{2} T^{14} - 5178888 p^{3} T^{15} - 563096 p^{4} T^{16} + 311700 p^{5} T^{17} - 8286 p^{6} T^{18} - 10536 p^{7} T^{19} + 1439 p^{8} T^{20} + 216 p^{9} T^{21} - 50 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 6 T + 51 T^{2} + 234 T^{3} + 887 T^{4} + 1680 T^{5} - 5660 T^{6} - 39600 T^{7} - 182487 T^{8} + 834 p^{2} T^{9} + 6939289 T^{10} + 52759566 T^{11} + 263892398 T^{12} + 52759566 p T^{13} + 6939289 p^{2} T^{14} + 834 p^{5} T^{15} - 182487 p^{4} T^{16} - 39600 p^{5} T^{17} - 5660 p^{6} T^{18} + 1680 p^{7} T^{19} + 887 p^{8} T^{20} + 234 p^{9} T^{21} + 51 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 12 T + 118 T^{2} + 840 T^{3} + 4771 T^{4} + 18960 T^{5} + 49586 T^{6} - 42168 T^{7} - 1397324 T^{8} - 9002076 T^{9} - 43390364 T^{10} - 165215580 T^{11} - 752536735 T^{12} - 165215580 p T^{13} - 43390364 p^{2} T^{14} - 9002076 p^{3} T^{15} - 1397324 p^{4} T^{16} - 42168 p^{5} T^{17} + 49586 p^{6} T^{18} + 18960 p^{7} T^{19} + 4771 p^{8} T^{20} + 840 p^{9} T^{21} + 118 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 200 T^{2} + 19654 T^{4} - 1282780 T^{6} + 62801839 T^{8} - 2446961612 T^{10} + 78075542648 T^{12} - 2446961612 p^{2} T^{14} + 62801839 p^{4} T^{16} - 1282780 p^{6} T^{18} + 19654 p^{8} T^{20} - 200 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 + 6 T + 99 T^{2} + 522 T^{3} + 5219 T^{4} + 25548 T^{5} + 206824 T^{6} + 1036272 T^{7} + 7091877 T^{8} + 36588498 T^{9} + 224451205 T^{10} + 1182465834 T^{11} + 6974158670 T^{12} + 1182465834 p T^{13} + 224451205 p^{2} T^{14} + 36588498 p^{3} T^{15} + 7091877 p^{4} T^{16} + 1036272 p^{5} T^{17} + 206824 p^{6} T^{18} + 25548 p^{7} T^{19} + 5219 p^{8} T^{20} + 522 p^{9} T^{21} + 99 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 2 T - 113 T^{2} - 10 T^{3} + 6533 T^{4} - 10556 T^{5} - 217390 T^{6} + 870184 T^{7} + 3167 p^{2} T^{8} - 32769190 T^{9} - 33804041 T^{10} + 517787714 T^{11} - 292082470 T^{12} + 517787714 p T^{13} - 33804041 p^{2} T^{14} - 32769190 p^{3} T^{15} + 3167 p^{6} T^{16} + 870184 p^{5} T^{17} - 217390 p^{6} T^{18} - 10556 p^{7} T^{19} + 6533 p^{8} T^{20} - 10 p^{9} T^{21} - 113 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + 4 T + 200 T^{2} + 744 T^{3} + 18359 T^{4} + 57292 T^{5} + 970430 T^{6} + 57292 p T^{7} + 18359 p^{2} T^{8} + 744 p^{3} T^{9} + 200 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 18 T + 233 T^{2} - 1918 T^{3} + 15792 T^{4} - 108898 T^{5} + 797957 T^{6} - 108898 p T^{7} + 15792 p^{2} T^{8} - 1918 p^{3} T^{9} + 233 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 16 T + 14 T^{2} - 256 T^{3} + 7637 T^{4} + 35904 T^{5} - 148150 T^{6} + 2568624 T^{7} + 17289050 T^{8} - 49123696 T^{9} + 898246854 T^{10} + 6861278848 T^{11} - 8007122355 T^{12} + 6861278848 p T^{13} + 898246854 p^{2} T^{14} - 49123696 p^{3} T^{15} + 17289050 p^{4} T^{16} + 2568624 p^{5} T^{17} - 148150 p^{6} T^{18} + 35904 