Dirichlet series
| L(s) = 1 | − 3·2-s − 3·3-s + 3·4-s − 12·5-s + 9·6-s − 7-s − 8-s + 3·9-s + 36·10-s − 5·11-s − 9·12-s − 3·13-s + 3·14-s + 36·15-s − 3·17-s − 9·18-s − 4·19-s − 36·20-s + 3·21-s + 15·22-s + 10·23-s + 3·24-s + 78·25-s + 9·26-s − 27-s − 3·28-s − 5·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3/2·4-s − 5.36·5-s + 3.67·6-s − 0.377·7-s − 0.353·8-s + 9-s + 11.3·10-s − 1.50·11-s − 2.59·12-s − 0.832·13-s + 0.801·14-s + 9.29·15-s − 0.727·17-s − 2.12·18-s − 0.917·19-s − 8.04·20-s + 0.654·21-s + 3.19·22-s + 2.08·23-s + 0.612·24-s + 78/5·25-s + 1.76·26-s − 0.192·27-s − 0.566·28-s − 0.928·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81268\times 10^{10}\) |
| Root analytic conductor: | \(2.72508\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 5^{12} \cdot 31^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.02147157450\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02147157450\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) |
| 3 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \) | |
| 5 | \( ( 1 + T )^{12} \) | |
| 31 | \( 1 + 8 T + 96 T^{2} + 605 T^{3} + 5220 T^{4} + 29008 T^{5} + 209059 T^{6} + 29008 p T^{7} + 5220 p^{2} T^{8} + 605 p^{3} T^{9} + 96 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 7 | \( 1 + T - 12 T^{2} - 23 T^{3} + 19 T^{4} + 51 T^{5} + 571 T^{6} + 788 T^{7} - 424 p T^{8} - 3991 T^{9} + 6943 T^{10} - 8562 T^{11} - 7271 T^{12} - 8562 p T^{13} + 6943 p^{2} T^{14} - 3991 p^{3} T^{15} - 424 p^{5} T^{16} + 788 p^{5} T^{17} + 571 p^{6} T^{18} + 51 p^{7} T^{19} + 19 p^{8} T^{20} - 23 p^{9} T^{21} - 12 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 + 5 T - 13 T^{2} - 65 T^{3} + 221 T^{4} + 670 T^{5} - 1135 T^{6} + 830 T^{7} - 5260 T^{8} - 87530 T^{9} + 141127 T^{10} + 254840 T^{11} - 3722901 T^{12} + 254840 p T^{13} + 141127 p^{2} T^{14} - 87530 p^{3} T^{15} - 5260 p^{4} T^{16} + 830 p^{5} T^{17} - 1135 p^{6} T^{18} + 670 p^{7} T^{19} + 221 p^{8} T^{20} - 65 p^{9} T^{21} - 13 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 3 T + 12 T^{2} + 29 T^{3} + 339 T^{4} + 643 T^{5} + 2999 T^{6} + 9326 T^{7} + 92822 T^{8} + 191297 T^{9} + 635257 T^{10} + 2067296 T^{11} + 16266169 T^{12} + 2067296 p T^{13} + 635257 p^{2} T^{14} + 191297 p^{3} T^{15} + 92822 p^{4} T^{16} + 9326 p^{5} T^{17} + 2999 p^{6} T^{18} + 643 p^{7} T^{19} + 339 p^{8} T^{20} + 29 p^{9} T^{21} + 12 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 3 T - 2 p T^{3} + 449 T^{4} + 78 T^{5} - 6131 T^{6} - 21900 T^{7} + 13374 p T^{8} + 46182 T^{9} - 1466585 T^{10} - 457765 T^{11} + 89814625 T^{12} - 457765 p T^{13} - 1466585 p^{2} T^{14} + 46182 p^{3} T^{15} + 13374 p^{5} T^{16} - 21900 p^{5} T^{17} - 6131 p^{6} T^{18} + 78 p^{7} T^{19} + 449 p^{8} T^{20} - 2 p^{10} T^{21} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 4 T + 2 T^{2} + 66 T^{3} + 736 T^{4} - 