Dirichlet series
| L(s) = 1 | − 4·3-s − 5-s + 2·7-s + 9·9-s − 11-s − 6·13-s + 4·15-s + 12·17-s − 14·19-s − 8·21-s + 19·23-s + 18·25-s − 17·27-s − 2·29-s + 8·31-s + 4·33-s − 2·35-s − 34·37-s + 24·39-s − 2·41-s + 15·43-s − 9·45-s − 9·47-s + 29·49-s − 48·51-s + 32·53-s + 55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.447·5-s + 0.755·7-s + 3·9-s − 0.301·11-s − 1.66·13-s + 1.03·15-s + 2.91·17-s − 3.21·19-s − 1.74·21-s + 3.96·23-s + 18/5·25-s − 3.27·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.338·35-s − 5.58·37-s + 3.84·39-s − 0.312·41-s + 2.28·43-s − 1.34·45-s − 1.31·47-s + 29/7·49-s − 6.72·51-s + 4.39·53-s + 0.134·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 3^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.03833\times 10^{10}\) |
| Root analytic conductor: | \(2.73386\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 3^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.741774387\) |
| \(L(\frac12)\) | \(\approx\) | \(1.741774387\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 4 T + 7 T^{2} + p^{2} T^{3} - p T^{4} - 5 p^{2} T^{5} - 11 p^{2} T^{6} - 5 p^{3} T^{7} - p^{3} T^{8} + p^{5} T^{9} + 7 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| 13 | \( ( 1 + T + T^{2} )^{6} \) | |
| good | 5 | \( 1 + T - 17 T^{2} - 6 p T^{3} + 144 T^{4} + 356 T^{5} - 664 T^{6} - 534 p T^{7} + 896 T^{8} + 2532 p T^{9} + 12814 T^{10} - 27063 T^{11} - 103651 T^{12} - 27063 p T^{13} + 12814 p^{2} T^{14} + 2532 p^{4} T^{15} + 896 p^{4} T^{16} - 534 p^{6} T^{17} - 664 p^{6} T^{18} + 356 p^{7} T^{19} + 144 p^{8} T^{20} - 6 p^{10} T^{21} - 17 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 2 T - 25 T^{2} + 12 p T^{3} + 303 T^{4} - 1467 T^{5} - 1663 T^{6} + 15712 T^{7} - 3322 T^{8} - 104066 T^{9} + 22695 p T^{10} + 305685 T^{11} - 1527553 T^{12} + 305685 p T^{13} + 22695 p^{3} T^{14} - 104066 p^{3} T^{15} - 3322 p^{4} T^{16} + 15712 p^{5} T^{17} - 1663 p^{6} T^{18} - 1467 p^{7} T^{19} + 303 p^{8} T^{20} + 12 p^{10} T^{21} - 25 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + T - 28 T^{2} - 145 T^{3} + 356 T^{4} + 3846 T^{5} + 2192 T^{6} - 60825 T^{7} - 147607 T^{8} + 615179 T^{9} + 3066540 T^{10} - 2917334 T^{11} - 40184823 T^{12} - 2917334 p T^{13} + 3066540 p^{2} T^{14} + 615179 p^{3} T^{15} - 147607 p^{4} T^{16} - 60825 p^{5} T^{17} + 2192 p^{6} T^{18} + 3846 p^{7} T^{19} + 356 p^{8} T^{20} - 145 p^{9} T^{21} - 28 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 6 T + 72 T^{2} - 235 T^{3} + 1812 T^{4} - 3435 T^{5} + 30155 T^{6} - 3435 p T^{7} + 1812 p^{2} T^{8} - 235 p^{3} T^{9} + 72 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( ( 1 + 7 T + 109 T^{2} + 552 T^{3} + 4881 T^{4} + 1000 p T^{5} + 120760 T^{6} + 1000 p^{2} T^{7} + 4881 p^{2} T^{8} + 552 p^{3} T^{9} + 109 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 19 T + 85 T^{2} + 76 T^{3} + 4236 T^{4} - 54722 T^{5} + 61228 T^{6} + 217791 T^{7} + 9916640 T^{8} - 48864976 T^{9} - 72985812 T^{10} - 431640815 T^{11} + 8784786536 T^{12} - 431640815 p T^{13} - 72985812 p^{2} T^{14} - 48864976 p^{3} T^{15} + 9916640 p^{4} T^{16} + 217791 p^{5} T^{17} + 61228 p^{6} T^{18} - 54722 p^{7} T^{19} + 4236 p^{8} T^{20} + 76 p^{9} T^{21} + 85 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 2 T - 71 T^{2} - 50 T^{3} + 2034 T^{4} - 2945 T^{5} - 38099 T^{6} + 174165 T^{7} + 729554 T^{8} - 3060184 T^{9} + 10301124 T^{10} + 14837524 T^{11} - 1050511345 T^{12} + 14837524 p T^{13} + 10301124 p^{2} T^{14} - 3060184 p^{3} T^{15} + 729554 p^{4} T^{16} + 174165 p^{5} T^{17} - 38099 p^{6} T^{18} - 2945 p^{7} T^{19} + 2034 