Dirichlet series
| L(s) = 1 | + 6·2-s + 15·4-s + 5·5-s − 3·7-s + 14·8-s − 4·9-s + 30·10-s + 6·11-s − 6·13-s − 18·14-s − 21·16-s − 8·17-s − 24·18-s + 4·19-s + 75·20-s + 36·22-s + 6·23-s + 28·25-s − 36·26-s − 3·27-s − 45·28-s + 12·29-s − 6·31-s − 84·32-s − 48·34-s − 15·35-s − 60·36-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 15/2·4-s + 2.23·5-s − 1.13·7-s + 4.94·8-s − 4/3·9-s + 9.48·10-s + 1.80·11-s − 1.66·13-s − 4.81·14-s − 5.25·16-s − 1.94·17-s − 5.65·18-s + 0.917·19-s + 16.7·20-s + 7.67·22-s + 1.25·23-s + 28/5·25-s − 7.06·26-s − 0.577·27-s − 8.50·28-s + 2.22·29-s − 1.07·31-s − 14.8·32-s − 8.23·34-s − 2.53·35-s − 10·36-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 23^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.70344\times 10^{6}\) |
| Root analytic conductor: | \(1.81818\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(37.28833649\) |
| \(L(\frac12)\) | \(\approx\) | \(37.28833649\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{6} \) |
| 3 | \( 1 + 4 T^{2} + p T^{3} + 22 T^{4} + p^{2} T^{5} + 23 p T^{6} + p^{3} T^{7} + 22 p^{2} T^{8} + p^{4} T^{9} + 4 p^{4} T^{10} + p^{6} T^{12} \) | |
| 23 | \( ( 1 - T + T^{2} )^{6} \) | |
| good | 5 | \( 1 - p T - 3 T^{2} + 44 T^{3} + 2 T^{4} - 6 p^{2} T^{5} - 406 T^{6} + 116 p T^{7} + 4662 T^{8} - 5406 T^{9} - 4336 p T^{10} + 19403 T^{11} + 70209 T^{12} + 19403 p T^{13} - 4336 p^{3} T^{14} - 5406 p^{3} T^{15} + 4662 p^{4} T^{16} + 116 p^{6} T^{17} - 406 p^{6} T^{18} - 6 p^{9} T^{19} + 2 p^{8} T^{20} + 44 p^{9} T^{21} - 3 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + 3 T - 2 p T^{2} + 15 T^{3} + 6 p^{2} T^{4} - 60 p T^{5} - 1254 T^{6} + 10533 T^{7} + 523 T^{8} - 65355 T^{9} + 206896 T^{10} + 317412 T^{11} - 1579091 T^{12} + 317412 p T^{13} + 206896 p^{2} T^{14} - 65355 p^{3} T^{15} + 523 p^{4} T^{16} + 10533 p^{5} T^{17} - 1254 p^{6} T^{18} - 60 p^{8} T^{19} + 6 p^{10} T^{20} + 15 p^{9} T^{21} - 2 p^{11} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T - 16 T^{2} + 138 T^{3} + 206 T^{4} - 1791 T^{5} - 141 p T^{6} + 9165 T^{7} + 20593 T^{8} + 26712 T^{9} - 363314 T^{10} - 393357 T^{11} + 4923130 T^{12} - 393357 p T^{13} - 363314 p^{2} T^{14} + 26712 p^{3} T^{15} + 20593 p^{4} T^{16} + 9165 p^{5} T^{17} - 141 p^{7} T^{18} - 1791 p^{7} T^{19} + 206 p^{8} T^{20} + 138 p^{9} T^{21} - 16 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 6 T - 20 T^{2} - 270 T^{3} - 243 T^{4} + 5100 T^{5} + 1149 p T^{6} - 51369 T^{7} - 284480 T^{8} + 289584 T^{9} + 3656542 T^{10} - 61863 p T^{11} - 45219281 T^{12} - 61863 p^{2} T^{13} + 3656542 p^{2} T^{14} + 289584 p^{3} T^{15} - 284480 p^{4} T^{16} - 51369 p^{5} T^{17} + 1149 p^{7} T^{18} + 5100 p^{7} T^{19} - 243 p^{8} T^{20} - 270 p^{9} T^{21} - 20 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 88 T^{2} + 283 T^{3} + 3404 T^{4} + 8815 T^{5} + 74779 T^{6} + 8815 p T^{7} + 3404 p^{2} T^{8} + 283 p^{3} T^{9} + 88 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( ( 1 - 2 T + 72 T^{2} - 60 T^{3} + 130 p T^{4} - 911 T^{5} + 56359 T^{6} - 911 p T^{7} + 130 p^{3} T^{8} - 60 p^{3} T^{9} + 72 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 12 T - 61 T^{2} + 894 T^{3} + 5957 T^{4} - 56541 T^{5} - 343047 T^{6} + 2065356 T^{7} + 18249040 T^{8} - 57765834 T^{9} - 704345789 T^{10} + 576476055 T^{11} + 23597087107 T^{12} + 576476055 p T^{13} - 704345789 p^{2} T^{14} - 57765834 p^{3} T^{15} + 18249040 p^{4} T^{16} + 2065356 p^{5} T^{17} - 343047 p^{6} T^{18} - 56541 p^{7} T^{19} + 5957 p^{8} T^{20} + 894 p^{9} T^{21} - 61 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 