556 T^{5} - 7539 T^{6} + 17398 T^{7} + 37464 T^{8} - 864566 T^{9} + 941232 T^{10} + 5565688 T^{11} - 4655341 T^{12} + 5565688 p T^{13} + 941232 p^{2} T^{14} - 864566 p^{3} T^{15} + 37464 p^{4} T^{16} + 17398 p^{5} T^{17} - 7539 p^{6} T^{18} - 556 p^{7} T^{19} + 736 p^{8} T^{20} + 66 p^{9} T^{21} + 2 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 10 T - 64 T^{2} + 965 T^{3} + 749 T^{4} - 36830 T^{5} + 30470 T^{6} + 552110 T^{7} + 642435 T^{8} + 108260 p T^{9} - 123604999 T^{10} - 107442165 T^{11} + 4381265071 T^{12} - 107442165 p T^{13} - 123604999 p^{2} T^{14} + 108260 p^{4} T^{15} + 642435 p^{4} T^{16} + 552110 p^{5} T^{17} + 30470 p^{6} T^{18} - 36830 p^{7} T^{19} + 749 p^{8} T^{20} + 965 p^{9} T^{21} - 64 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 5 T - 7 T^{2} - 220 T^{3} - 1279 T^{4} + 1660 T^{5} + 29285 T^{6} + 271760 T^{7} + 194790 T^{8} - 4975940 T^{9} - 17888857 T^{10} + 12093675 T^{11} + 1104292939 T^{12} + 12093675 p T^{13} - 17888857 p^{2} T^{14} - 4975940 p^{3} T^{15} + 194790 p^{4} T^{16} + 271760 p^{5} T^{17} + 29285 p^{6} T^{18} + 1660 p^{7} T^{19} - 1279 p^{8} T^{20} - 220 p^{9} T^{21} - 7 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - T + 162 T^{2} - 150 T^{3} + 12815 T^{4} - 10106 T^{5} + 598831 T^{6} - 10106 p T^{7} + 12815 p^{2} T^{8} - 150 p^{3} T^{9} + 162 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 13 T - 62 T^{2} + 2017 T^{3} - 7049 T^{4} - 116133 T^{5} + 1043089 T^{6} + 1487366 T^{7} - 52534676 T^{8} + 134754597 T^{9} + 1063831863 T^{10} - 4215959524 T^{11} - 9593707671 T^{12} - 4215959524 p T^{13} + 1063831863 p^{2} T^{14} + 134754597 p^{3} T^{15} - 52534676 p^{4} T^{16} + 1487366 p^{5} T^{17} + 1043089 p^{6} T^{18} - 116133 p^{7} T^{19} - 7049 p^{8} T^{20} + 2017 p^{9} T^{21} - 62 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 16 T - 5 T^{2} + 1542 T^{3} - 8046 T^{4} - 45546 T^{5} + 580106 T^{6} - 547310 T^{7} - 21708607 T^{8} + 109977131 T^{9} + 259177200 T^{10} - 3059676195 T^{11} + 10168795670 T^{12} - 3059676195 p T^{13} + 259177200 p^{2} T^{14} + 109977131 p^{3} T^{15} - 21708607 p^{4} T^{16} - 547310 p^{5} T^{17} + 580106 p^{6} T^{18} - 45546 p^{7} T^{19} - 8046 p^{8} T^{20} + 1542 p^{9} T^{21} - 5 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - T - 152 T^{2} - 92 T^{3} + 9399 T^{4} + 24974 T^{5} - 259479 T^{6} - 1962678 T^{7} - 4981228 T^{8} + 100761596 T^{9} + 979017553 T^{10} - 2318847463 T^{11} - 59651566401 T^{12} - 2318847463 p T^{13} + 979017553 p^{2} T^{14} + 100761596 p^{3} T^{15} - 4981228 p^{4} T^{16} - 1962678 p^{5} T^{17} - 259479 p^{6} T^{18} + 24974 p^{7} T^{19} + 9399 p^{8} T^{20} - 92 p^{9} T^{21} - 152 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 3 T + 32 T^{2} - 24 T^{3} + 5689 T^{4} - 24468 T^{5} + 129999 T^{6} - 1301316 T^{7} + 16077832 T^{8} - 108310932 T^{9} - 142795873 T^{10} - 