p^{7} T^{19} + 7637 p^{8} T^{20} - 256 p^{9} T^{21} + 14 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 12 T + 306 T^{2} - 3096 T^{3} + 46301 T^{4} - 403440 T^{5} + 4780934 T^{6} - 37330380 T^{7} + 387155802 T^{8} - 2784330036 T^{9} + 26090681306 T^{10} - 173997632736 T^{11} + 1495008560357 T^{12} - 173997632736 p T^{13} + 26090681306 p^{2} T^{14} - 2784330036 p^{3} T^{15} + 387155802 p^{4} T^{16} - 37330380 p^{5} T^{17} + 4780934 p^{6} T^{18} - 403440 p^{7} T^{19} + 46301 p^{8} T^{20} - 3096 p^{9} T^{21} + 306 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 188 T^{2} - 456 T^{3} + 19205 T^{4} + 77412 T^{5} - 974760 T^{6} - 7272024 T^{7} + 7716046 T^{8} + 389439300 T^{9} + 3375770258 T^{10} - 9505286676 T^{11} - 292267488731 T^{12} - 9505286676 p T^{13} + 3375770258 p^{2} T^{14} + 389439300 p^{3} T^{15} + 7716046 p^{4} T^{16} - 7272024 p^{5} T^{17} - 974760 p^{6} T^{18} + 77412 p^{7} T^{19} + 19205 p^{8} T^{20} - 456 p^{9} T^{21} - 188 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 186 T^{2} + 245 p T^{4} + 62820 T^{5} + 788542 T^{6} + 9313320 T^{7} + 46135038 T^{8} + 497801700 T^{9} + 5359273486 T^{10} + 14137675260 T^{11} + 461910843149 T^{12} + 14137675260 p T^{13} + 5359273486 p^{2} T^{14} + 497801700 p^{3} T^{15} + 46135038 p^{4} T^{16} + 9313320 p^{5} T^{17} + 788542 p^{6} T^{18} + 62820 p^{7} T^{19} + 245 p^{9} T^{20} + 186 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 - 2 T - 219 T^{2} + 414 T^{3} + 21547 T^{4} - 34460 T^{5} - 1616364 T^{6} + 2002464 T^{7} + 126910963 T^{8} - 122742450 T^{9} - 9073876249 T^{10} + 3928073010 T^{11} + 592194338278 T^{12} + 3928073010 p T^{13} - 9073876249 p^{2} T^{14} - 122742450 p^{3} T^{15} + 126910963 p^{4} T^{16} + 2002464 p^{5} T^{17} - 1616364 p^{6} T^{18} - 34460 p^{7} T^{19} + 21547 p^{8} T^{20} + 414 p^{9} T^{21} - 219 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 396 T^{2} + 82922 T^{4} - 11857216 T^{6} + 1291830615 T^{8} - 115038320068 T^{10} + 8757474588608 T^{12} - 115038320068 p^{2} T^{14} + 1291830615 p^{4} T^{16} - 11857216 p^{6} T^{18} + 82922 p^{8} T^{20} - 396 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 6 T + 407 T^{2} - 2370 T^{3} + 92065 T^{4} - 493560 T^{5} + 14539942 T^{6} - 71788392 T^{7} + 1770614587 T^{8} - 8056540398 T^{9} + 173420485499 T^{10} - 724110186570 T^{11} + 13947527013530 T^{12} - 724110186570 p T^{13} + 173420485499 p^{2} T^{14} - 8056540398 p^{3} T^{15} + 1770614587 p^{4} T^{16} - 71788392 p^{5} T^{17} + 14539942 p^{6} T^{18} - 493560 p^{7} T^{19} + 92065 p^{8} T^{20} - 2370 p^{9} T^{21} + 407 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 14 T - 65 T^{2} + 2974 T^{3} - 18589 T^{4} - 123232 T^{5} + 2904128 T^{6} - 17165848 T^{7} - 78984031 T^{8} + 2216343070 T^{9} - 11703769439 T^{10} - 74865612950 T^{11} + 1535021723078 T^{12} - 74865612950 p T^{13} - 11703769439 p^{2} T^{14} + 2216343070 