Dirichlet series
| L(s) = 1 | + 3·3-s − 12·5-s + 5·7-s − 9·9-s + 10·11-s + 11·13-s − 36·15-s − 4·17-s + 5·19-s + 15·21-s + 18·23-s + 78·25-s − 36·27-s + 16·29-s − 31-s + 30·33-s − 60·35-s + 2·37-s + 33·39-s + 4·41-s + 7·43-s + 108·45-s − 28·49-s − 12·51-s + 39·53-s − 120·55-s + 15·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 5.36·5-s + 1.88·7-s − 3·9-s + 3.01·11-s + 3.05·13-s − 9.29·15-s − 0.970·17-s + 1.14·19-s + 3.27·21-s + 3.75·23-s + 78/5·25-s − 6.92·27-s + 2.97·29-s − 0.179·31-s + 5.22·33-s − 10.1·35-s + 0.328·37-s + 5.28·39-s + 0.624·41-s + 1.06·43-s + 16.0·45-s − 4·49-s − 1.68·51-s + 5.35·53-s − 16.1·55-s + 1.98·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12} \cdot 151^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12} \cdot 151^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 5^{12} \cdot 151^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.58404\times 10^{20}\) |
| Root analytic conductor: | \(6.94475\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 5^{12} \cdot 151^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(75.94034083\) |
| \(L(\frac12)\) | \(\approx\) | \(75.94034083\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T )^{12} \) | |
| 151 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 - p T + 2 p^{2} T^{2} - 5 p^{2} T^{3} + 164 T^{4} - 362 T^{5} + 344 p T^{6} - 77 p^{3} T^{7} + 1673 p T^{8} - 3109 p T^{9} + 6607 p T^{10} - 33914 T^{11} + 65018 T^{12} - 33914 p T^{13} + 6607 p^{3} T^{14} - 3109 p^{4} T^{15} + 1673 p^{5} T^{16} - 77 p^{8} T^{17} + 344 p^{7} T^{18} - 362 p^{7} T^{19} + 164 p^{8} T^{20} - 5 p^{11} T^{21} + 2 p^{12} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 5 T + 53 T^{2} - 237 T^{3} + 1423 T^{4} - 5728 T^{5} + 25292 T^{6} - 92019 T^{7} + 329208 T^{8} - 1082072 T^{9} + 3293391 T^{10} - 9729941 T^{11} + 25909424 T^{12} - 9729941 p T^{13} + 3293391 p^{2} T^{14} - 1082072 p^{3} T^{15} + 329208 p^{4} T^{16} - 92019 p^{5} T^{17} + 25292 p^{6} T^{18} - 5728 p^{7} T^{19} + 1423 p^{8} T^{20} - 237 p^{9} T^{21} + 53 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 10 T + 141 T^{2} - 1004 T^{3} + 8314 T^{4} - 46853 T^{5} + 26276 p T^{6} - 123268 p T^{7} + 6788446 T^{8} - 27225939 T^{9} + 10471133 p T^{10} - 400077789 T^{11} + 1459053450 T^{12} - 400077789 p T^{13} + 10471133 p^{3} T^{14} - 27225939 p^{3} T^{15} + 6788446 p^{4} T^{16} - 123268 p^{6} T^{17} + 26276 p^{7} T^{18} - 46853 p^{7} T^{19} + 8314 p^{8} T^{20} - 1004 p^{9} T^{21} + 141 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 11 T + 155 T^{2} - 1205 T^{3} + 10213 T^{4} - 62872 T^{5} + 31035 p T^{6} - 2074409 T^{7} + 840491 p T^{8} - 48212982 T^{9} + 216272033 T^{10} - 829548582 T^{11} + 3224799856 T^{12} - 829548582 p T^{13} + 216272033 p^{2} T^{14} - 48212982 p^{3} T^{15} + 840491 p^{5} T^{16} - 2074409 p^{5} T^{17} + 31035 p^{7} T^{18} - 62872 p^{7} T^{19} + 10213 p^{8} T^{20} - 1205 p^{9} T^{21} + 155 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T + 118 T^{2} + 31 p T^{3} + 7449 T^{4} + 32803 T^{5} + 320129 T^{6} + 1321661 T^{7} + 10143121 T^{8} + 38481525 T^{9} + 247165225 T^{10} + 847543192 T^{11} + 4728920234 T^{12} + 847543192 p T^{13} + 247165225 p^{2} T^{14} + 38481525 p^{3} T^{15} + 10143121 p^{4} T^{16} + 1321661 p^{5} T^{17} + 320129 p^{6} T^{18} + 32803 p^{7} T^{19} + 7449 p^{8} T^{20} + 31 p^{10} T^{21} + 118 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 5 T + 