6959907621 T^{11} + 40412768229 T^{12} - 6959907621 p T^{13} - 142795873 p^{2} T^{14} - 108310932 p^{3} T^{15} + 16077832 p^{4} T^{16} - 1301316 p^{5} T^{17} + 129999 p^{6} T^{18} - 24468 p^{7} T^{19} + 5689 p^{8} T^{20} - 24 p^{9} T^{21} + 32 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 23 T + 19 T^{2} + 3433 T^{3} - 25449 T^{4} - 96248 T^{5} + 1686299 T^{6} - 2625142 T^{7} - 24357394 T^{8} - 465768818 T^{9} + 6139289829 T^{10} + 527943162 p T^{11} - 12560125211 p T^{12} + 527943162 p^{2} T^{13} + 6139289829 p^{2} T^{14} - 465768818 p^{3} T^{15} - 24357394 p^{4} T^{16} - 2625142 p^{5} T^{17} + 1686299 p^{6} T^{18} - 96248 p^{7} T^{19} - 25449 p^{8} T^{20} + 3433 p^{9} T^{21} + 19 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 23 T + 326 T^{2} + 3790 T^{3} + 37395 T^{4} + 337858 T^{5} + 2841639 T^{6} + 337858 p T^{7} + 37395 p^{2} T^{8} + 3790 p^{3} T^{9} + 326 p^{4} T^{10} + 23 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 33 T + 787 T^{2} + 12610 T^{3} + 167625 T^{4} + 1758413 T^{5} + 15936859 T^{6} + 1758413 p T^{7} + 167625 p^{2} T^{8} + 12610 p^{3} T^{9} + 787 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 23 T + 438 T^{2} - 6048 T^{3} + 76391 T^{4} - 843908 T^{5} + 9063259 T^{6} - 91504744 T^{7} + 921121074 T^{8} - 8829390278 T^{9} + 82962177493 T^{10} - 738400887829 T^{11} + 6398727347719 T^{12} - 738400887829 p T^{13} + 82962177493 p^{2} T^{14} - 8829390278 p^{3} T^{15} + 921121074 p^{4} T^{16} - 91504744 p^{5} T^{17} + 9063259 p^{6} T^{18} - 843908 p^{7} T^{19} + 76391 p^{8} T^{20} - 6048 p^{9} T^{21} + 438 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 4 T - 100 T^{2} - 1008 T^{3} - 1316 T^{4} + 52304 T^{5} + 793041 T^{6} + 1837300 T^{7} - 39402432 T^{8} - 255556724 T^{9} - 95378700 T^{10} + 4503434260 T^{11} + 97735822325 T^{12} + 4503434260 p T^{13} - 95378700 p^{2} T^{14} - 255556724 p^{3} T^{15} - 39402432 p^{4} T^{16} + 1837300 p^{5} T^{17} + 793041 p^{6} T^{18} + 52304 p^{7} T^{19} - 1316 p^{8} T^{20} - 1008 p^{9} T^{21} - 100 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 21 T + 262 T^{2} - 3139 T^{3} + 31171 T^{4} - 317011 T^{5} + 3604231 T^{6} - 37147467 T^{7} + 382043264 T^{8} - 4087935141 T^{9} + 36954724892 T^{10} - 316482025367 T^{11} + 2900323342294 T^{12} - 316482025367 p T^{13} + 36954724892 p^{2} T^{14} - 4087935141 p^{3} T^{15} + 382043264 p^{4} T^{16} - 37147467 p^{5} T^{17} + 3604231 p^{6} T^{18} - 317011 p^{7} T^{19} + 31171 p^{8} T^{20} - 3139 p^{9} T^{21} + 262 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 4 T - 240 T^{2} + 1818 T^{3} + 23159 T^{4} - 414424 T^{5} - 383864 T^{6} + 57894580 T^{7} - 290598612 T^{8} - 4888482676 T^{9} + 53349410020 T^{10} + 173313419650 T^{11} - 5373498848240 T^{12} + 173313419650 p T^{13} + 53349410020 p^{2} T^{14} - 4888482676 p^{3} T^{15} - 290598612 p^{4} T^{16} + 57894580 p^{5} T^{17} - 