6 T - 82 T^{2} - 126 T^{3} + 6003 T^{4} - 5074 T^{5} - 190373 T^{6} + 1075387 T^{7} + 3633830 T^{8} - 37681700 T^{9} + 101744626 T^{10} + 715843207 T^{11} - 4711915269 T^{12} + 715843207 p T^{13} + 101744626 p^{2} T^{14} - 37681700 p^{3} T^{15} + 3633830 p^{4} T^{16} + 1075387 p^{5} T^{17} - 190373 p^{6} T^{18} - 5074 p^{7} T^{19} + 6003 p^{8} T^{20} - 126 p^{9} T^{21} - 82 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 4 T + 131 T^{2} - 675 T^{3} + 258 p T^{4} - 41530 T^{5} + 453040 T^{6} - 41530 p T^{7} + 258 p^{3} T^{8} - 675 p^{3} T^{9} + 131 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 - 15 T - 19 T^{2} + 1344 T^{3} - 2744 T^{4} - 53796 T^{5} + 130630 T^{6} + 906081 T^{7} + 5606962 T^{8} + 10496694 T^{9} - 881147956 T^{10} - 664954827 T^{11} + 50972584934 T^{12} - 664954827 p T^{13} - 881147956 p^{2} T^{14} + 10496694 p^{3} T^{15} + 5606962 p^{4} T^{16} + 906081 p^{5} T^{17} + 130630 p^{6} T^{18} - 53796 p^{7} T^{19} - 2744 p^{8} T^{20} + 1344 p^{9} T^{21} - 19 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 14 T + 63 T^{2} + 876 T^{3} + 10676 T^{4} + 30990 T^{5} + 200670 T^{6} + 3795286 T^{7} + 8567166 T^{8} - 655092 p T^{9} + 734698952 T^{10} + 2308821209 T^{11} - 25803761600 T^{12} + 2308821209 p T^{13} + 734698952 p^{2} T^{14} - 655092 p^{4} T^{15} + 8567166 p^{4} T^{16} + 3795286 p^{5} T^{17} + 200670 p^{6} T^{18} + 30990 p^{7} T^{19} + 10676 p^{8} T^{20} + 876 p^{9} T^{21} + 63 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 9 T - 37 T^{2} + 852 T^{3} - 6041 T^{4} + 20649 T^{5} + 110176 T^{6} - 2758239 T^{7} + 26864857 T^{8} - 102319884 T^{9} - 372561019 T^{10} + 6665583759 T^{11} - 52152963586 T^{12} + 6665583759 p T^{13} - 372561019 p^{2} T^{14} - 102319884 p^{3} T^{15} + 26864857 p^{4} T^{16} - 2758239 p^{5} T^{17} + 110176 p^{6} T^{18} + 20649 p^{7} T^{19} - 6041 p^{8} T^{20} + 852 p^{9} T^{21} - 37 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 5 T + 221 T^{2} + 562 T^{3} + 20687 T^{4} + 21500 T^{5} + 1248128 T^{6} + 21500 p T^{7} + 20687 p^{2} T^{8} + 562 p^{3} T^{9} + 221 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 18 T + 2 T^{2} - 144 T^{3} + 26126 T^{4} - 88983 T^{5} - 561132 T^{6} - 16689915 T^{7} + 108454318 T^{8} + 345065940 T^{9} + 7506276463 T^{10} - 60696764073 T^{11} - 113236138244 T^{12} - 60696764073 p T^{13} + 7506276463 p^{2} T^{14} + 345065940 p^{3} T^{15} + 108454318 p^{4} T^{16} - 16689915 p^{5} T^{17} - 561132 p^{6} T^{18} - 88983 p^{7} T^{19} + 26126 p^{8} T^{20} - 144 p^{9} T^{21} + 2 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 3 T - 158 T^{2} + 819 T^{3} + 18228 T^{4} - 166086 T^{5} - 544206 T^{6} + 21205689 T^{7} - 50366153 T^{8} - 1273225077 T^{9} + 11328326416 T^{10} + 39057230916 T^{11} - 842645372279 T^{12} + 39057230916 p T^{13} + 11328326416 p^{2} T^{14} - 1273225077 p^{3} T^{15} - 50366153 p^{4} T^{16} + 21205689 p^{5} T^{17} - 544206 p^{6} T^{18} - 166086 p^{7} T^{19} + 18228 p^{8} T^{20} + 819 p^{9} T^{21} - 158 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 8 T - 150 T^{2} + 1770 T^{3} + 9460 T^{4} - 2891 p T^{5} + 140667 T^{6} + 12473259 T^{7} - 76262909 T^{8} - 540308430 T^{9} + 8424720158 T^{10} + 10240308063 T^{11} - 629039886974 T^{12} + 10240308063 p T^{13} + 8424720158 p^{2} T^{14} - 540308430 p^{3} T^{15} - 76262909 p^{4} T^{16} + 12473259 p^{5} T^{17} + 140667 p^{6} T^{18} - 2891 p^{8} T^{19} + 9460 p^{8} T^{20} + 1770 p^{9} T^{21} - 150 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 9 T + 100 T^{2} + 252 T^{3} + 3338 T^{4} - 18720 T^{5} - 90110 T^{6} - 18720 p