9 p T^{2} - 851 T^{3} + 14393 T^{4} - 68203 T^{5} + 778818 T^{6} - 3416354 T^{7} + 29913432 T^{8} - 119049473 T^{9} + 854481504 T^{10} - 3032211964 T^{11} + 18557844314 T^{12} - 3032211964 p T^{13} + 854481504 p^{2} T^{14} - 119049473 p^{3} T^{15} + 29913432 p^{4} T^{16} - 3416354 p^{5} T^{17} + 778818 p^{6} T^{18} - 68203 p^{7} T^{19} + 14393 p^{8} T^{20} - 851 p^{9} T^{21} + 9 p^{11} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 18 T + 367 T^{2} - 4553 T^{3} + 54891 T^{4} - 522034 T^{5} + 4669378 T^{6} - 35885385 T^{7} + 257511339 T^{8} - 1645483248 T^{9} + 426523561 p T^{10} - 52889965641 T^{11} + 266146274438 T^{12} - 52889965641 p T^{13} + 426523561 p^{3} T^{14} - 1645483248 p^{3} T^{15} + 257511339 p^{4} T^{16} - 35885385 p^{5} T^{17} + 4669378 p^{6} T^{18} - 522034 p^{7} T^{19} + 54891 p^{8} T^{20} - 4553 p^{9} T^{21} + 367 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 16 T + 332 T^{2} - 4127 T^{3} + 50410 T^{4} - 505365 T^{5} + 4664711 T^{6} - 38760038 T^{7} + 294317772 T^{8} - 2067630380 T^{9} + 13372644504 T^{10} - 80514659399 T^{11} + 449227882818 T^{12} - 80514659399 p T^{13} + 13372644504 p^{2} T^{14} - 2067630380 p^{3} T^{15} + 294317772 p^{4} T^{16} - 38760038 p^{5} T^{17} + 4664711 p^{6} T^{18} - 505365 p^{7} T^{19} + 50410 p^{8} T^{20} - 4127 p^{9} T^{21} + 332 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + T + 240 T^{2} + 34 T^{3} + 28362 T^{4} - 12186 T^{5} + 71186 p T^{6} - 1678769 T^{7} + 126135114 T^{8} - 113523695 T^{9} + 5564527804 T^{10} - 5007076406 T^{11} + 193626966370 T^{12} - 5007076406 p T^{13} + 5564527804 p^{2} T^{14} - 113523695 p^{3} T^{15} + 126135114 p^{4} T^{16} - 1678769 p^{5} T^{17} + 71186 p^{7} T^{18} - 12186 p^{7} T^{19} + 28362 p^{8} T^{20} + 34 p^{9} T^{21} + 240 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 2 T + 255 T^{2} - 590 T^{3} + 32982 T^{4} - 86555 T^{5} + 2859105 T^{6} - 8108526 T^{7} + 184958463 T^{8} - 535864322 T^{9} + 9410474584 T^{10} - 26222253969 T^{11} + 386206617076 T^{12} - 26222253969 p T^{13} + 9410474584 p^{2} T^{14} - 535864322 p^{3} T^{15} + 184958463 p^{4} T^{16} - 8108526 p^{5} T^{17} + 2859105 p^{6} T^{18} - 86555 p^{7} T^{19} + 32982 p^{8} T^{20} - 590 p^{9} T^{21} + 255 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 4 T + 267 T^{2} - 700 T^{3} + 33375 T^{4} - 56845 T^{5} + 2758533 T^{6} - 3467588 T^{7} + 176379019 T^{8} - 202640590 T^{9} + 9313440944 T^{10} - 10497664985 T^{11} + 414700967562 T^{12} - 10497664985 p T^{13} + 9313440944 p^{2} T^{14} - 202640590 p^{3} T^{15} + 176379019 p^{4} T^{16} - 3467588 p^{5} T^{17} + 2758533 p^{6} T^{18} - 56845 p^{7} T^{19} + 33375 p^{8} T^{20} - 700 p^{9} T^{21} + 267 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 7 T + 367 T^{2} - 2305 T^{3} + 62018 T^{4} - 351425 T^{5} + 6465474 T^{6} - 33306226 T^{7} + 474553429 T^{8} - 2245965849 T^{9} + 26849890287 T^{10} - 117795194122 T^{11} + 1253499992400 T^{12} - 117795194122 p T^{13} + 26849890287 p^{2} T^{14} - 2245965849 p^{3} T^{15} + 474553429 p^{4} T^{16} - 33306226 p^{5} T^{17} + 6465474 p^{6} T^{18} - 351425 p^{7} T^{19} + 62018 p^{8} T^{20} - 2305 p^{9} T^{21} + 367 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 373 T^{2} - 54 T^{3} + 68901 T^{4} - 17235 T^{5} + 8327671 T^{6} - 2584696 T^{7} + 732655165 T^{8} - 242912206 T^{9} + 49368235556 T^{10} - 15844207299 T^{11} + 2610053872218 T^{12} - 15844207299 p T^{13} + 49368235556 