p^{3} T^{15} - 78984031 p^{4} T^{16} - 17165848 p^{5} T^{17} + 2904128 p^{6} T^{18} - 123232 p^{7} T^{19} - 18589 p^{8} T^{20} + 2974 p^{9} T^{21} - 65 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 20 T + 356 T^{2} - 3156 T^{3} + 24185 T^{4} - 42500 T^{5} + 239906 T^{6} - 42500 p T^{7} + 24185 p^{2} T^{8} - 3156 p^{3} T^{9} + 356 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 20 T + 62 T^{2} - 3032 T^{3} - 49315 T^{4} - 263916 T^{5} + 1563782 T^{6} + 45627972 T^{7} + 492659654 T^{8} + 3023500120 T^{9} - 3867656028 T^{10} - 364952721772 T^{11} - 4792682445183 T^{12} - 364952721772 p T^{13} - 3867656028 p^{2} T^{14} + 3023500120 p^{3} T^{15} + 492659654 p^{4} T^{16} + 45627972 p^{5} T^{17} + 1563782 p^{6} T^{18} - 263916 p^{7} T^{19} - 49315 p^{8} T^{20} - 3032 p^{9} T^{21} + 62 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 1072 T^{2} + 535030 T^{4} - 164321096 T^{6} + 34577082223 T^{8} - 5250510233512 T^{10} + 589883979492356 T^{12} - 5250510233512 p^{2} T^{14} + 34577082223 p^{4} T^{16} - 164321096 p^{6} T^{18} + 535030 p^{8} T^{20} - 1072 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−3.18903067517040979080435561602, −3.04948731888906764376383330063, −2.90414780137521829571374971136, −2.68342521692346637306424478322, −2.57996370486645748564964628402, −2.56768591896354278491093101912, −2.41287169003591201594366650550, −2.40757077019680392370627665828, −2.34247982048468271756613955839, −2.24740338315503936972624849098, −2.03598900570545648113661957453, −1.88207776826414845843588244673, −1.83194800821829077004963967053, −1.82656247142143032583366019740, −1.72836571568253578051009220480, −1.37126956689796223501590836183, −1.26387685640548564883786819300, −1.24077643666780852839204589114, −1.19929742259199123155535753915, −0.998119258182875233386444652113, −0.858505676355158165341798185869, −0.77308569509717567737530296493, −0.54447775055199667967733542902, −0.07169123703916983149821033226, −0.06937353519417908345976905852, 0.06937353519417908345976905852, 0.07169123703916983149821033226, 0.54447775055199667967733542902, 0.77308569509717567737530296493, 0.858505676355158165341798185869, 0.998119258182875233386444652113, 1.19929742259199123155535753915, 1.24077643666780852839204589114, 1.26387685640548564883786819300, 1.37126956689796223501590836183, 1.72836571568253578051009220480, 1.82656247142143032583366019740, 1.83194800821829077004963967053, 1.88207776826414845843588244673, 2.03598900570545648113661957453, 2.24740338315503936972624849098, 2.34247982048468271756613955839, 2.40757077019680392370627665828, 2.41287169003591201594366650550, 2.56768591896354278491093101912, 2.57996370486645748564964628402, 2.68342521692346637306424478322, 2.90414780137521829571374971136, 3.04948731888906764376383330063, 3.18903067517040979080435561602
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.