383864 p^{6} T^{18} - 414424 p^{7} T^{19} + 23159 p^{8} T^{20} + 1818 p^{9} T^{21} - 240 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 5 T - 157 T^{2} - 70 T^{3} + 33341 T^{4} + 76690 T^{5} - 4754095 T^{6} - 4599850 T^{7} + 576745920 T^{8} + 107030440 T^{9} - 65436427897 T^{10} - 47156613825 T^{11} + 5858579950339 T^{12} - 47156613825 p T^{13} - 65436427897 p^{2} T^{14} + 107030440 p^{3} T^{15} + 576745920 p^{4} T^{16} - 4599850 p^{5} T^{17} - 4754095 p^{6} T^{18} + 76690 p^{7} T^{19} + 33341 p^{8} T^{20} - 70 p^{9} T^{21} - 157 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 25 T + 224 T^{2} - 1500 T^{3} + 13659 T^{4} + 36210 T^{5} - 19280 p T^{6} + 10670315 T^{7} - 80299070 T^{8} + 272435015 T^{9} + 24813523689 T^{10} - 339806282575 T^{11} + 2617104620386 T^{12} - 339806282575 p T^{13} + 24813523689 p^{2} T^{14} + 272435015 p^{3} T^{15} - 80299070 p^{4} T^{16} + 10670315 p^{5} T^{17} - 19280 p^{7} T^{18} + 36210 p^{7} T^{19} + 13659 p^{8} T^{20} - 1500 p^{9} T^{21} + 224 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 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
−3.21755500681510148771612444127, −3.21496144414825445098664958167, −3.05273870878946778924539842559, −3.04624775756078080635258034876, −2.82152371080061491102832297244, −2.66397034949611589734120704719, −2.60150753995296117633456049836, −2.57146079879295263548379184266, −2.43408819854790916584842249810, −2.39291729415706435644086157856, −2.28919546546848874497022329369, −2.11041143285199463116146997403, −1.96690041089795765570622241324, −1.68379059480303897281956784847, −1.64630047862879363692283775889, −1.53324783842402902794830311220, −1.28534684790867508801984459559, −1.18490246431982434489720430461, −0.824095338247814709825079315231, −0.816019940188614360441176854965, −0.64842271840456376699459373171, −0.60454462381447890905405069748, −0.43438556903692472001332084362, −0.24237805806458248703482017090, −0.15185818059642621798995167584, 0.15185818059642621798995167584, 0.24237805806458248703482017090, 0.43438556903692472001332084362, 0.60454462381447890905405069748, 0.64842271840456376699459373171, 0.816019940188614360441176854965, 0.824095338247814709825079315231, 1.18490246431982434489720430461, 1.28534684790867508801984459559, 1.53324783842402902794830311220, 1.64630047862879363692283775889, 1.68379059480303897281956784847, 1.96690041089795765570622241324, 2.11041143285199463116146997403, 2.28919546546848874497022329369, 2.39291729415706435644086157856, 2.43408819854790916584842249810, 2.57146079879295263548379184266, 2.60150753995296117633456049836, 2.66397034949611589734120704719, 2.82152371080061491102832297244, 3.04624775756078080635258034876, 3.05273870878946778924539842559, 3.21496144414825445098664958167, 3.21755500681510148771612444127
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.