p^{2} T^{14} - 242912206 p^{3} T^{15} + 732655165 p^{4} T^{16} - 2584696 p^{5} T^{17} + 8327671 p^{6} T^{18} - 17235 p^{7} T^{19} + 68901 p^{8} T^{20} - 54 p^{9} T^{21} + 373 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( 1 - 39 T + 1073 T^{2} - 21235 T^{3} + 352653 T^{4} - 4948457 T^{5} + 61879929 T^{6} - 691394072 T^{7} + 7085505731 T^{8} - 66533059143 T^{9} + 580845739822 T^{10} - 4696033701050 T^{11} + 35475983531998 T^{12} - 4696033701050 p T^{13} + 580845739822 p^{2} T^{14} - 66533059143 p^{3} T^{15} + 7085505731 p^{4} T^{16} - 691394072 p^{5} T^{17} + 61879929 p^{6} T^{18} - 4948457 p^{7} T^{19} + 352653 p^{8} T^{20} - 21235 p^{9} T^{21} + 1073 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 4 T + 350 T^{2} + 1535 T^{3} + 57100 T^{4} + 225195 T^{5} + 5493801 T^{6} + 15560780 T^{7} + 319966674 T^{8} + 270969040 T^{9} + 10950635520 T^{10} - 35171673695 T^{11} + 353469201160 T^{12} - 35171673695 p T^{13} + 10950635520 p^{2} T^{14} + 270969040 p^{3} T^{15} + 319966674 p^{4} T^{16} + 15560780 p^{5} T^{17} + 5493801 p^{6} T^{18} + 225195 p^{7} T^{19} + 57100 p^{8} T^{20} + 1535 p^{9} T^{21} + 350 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 32 T + 995 T^{2} + 20277 T^{3} + 380678 T^{4} + 5818630 T^{5} + 81958011 T^{6} + 1008758892 T^{7} + 11507197045 T^{8} + 118259200130 T^{9} + 1131558630826 T^{10} + 9889032493271 T^{11} + 80655507146600 T^{12} + 9889032493271 p T^{13} + 1131558630826 p^{2} T^{14} + 118259200130 p^{3} T^{15} + 11507197045 p^{4} T^{16} + 1008758892 p^{5} T^{17} + 81958011 p^{6} T^{18} + 5818630 p^{7} T^{19} + 380678 p^{8} T^{20} + 20277 p^{9} T^{21} + 995 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 4 T + 495 T^{2} - 2334 T^{3} + 120862 T^{4} - 613667 T^{5} + 19567446 T^{6} - 100204094 T^{7} + 2349488220 T^{8} - 11605195795 T^{9} + 219970063029 T^{10} - 1011683317879 T^{11} + 16437292090748 T^{12} - 1011683317879 p T^{13} + 219970063029 p^{2} T^{14} - 11605195795 p^{3} T^{15} + 2349488220 p^{4} T^{16} - 100204094 p^{5} T^{17} + 19567446 p^{6} T^{18} - 613667 p^{7} T^{19} + 120862 p^{8} T^{20} - 2334 p^{9} T^{21} + 495 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 24 T + 573 T^{2} - 8229 T^{3} + 122860 T^{4} - 1394684 T^{5} + 16813349 T^{6} - 166412378 T^{7} + 1764351545 T^{8} - 15998237484 T^{9} + 155570574382 T^{10} - 1313986932737 T^{11} + 11848956998900 T^{12} - 1313986932737 p T^{13} + 155570574382 p^{2} T^{14} - 15998237484 p^{3} T^{15} + 1764351545 p^{4} T^{16} - 166412378 p^{5} T^{17} + 16813349 p^{6} T^{18} - 1394684 p^{7} T^{19} + 122860 p^{8} T^{20} - 8229 p^{9} T^{21} + 573 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 10 T + 644 T^{2} + 5195 T^{3} + 192240 T^{4} + 1274477 T^{5} + 35994559 T^{6} + 199038168 T^{7} + 4807466626 T^{8} + 22589788876 T^{9} + 6733100160 p T^{10} + 2014694402495 T^{11} + 39977243151166 T^{12} + 2014694402495 p T^{13} + 6733100160 p^{3} T^{14} + 22589788876 p^{3} T^{15} + 4807466626 p^{4} T^{16} + 199038168 p^{5} T^{17} + 35994559 p^{6} T^{18} + 1274477 p^{7} T^{19} + 192240 p^{8} T^{20} + 5195 p^{9} T^{21} + 644 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 32 T + 1208 T^{2} - 26731 T^{3} + 598942 T^{4} - 10225332 T^{5} + 170346951 T^{6} - 2364896674 T^{7} + 31669466693 T^{8} - 367543872768 T^{9} + 4098646312177 T^{10} - 40314255770695 T^{11} + 380553530514200 T^{12} - 40314255770695 p T^{13} + 4098646312177 p^{2} T^{14} - 367543872768 