Dirichlet series
| L(s) = 1 | − 2-s − 8·4-s + 14·5-s − 12·7-s + 6·8-s − 14·10-s + 14·11-s − 10·13-s + 12·14-s + 32·16-s − 16·17-s − 14·19-s − 112·20-s − 14·22-s − 12·23-s + 105·25-s + 10·26-s + 96·28-s − 4·31-s − 15·32-s + 16·34-s − 168·35-s − 14·37-s + 14·38-s + 84·40-s − 18·41-s − 20·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4·4-s + 6.26·5-s − 4.53·7-s + 2.12·8-s − 4.42·10-s + 4.22·11-s − 2.77·13-s + 3.20·14-s + 8·16-s − 3.88·17-s − 3.21·19-s − 25.0·20-s − 2.98·22-s − 2.50·23-s + 21·25-s + 1.96·26-s + 18.1·28-s − 0.718·31-s − 2.65·32-s + 2.74·34-s − 28.3·35-s − 2.30·37-s + 2.27·38-s + 13.2·40-s − 2.81·41-s − 3.04·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{14} \cdot 11^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 5^{14} \cdot 11^{14} \cdot 19^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{28} \cdot 5^{14} \cdot 11^{14} \cdot 19^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.81510\times 10^{26}\) |
| Root analytic conductor: | \(8.66598\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 3^{28} \cdot 5^{14} \cdot 11^{14} \cdot 19^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{14} \) | |
| 11 | \( ( 1 - T )^{14} \) | |
| 19 | \( ( 1 + T )^{14} \) | |
| good | 2 | \( 1 + T + 9 T^{2} + 11 T^{3} + 45 T^{4} + 31 p T^{5} + 169 T^{6} + 235 T^{7} + 265 p T^{8} + 349 p T^{9} + 1425 T^{10} + 895 p T^{11} + 1663 p T^{12} + 4077 T^{13} + 6961 T^{14} + 4077 p T^{15} + 1663 p^{3} T^{16} + 895 p^{4} T^{17} + 1425 p^{4} T^{18} + 349 p^{6} T^{19} + 265 p^{7} T^{20} + 235 p^{7} T^{21} + 169 p^{8} T^{22} + 31 p^{10} T^{23} + 45 p^{10} T^{24} + 11 p^{11} T^{25} + 9 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) |
| 7 | \( 1 + 12 T + 116 T^{2} + 820 T^{3} + 4997 T^{4} + 26148 T^{5} + 17592 p T^{6} + 524102 T^{7} + 2058954 T^{8} + 152898 p^{2} T^{9} + 25552643 T^{10} + 81821406 T^{11} + 247765708 T^{12} + 709291614 T^{13} + 1927987534 T^{14} + 709291614 p T^{15} + 247765708 p^{2} T^{16} + 81821406 p^{3} T^{17} + 25552643 p^{4} T^{18} + 152898 p^{7} T^{19} + 2058954 p^{6} T^{20} + 524102 p^{7} T^{21} + 17592 p^{9} T^{22} + 26148 p^{9} T^{23} + 4997 p^{10} T^{24} + 820 p^{11} T^{25} + 116 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( 1 + 10 T + 142 T^{2} + 1108 T^{3} + 9128 T^{4} + 59010 T^{5} + 366025 T^{6} + 2028586 T^{7} + 10498974 T^{8} + 50948072 T^{9} + 231740754 T^{10} + 998588418 T^{11} + 4091669521 T^{12} + 15820842988 T^{13} + 58812495886 T^{14} + 15820842988 p T^{15} + 4091669521 p^{2} T^{16} + 998588418 p^{3} T^{17} + 231740754 p^{4} T^{18} + 50948072 p^{5} T^{19} + 10498974 p^{6} T^{20} + 2028586 p^{7} T^{21} + 366025 p^{8} T^{22} + 59010 p^{9} T^{23} + 9128 p^{10} T^{24} + 1108 p^{11} T^{25} + 142 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 16 T + 244 T^{2} + 2594 T^{3} + 24588 T^{4} + 198116 T^{5} + 85039 p T^{6} + 9485940 T^{7} + 57467570 T^{8} + 321349690 T^{9} + 1684269916 T^{10} + 487423764 p T^{11} + 38646829057 T^{12} + 171049031368 T^{13} + 722538511906 T^{14} + 171049031368 p T^{15} + 38646829057 