T^{7} + 3338 p^{2} T^{8} + 252 p^{3} T^{9} + 100 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 16 T + 340 T^{2} + 4138 T^{3} + 54182 T^{4} + 516269 T^{5} + 5032649 T^{6} + 516269 p T^{7} + 54182 p^{2} T^{8} + 4138 p^{3} T^{9} + 340 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 + 7 T - 295 T^{2} - 1252 T^{3} + 52630 T^{4} + 104740 T^{5} - 6258032 T^{6} + 748144 T^{7} + 574511664 T^{8} - 699341380 T^{9} - 44343565990 T^{10} + 34767579385 T^{11} + 3389151098235 T^{12} + 34767579385 p T^{13} - 44343565990 p^{2} T^{14} - 699341380 p^{3} T^{15} + 574511664 p^{4} T^{16} + 748144 p^{5} T^{17} - 6258032 p^{6} T^{18} + 104740 p^{7} T^{19} + 52630 p^{8} T^{20} - 1252 p^{9} T^{21} - 295 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 3 T - 221 T^{2} + 514 T^{3} + 20104 T^{4} - 26380 T^{5} - 773432 T^{6} - 130926 T^{7} + 24662490 T^{8} + 89545706 T^{9} - 8255114492 T^{10} - 3516847709 T^{11} + 1105193201531 T^{12} - 3516847709 p T^{13} - 8255114492 p^{2} T^{14} + 89545706 p^{3} T^{15} + 24662490 p^{4} T^{16} - 130926 p^{5} T^{17} - 773432 p^{6} T^{18} - 26380 p^{7} T^{19} + 20104 p^{8} T^{20} + 514 p^{9} T^{21} - 221 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 21 T + 421 T^{2} + 4146 T^{3} + 48983 T^{4} + 340626 T^{5} + 4104388 T^{6} + 340626 p T^{7} + 48983 p^{2} T^{8} + 4146 p^{3} T^{9} + 421 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 13 T + 6 T^{2} - 117 T^{3} + 905 T^{4} + 144051 T^{5} - 960618 T^{6} - 10172978 T^{7} + 42925869 T^{8} + 43903785 T^{9} + 8446437137 T^{10} + 44628265907 T^{11} - 1823635798670 T^{12} + 44628265907 p T^{13} + 8446437137 p^{2} T^{14} + 43903785 p^{3} T^{15} + 42925869 p^{4} T^{16} - 10172978 p^{5} T^{17} - 960618 p^{6} T^{18} + 144051 p^{7} T^{19} + 905 p^{8} T^{20} - 117 p^{9} T^{21} + 6 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.84749641651644465478239896975, −3.83600457782168726223911183363, −3.56442220251893292501515272292, −3.37852302479900977744399803516, −3.20077721683084625728568914072, −3.17977067252652157861186018698, −3.12844248591848755962284186718, −3.08586056539284702694250542440, −3.00626048520851179876570061752, −2.97866925575516916108092759497, −2.70889858592709767109163011116, −2.69075626789017502521112699740, −2.63112847257156799230853844885, −2.42529945161547474542759151456, −2.24826834372547435527326661480, −2.11741945716828932068899389933, −2.04812929752584709453442730130, −1.99653120250971840541128952589, −1.83527039919616539746579362590, −1.34148018563562391188365159611, −1.23519086375632857922026536783, −1.06354464266239460361355464894, −1.00149511884867052719760985515, −0.60596705401694030222323140460, −0.33895329643314065577311910062, 0.33895329643314065577311910062, 0.60596705401694030222323140460, 1.00149511884867052719760985515, 1.06354464266239460361355464894, 1.23519086375632857922026536783, 1.34148018563562391188365159611, 1.83527039919616539746579362590, 1.99653120250971840541128952589, 2.04812929752584709453442730130, 2.11741945716828932068899389933, 2.24826834372547435527326661480, 2.42529945161547474542759151456, 2.63112847257156799230853844885, 2.69075626789017502521112699740, 2.70889858592709767109163011116, 2.97866925575516916108092759497, 3.00626048520851179876570061752, 3.08586056539284702694250542440, 3.12844248591848755962284186718, 3.17977067252652157861186018698, 3.20077721683084625728568914072, 3.37852302479900977744399803516, 3.56442220251893292501515272292, 3.83600457782168726223911183363, 3.84749641651644465478239896975
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.