p^{3} T^{15} + 31669466693 p^{4} T^{16} - 2364896674 p^{5} T^{17} + 170346951 p^{6} T^{18} - 10225332 p^{7} T^{19} + 598942 p^{8} T^{20} - 26731 p^{9} T^{21} + 1208 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 9 T + 767 T^{2} - 6769 T^{3} + 286605 T^{4} - 2407178 T^{5} + 68318501 T^{6} - 534452349 T^{7} + 11506512135 T^{8} - 82238330606 T^{9} + 1437899185287 T^{10} - 9202188223588 T^{11} + 136466743467514 T^{12} - 9202188223588 p T^{13} + 1437899185287 p^{2} T^{14} - 82238330606 p^{3} T^{15} + 11506512135 p^{4} T^{16} - 534452349 p^{5} T^{17} + 68318501 p^{6} T^{18} - 2407178 p^{7} T^{19} + 286605 p^{8} T^{20} - 6769 p^{9} T^{21} + 767 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 15 T + 483 T^{2} - 5211 T^{3} + 96324 T^{4} - 826853 T^{5} + 11988900 T^{6} - 93547398 T^{7} + 1258290761 T^{8} - 10540139229 T^{9} + 134590686313 T^{10} - 1182018629254 T^{11} + 13237557681668 T^{12} - 1182018629254 p T^{13} + 134590686313 p^{2} T^{14} - 10540139229 p^{3} T^{15} + 1258290761 p^{4} T^{16} - 93547398 p^{5} T^{17} + 11988900 p^{6} T^{18} - 826853 p^{7} T^{19} + 96324 p^{8} T^{20} - 5211 p^{9} T^{21} + 483 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 15 T + 647 T^{2} - 10235 T^{3} + 217588 T^{4} - 3311979 T^{5} + 50610296 T^{6} - 690685452 T^{7} + 8896369343 T^{8} - 105921681527 T^{9} + 1219816901553 T^{10} - 12751757948264 T^{11} + 132534100467000 T^{12} - 12751757948264 p T^{13} + 1219816901553 p^{2} T^{14} - 105921681527 p^{3} T^{15} + 8896369343 p^{4} T^{16} - 690685452 p^{5} T^{17} + 50610296 p^{6} T^{18} - 3311979 p^{7} T^{19} + 217588 p^{8} T^{20} - 10235 p^{9} T^{21} + 647 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.65811639143159781183622291006, −2.32642708113093490504242419636, −2.10148862225871136020207191352, −2.09855917572735404326542290917, −1.92552010021362747427769266248, −1.90961036462814447028481754232, −1.90820976874329834887466705948, −1.89119088535350945858078415221, −1.76578745653720922152945951517, −1.73704891052502718138443343572, −1.55660227918472178489867373125, −1.54621353299056100598728099585, −1.28480568468355410586057962471, −1.14040302413683845109450887582, −1.13887767084633035989377945430, −0.884811446252608357616756605657, −0.825225300189948389669420287825, −0.824022758245089279489926625257, −0.77168841468397442370864762999, −0.71610130604449663879847470545, −0.69426452487163097349480740355, −0.64703771349796756909075319573, −0.45063224972327728119954091462, −0.26259260167977497345298837692, −0.22748610789417452379423566081, 0.22748610789417452379423566081, 0.26259260167977497345298837692, 0.45063224972327728119954091462, 0.64703771349796756909075319573, 0.69426452487163097349480740355, 0.71610130604449663879847470545, 0.77168841468397442370864762999, 0.824022758245089279489926625257, 0.825225300189948389669420287825, 0.884811446252608357616756605657, 1.13887767084633035989377945430, 1.14040302413683845109450887582, 1.28480568468355410586057962471, 1.54621353299056100598728099585, 1.55660227918472178489867373125, 1.73704891052502718138443343572, 1.76578745653720922152945951517, 1.89119088535350945858078415221, 1.90820976874329834887466705948, 1.90961036462814447028481754232, 1.92552010021362747427769266248, 2.09855917572735404326542290917, 2.10148862225871136020207191352, 2.32642708113093490504242419636, 2.65811639143159781183622291006
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.