p^{2} T^{16} + 487423764 p^{4} T^{17} + 1684269916 p^{4} T^{18} + 321349690 p^{5} T^{19} + 57467570 p^{6} T^{20} + 9485940 p^{7} T^{21} + 85039 p^{9} T^{22} + 198116 p^{9} T^{23} + 24588 p^{10} T^{24} + 2594 p^{11} T^{25} + 244 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 + 12 T + 223 T^{2} + 2224 T^{3} + 1101 p T^{4} + 213136 T^{5} + 1872966 T^{6} + 13651532 T^{7} + 100306801 T^{8} + 643478840 T^{9} + 4102585738 T^{10} + 23412256556 T^{11} + 131962656455 T^{12} + 673662157972 T^{13} + 3389047753306 T^{14} + 673662157972 p T^{15} + 131962656455 p^{2} T^{16} + 23412256556 p^{3} T^{17} + 4102585738 p^{4} T^{18} + 643478840 p^{5} T^{19} + 100306801 p^{6} T^{20} + 13651532 p^{7} T^{21} + 1872966 p^{8} T^{22} + 213136 p^{9} T^{23} + 1101 p^{11} T^{24} + 2224 p^{11} T^{25} + 223 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 + 175 T^{2} - 122 T^{3} + 15563 T^{4} - 23682 T^{5} + 942593 T^{6} - 2359410 T^{7} + 43739769 T^{8} - 156814202 T^{9} + 1667200805 T^{10} - 7628836372 T^{11} + 55219798891 T^{12} - 283804168356 T^{13} + 1663386785686 T^{14} - 283804168356 p T^{15} + 55219798891 p^{2} T^{16} - 7628836372 p^{3} T^{17} + 1667200805 p^{4} T^{18} - 156814202 p^{5} T^{19} + 43739769 p^{6} T^{20} - 2359410 p^{7} T^{21} + 942593 p^{8} T^{22} - 23682 p^{9} T^{23} + 15563 p^{10} T^{24} - 122 p^{11} T^{25} + 175 p^{12} T^{26} + p^{14} T^{28} \) | |
| 31 | \( 1 + 4 T + 252 T^{2} + 738 T^{3} + 30126 T^{4} + 54606 T^{5} + 2290817 T^{6} + 1622074 T^{7} + 127140340 T^{8} - 37319378 T^{9} + 5645226616 T^{10} - 5875759856 T^{11} + 213039867741 T^{12} - 296494187988 T^{13} + 7035063198614 T^{14} - 296494187988 p T^{15} + 213039867741 p^{2} T^{16} - 5875759856 p^{3} T^{17} + 5645226616 p^{4} T^{18} - 37319378 p^{5} T^{19} + 127140340 p^{6} T^{20} + 1622074 p^{7} T^{21} + 2290817 p^{8} T^{22} + 54606 p^{9} T^{23} + 30126 p^{10} T^{24} + 738 p^{11} T^{25} + 252 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 + 14 T + 360 T^{2} + 4242 T^{3} + 62557 T^{4} + 631408 T^{5} + 7004178 T^{6} + 61675372 T^{7} + 568520112 T^{8} + 4439887542 T^{9} + 35611305429 T^{10} + 249756202946 T^{11} + 1784227901798 T^{12} + 11318800536812 T^{13} + 72888128885338 T^{14} + 11318800536812 p T^{15} + 1784227901798 p^{2} T^{16} + 249756202946 p^{3} T^{17} + 35611305429 p^{4} T^{18} + 4439887542 p^{5} T^{19} + 568520112 p^{6} T^{20} + 61675372 p^{7} T^{21} + 7004178 p^{8} T^{22} + 631408 p^{9} T^{23} + 62557 p^{10} T^{24} + 4242 p^{11} T^{25} + 360 p^{12} T^{26} + 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 + 18 T + 427 T^{2} + 5866 T^{3} + 86314 T^{4} + 966902 T^{5} + 11001639 T^{6} + 105203298 T^{7} + 1002314608 T^{8} + 8408805954 T^{9} + 69830561673 T^{10} + 523389110338 T^{11} + 3875256734901 T^{12} + 26228877655384 T^{13} + 175316974128074 T^{14} + 26228877655384 p T^{15} + 3875256734901 p^{2} T^{16} + 523389110338 p^{3} T^{17} + 69830561673 p^{4} T^{18} + 8408805954 p^{5} T^{19} + 1002314608 p^{6} T^{20} + 105203298 p^{7} T^{21} + 11001639 p^{8} T^{22} + 966902 p^{9} T^{23} + 86314 p^{10} T^{24} + 