p^{8} T^{20} - 50 p^{9} T^{21} - 71 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 8 T - 77 T^{2} + 468 T^{3} + 5264 T^{4} - 541 p T^{5} - 212981 T^{6} + 65633 T^{7} + 6780536 T^{8} + 8681576 T^{9} - 150146070 T^{10} - 219740916 T^{11} + 3926987469 T^{12} - 219740916 p T^{13} - 150146070 p^{2} T^{14} + 8681576 p^{3} T^{15} + 6780536 p^{4} T^{16} + 65633 p^{5} T^{17} - 212981 p^{6} T^{18} - 541 p^{8} T^{19} + 5264 p^{8} T^{20} + 468 p^{9} T^{21} - 77 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 17 T + 155 T^{2} + 747 T^{3} + 1666 T^{4} - 4493 T^{5} - 49898 T^{6} - 4493 p T^{7} + 1666 p^{2} T^{8} + 747 p^{3} T^{9} + 155 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 2 T - 37 T^{2} + 52 T^{3} - 1099 T^{4} - 14289 T^{5} + 36959 T^{6} + 342990 T^{7} - 546304 T^{8} - 12245894 T^{9} + 83979345 T^{10} + 483815711 T^{11} - 4001736225 T^{12} + 483815711 p T^{13} + 83979345 p^{2} T^{14} - 12245894 p^{3} T^{15} - 546304 p^{4} T^{16} + 342990 p^{5} T^{17} + 36959 p^{6} T^{18} - 14289 p^{7} T^{19} - 1099 p^{8} T^{20} + 52 p^{9} T^{21} - 37 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 15 T - 44 T^{2} + 1605 T^{3} + 436 T^{4} - 96150 T^{5} + 13421 T^{6} + 2798805 T^{7} + 15939163 T^{8} - 37455615 T^{9} - 1767812744 T^{10} - 358014495 T^{11} + 104035237766 T^{12} - 358014495 p T^{13} - 1767812744 p^{2} T^{14} - 37455615 p^{3} T^{15} + 15939163 p^{4} T^{16} + 2798805 p^{5} T^{17} + 13421 p^{6} T^{18} - 96150 p^{7} T^{19} + 436 p^{8} T^{20} + 1605 p^{9} T^{21} - 44 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 9 T - 96 T^{2} - 791 T^{3} + 8358 T^{4} + 63018 T^{5} - 231163 T^{6} - 2385648 T^{7} - 5455962 T^{8} + 94088395 T^{9} + 1640985885 T^{10} - 432734379 T^{11} - 92112254279 T^{12} - 432734379 p T^{13} + 1640985885 p^{2} T^{14} + 94088395 p^{3} T^{15} - 5455962 p^{4} T^{16} - 2385648 p^{5} T^{17} - 231163 p^{6} T^{18} + 63018 p^{7} T^{19} + 8358 p^{8} T^{20} - 791 p^{9} T^{21} - 96 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 16 T + 341 T^{2} - 3597 T^{3} + 44717 T^{4} - 351571 T^{5} + 3140075 T^{6} - 351571 p T^{7} + 44717 p^{2} T^{8} - 3597 p^{3} T^{9} + 341 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 5 T - 219 T^{2} + 2256 T^{3} + 24682 T^{4} - 382892 T^{5} - 866756 T^{6} + 40107018 T^{7} - 98677820 T^{8} - 2428428608 T^{9} + 18619322966 T^{10} + 64533742905 T^{11} - 1426618140149 T^{12} + 64533742905 p T^{13} + 18619322966 p^{2} T^{14} - 2428428608 p^{3} T^{15} - 98677820 p^{4} T^{16} + 40107018 p^{5} T^{17} - 866756 p^{6} T^{18} - 382892 p^{7} T^{19} + 24682 p^{8} T^{20} + 2256 p^{9} T^{21} - 219 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 6 T - 158 T^{2} - 452 T^{3} + 25558 T^{4} + 100947 T^{5} - 1054744 T^{6} - 21449619 T^{7} - 190172 T^{8} + 1201558268 T^{9} + 12652888861 T^{10} - 51952082111 T^{11} - 866504801022 T^{12} - 51952082111 p T^{13} + 12652888861 p^{2} T^{14} + 1201558268 p^{3} T^{15} - 190172 p^{4} T^{16} - 21449619 p^{5} T^{17} - 1054744 p^{6} T^{18} + 100947 p^{7} T^{19} + 25558 p^{8} T^{20} - 452 p^{9} T^{21} - 158 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 8 T - 87 T^{2} + 2086 T^{3} - 7511 T^{4} - 104423 T^{5} + 2025145 T^{6} - 9453174 T^{7} - 79890902 T^{8} + 1676113420 T^{9} - 6418620129 T^{10} - 56233364259 T^{11} + 977846727627 T^{12} - 56233364259 p T^{13} - 6418620129 p^{2} T^{14} + 1676113420 p^{3} T^{15} - 79890902 p^{4} T^{16} - 9453174 p^{5} T^{17} + 2025145 p^{6} T^{18} - 104423 p^{7} T^{19} - 7511 p^{8} T^{20} + 2086 p^{9} T^{21} - 87 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 21 T + 552 T^{2} + 7365 T^{3} + 