5866 p^{11} T^{25} + 427 p^{12} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 + 20 T + 389 T^{2} + 4930 T^{3} + 62531 T^{4} + 639422 T^{5} + 6621280 T^{6} + 1373462 p T^{7} + 532364903 T^{8} + 4256675402 T^{9} + 34436990152 T^{10} + 251571978808 T^{11} + 1866045432813 T^{12} + 12612039985176 T^{13} + 86564086171866 T^{14} + 12612039985176 p T^{15} + 1866045432813 p^{2} T^{16} + 251571978808 p^{3} T^{17} + 34436990152 p^{4} T^{18} + 4256675402 p^{5} T^{19} + 532364903 p^{6} T^{20} + 1373462 p^{8} T^{21} + 6621280 p^{8} T^{22} + 639422 p^{9} T^{23} + 62531 p^{10} T^{24} + 4930 p^{11} T^{25} + 389 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 + 8 T + 151 T^{2} + 1100 T^{3} + 14908 T^{4} + 89804 T^{5} + 1040744 T^{6} + 5892148 T^{7} + 62494454 T^{8} + 347058308 T^{9} + 3428881969 T^{10} + 18820660800 T^{11} + 177748533549 T^{12} + 956841947752 T^{13} + 8445314129440 T^{14} + 956841947752 p T^{15} + 177748533549 p^{2} T^{16} + 18820660800 p^{3} T^{17} + 3428881969 p^{4} T^{18} + 347058308 p^{5} T^{19} + 62494454 p^{6} T^{20} + 5892148 p^{7} T^{21} + 1040744 p^{8} T^{22} + 89804 p^{9} T^{23} + 14908 p^{10} T^{24} + 1100 p^{11} T^{25} + 151 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 + 20 T + 515 T^{2} + 7232 T^{3} + 112626 T^{4} + 1280500 T^{5} + 15477323 T^{6} + 153394948 T^{7} + 1579090444 T^{8} + 14133136440 T^{9} + 129034681177 T^{10} + 1057692120836 T^{11} + 8745024816857 T^{12} + 66094574365784 T^{13} + 501710602078514 T^{14} + 66094574365784 p T^{15} + 8745024816857 p^{2} T^{16} + 1057692120836 p^{3} T^{17} + 129034681177 p^{4} T^{18} + 14133136440 p^{5} T^{19} + 1579090444 p^{6} T^{20} + 153394948 p^{7} T^{21} + 15477323 p^{8} T^{22} + 1280500 p^{9} T^{23} + 112626 p^{10} T^{24} + 7232 p^{11} T^{25} + 515 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 - 2 T + 415 T^{2} - 710 T^{3} + 89421 T^{4} - 138144 T^{5} + 13211842 T^{6} - 19079814 T^{7} + 1489824299 T^{8} - 2049564936 T^{9} + 135451313800 T^{10} - 177827116140 T^{11} + 10244614888931 T^{12} - 12678595089102 T^{13} + 655446846346518 T^{14} - 12678595089102 p T^{15} + 10244614888931 p^{2} T^{16} - 177827116140 p^{3} T^{17} + 135451313800 p^{4} T^{18} - 2049564936 p^{5} T^{19} + 1489824299 p^{6} T^{20} - 19079814 p^{7} T^{21} + 13211842 p^{8} T^{22} - 138144 p^{9} T^{23} + 89421 p^{10} T^{24} - 710 p^{11} T^{25} + 415 p^{12} T^{26} - 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 4 T + 365 T^{2} - 1724 T^{3} + 69201 T^{4} - 388124 T^{5} + 8940771 T^{6} - 60078552 T^{7} + 881744051 T^{8} - 7052829816 T^{9} + 71247411127 T^{10} - 655243962024 T^{11} + 4994755780219 T^{12} - 49127273977020 T^{13} + 317311549823538 T^{14} - 49127273977020 p T^{15} + 4994755780219 p^{2} T^{16} - 655243962024 p^{3} T^{17} + 71247411127 p^{4} T^{18} - 7052829816 p^{5} T^{19} + 881744051 p^{6} T^{20} - 60078552 p^{7} T^{21} + 8940771 p^{8} T^{22} - 388124 p^{9} T^{23} + 69201 p^{10} T^{24} - 1724 p^{11} T^{25} + 365 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 + 22 T + 584 T^{2} + 10492 T^{3} + 180291 T^{4} + 2628490 T^{5} + 36266834 T^{6} + 449491700 T^{7} + 5303869322 T^{8} + 861171024 p T^{9} + 599710429137 T^{10} + 5832984508614 T^{11} + 54345267093934 T^{12} + 476957999887486 T^{13} + 4019354062363282 T^{14} + 476957999887486 p T^{15} + 54345267093934 p^{2} T^{16} + 5832984508614 p^{3} T^{17} + 599710429137 p^{4} T^{18} + 861171024 p^{6} T^{19} + 5303869322 p^{6} T^{20} + 449491700 p^{7} T^{21} + 36266834 p^{8} T^{22} + 2628490 p^{9} T^{23} + 180291 p^{10} T^{24} + 10492 p^{11} T^{25} + 584 p^{12} T^{26} + 22 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 430 T^{2} + 826 T^{3} + 94623 T^{4} + 356458 T^{5} + 208338 p T^{6} + 74736498 T^{7} + 1867970076 T^{8} + 10443381740 T^{9} + 199121650979 T^{10} + 1107336160806 T^{11} + 18061986639280 T^{12} + 94753242850124 T^{13} + 1392076778540154 T^{14} + 94753242850124 p T^{15} + 18061986639280 p^{2} T^{16} + 1107336160806 p^{3} T^{17} + 199121650979 p^{4} T^{18} + 10443381740 p^{5} T^{19} + 1867970076 p^{6} T^{20} + 74736498 p^{7} T^{21} + 208338 p^{9} T^{22} + 356458 p^{9} T^{23} + 94623 p^{10} T^{24} + 826 p^{11} T^{25} + 430 p^{12} T^{26} + p^{14} T^{28} \) | |
| 73 | \( 1 + 28 T + 703 T^{2} + 172 p T^{3} + 210128 T^{4} + 3030868 T^{5} + 41398984 T^{6} + 515851724 T^{7} + 6107594926 T^{8} + 67640279820 T^{9} + 714633075233 T^{10} + 7156904445836 T^{11} + 68535023424753 T^{12} + 625878421922112 T^{13} + 5470821225504368 T^{14} + 625878421922112 p T^{15} + 68535023424753 p^{2} T^{16} + 7156904445836 p^{3} T^{17} + 714633075233 p^{4} T^{18} + 67640279820 p^{5} T^{19} + 6107594926 p^{6} T^{20} + 515851724 p^{7} T^{21} + 41398984 p^{8} T^{22} + 3030868 p^{9} T^{23} + 210128 p^{10} T^{24} + 172 p^{12} T^{25} + 703 p^{12} T^{26} + 28 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 + 14 T + 677 T^{2} + 9200 T^{3} + 236857 T^{4} + 3018618 T^{5} + 55417812 T^{6} + 653356732 T^{7} + 9585263049 T^{8} + 103827892002 T^{9} + 1289897437122 T^{10} + 12776763926888 T^{11} + 138923696276129 T^{12} + 1252043754275050 T^{13} + 12150804702244866 T^{14} + 1252043754275050 p T^{15} + 138923696276129 p^{2} T^{16} + 12776763926888 p^{3} T^{17} + 1289897437122 p^{4} T^{18} + 103827892002 p^{5} T^{19} + 9585263049 p^{6} T^{20} + 653356732 p^{7} T^{21} + 55417812 p^{8} T^{22} + 3018618 p^{9} T^{23} + 236857 p^{10} T^{24} + 9200 p^{11} T^{25} + 677 p^{12} T^{26} + 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 + 10 T + 583 T^{2} + 4888 T^{3} + 173601 T^{4} + 1255954 T^{5} + 34689815 T^{6} + 215308982 T^{7} + 5192045367 T^{8} + 27612231180 T^{9} + 623297226057 T^{10} + 2864484925958 T^{11} + 63026269499143 T^{12} + 258872325963740 T^{13} + 5563480500659250 T^{14} + 258872325963740 p T^{15} + 63026269499143 p^{2} T^{16} + 2864484925958 p^{3} T^{17} + 623297226057 p^{4} T^{18} + 27612231180 p^{5} T^{19} + 5192045367 p^{6} T^{20} + 215308982 p^{7} T^{21} + 34689815 p^{8} T^{22} + 1255954 p^{9} T^{23} + 173601 p^{10} T^{24} + 4888 p^{11} T^{25} + 583 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 - 26 T + 972 T^{2} - 17104 T^{3} + 370649 T^{4} - 4956720 T^{5} + 80636282 T^{6} - 884378586 T^{7} + 12257402080 T^{8} - 118767998590 T^{9} + 1523098148401 T^{10} - 13810352955408 T^{11} + 167616120674106 T^{12} - 1431417127240110 T^{13} + 16132113894723754 T^{14} - 1431417127240110 p T^{15} + 167616120674106 p^{2} T^{16} - 13810352955408 p^{3} T^{17} + 1523098148401 p^{4} T^{18} - 118767998590 p^{5} T^{19} + 12257402080 p^{6} T^{20} - 884378586 p^{7} T^{21} + 80636282 p^{8} T^{22} - 4956720 p^{9} T^{23} + 370649 p^{10} T^{24} - 17104 p^{11} T^{25} + 972 p^{12} T^{26} - 26 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 + 32 T + 1155 T^{2} + 25092 T^{3} + 536909 T^{4} + 8832306 T^{5} + 138904420 T^{6} + 1807808392 T^{7} + 22338677699 T^{8} + 233063215566 T^{9} + 2324632462044 T^{10} + 19540822223166 T^{11} + 166497232232971 T^{12} + 1258046984487990 T^{13} + 12196049139392930 T^{14} + 1258046984487990 p T^{15} + 166497232232971 p^{2} T^{16} + 19540822223166 p^{3} T^{17} + 2324632462044 p^{4} T^{18} + 233063215566 p^{5} T^{19} + 22338677699 p^{6} T^{20} + 1807808392 p^{7} T^{21} + 138904420 p^{8} T^{22} + 8832306 p^{9} T^{23} + 536909 p^{10} T^{24} + 25092 p^{11} T^{25} + 1155 p^{12} T^{26} + 32 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.33491927350786017855581882996, −2.26721012731061809683319181513, −2.26028407335118436136574840774, −2.23613460157560222939878683601, −2.15546509380873580619257560888, −2.05693051486334629572868752875, −2.01324083489220055138662663858, −2.00248438497401070321592658089, −1.97516814421972413266331947365, −1.81798186979262811800834887443, −1.77595070795304189126176330816, −1.76246654963177116995982911633, −1.57831806427752853897318185186, −1.55588059384242653121682515228, −1.43017705098054620974417056100, −1.33597019975702664868767514757, −1.32940947097198325822409675070, −1.30812879160181593139783835375, −1.19781858029161174775386007227, −1.16032314729191558403063670878, −1.09791060963580020489787282236, −1.05614221404034277179471356342, −0.996063886915601292972718473260, −0.990434569330295818095821615409, −0.879334113313874324484872498731, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.879334113313874324484872498731, 0.990434569330295818095821615409, 0.996063886915601292972718473260, 1.05614221404034277179471356342, 1.09791060963580020489787282236, 1.16032314729191558403063670878, 1.19781858029161174775386007227, 1.30812879160181593139783835375, 1.32940947097198325822409675070, 1.33597019975702664868767514757, 1.43017705098054620974417056100, 1.55588059384242653121682515228, 1.57831806427752853897318185186, 1.76246654963177116995982911633, 1.77595070795304189126176330816, 1.81798186979262811800834887443, 1.97516814421972413266331947365, 2.00248438497401070321592658089, 2.01324083489220055138662663858, 2.05693051486334629572868752875, 2.15546509380873580619257560888, 2.23613460157560222939878683601, 2.26028407335118436136574840774, 2.26721012731061809683319181513, 2.33491927350786017855581882996
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.