108168 T^{4} + 1029909 T^{5} + 10503502 T^{6} + 1029909 p T^{7} + 108168 p^{2} T^{8} + 7365 p^{3} T^{9} + 552 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( ( 1 + T + 181 T^{2} + 797 T^{3} + 15046 T^{4} + 126001 T^{5} + 1041188 T^{6} + 126001 p T^{7} + 15046 p^{2} T^{8} + 797 p^{3} T^{9} + 181 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 - 34 T + 274 T^{2} + 1008 T^{3} + 24830 T^{4} - 899081 T^{5} + 3408808 T^{6} + 17123755 T^{7} + 654386474 T^{8} - 7963303748 T^{9} - 4308732765 T^{10} - 202867350615 T^{11} + 6976154776464 T^{12} - 202867350615 p T^{13} - 4308732765 p^{2} T^{14} - 7963303748 p^{3} T^{15} + 654386474 p^{4} T^{16} + 17123755 p^{5} T^{17} + 3408808 p^{6} T^{18} - 899081 p^{7} T^{19} + 24830 p^{8} T^{20} + 1008 p^{9} T^{21} + 274 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 44 T + 709 T^{2} - 6594 T^{3} + 101865 T^{4} - 1690345 T^{5} + 14037545 T^{6} - 106393506 T^{7} + 1871755556 T^{8} - 19587962148 T^{9} + 104769987163 T^{10} - 1328892312861 T^{11} + 18520826492279 T^{12} - 1328892312861 p T^{13} + 104769987163 p^{2} T^{14} - 19587962148 p^{3} T^{15} + 1871755556 p^{4} T^{16} - 106393506 p^{5} T^{17} + 14037545 p^{6} T^{18} - 1690345 p^{7} T^{19} + 101865 p^{8} T^{20} - 6594 p^{9} T^{21} + 709 p^{10} T^{22} - 44 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - 11 T + 243 T^{2} - 2974 T^{3} + 41947 T^{4} - 395342 T^{5} + 4515440 T^{6} - 395342 p T^{7} + 41947 p^{2} T^{8} - 2974 p^{3} T^{9} + 243 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 - 12 T - 350 T^{2} + 3834 T^{3} + 79315 T^{4} - 724890 T^{5} - 12688285 T^{6} + 91701159 T^{7} + 1583812456 T^{8} - 7590111246 T^{9} - 170176594430 T^{10} + 298340892537 T^{11} + 16744508198807 T^{12} + 298340892537 p T^{13} - 170176594430 p^{2} T^{14} - 7590111246 p^{3} T^{15} + 1583812456 p^{4} T^{16} + 91701159 p^{5} T^{17} - 12688285 p^{6} T^{18} - 724890 p^{7} T^{19} + 79315 p^{8} T^{20} + 3834 p^{9} T^{21} - 350 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.33858123952618020793268587630, −3.19867428819935466716384318663, −2.94032196507401585094190844429, −2.93945740696852761018596005840, −2.81662922118837029766478487300, −2.77558778922723905447479531265, −2.67214527728549115875220848834, −2.44038252360750651613073582100, −2.35532120679122174166424388172, −2.34099311866090706441336826909, −2.24315646565439912957307637623, −2.13928582058603408450679116973, −1.91854730772275985281308253331, −1.80885900056585596825402851655, −1.72396290357897732497574708652, −1.61178769264283595394519327053, −1.53408320983449606364415441197, −1.14452602568307500266244192473, −1.06565837781319769034503606390, −0.926647455697108414610529628188, −0.78438853741718652349538564293, −0.77705436140159678214966653067, −0.75813660728091640594898891803, −0.51359444185754010523978858435, −0.12938571558753131887441652676, 0.12938571558753131887441652676, 0.51359444185754010523978858435, 0.75813660728091640594898891803, 0.77705436140159678214966653067, 0.78438853741718652349538564293, 0.926647455697108414610529628188, 1.06565837781319769034503606390, 1.14452602568307500266244192473, 1.53408320983449606364415441197, 1.61178769264283595394519327053, 1.72396290357897732497574708652, 1.80885900056585596825402851655, 1.91854730772275985281308253331, 2.13928582058603408450679116973, 2.24315646565439912957307637623, 2.34099311866090706441336826909, 2.35532120679122174166424388172, 2.44038252360750651613073582100, 2.67214527728549115875220848834, 2.77558778922723905447479531265, 2.81662922118837029766478487300, 2.93945740696852761018596005840, 2.94032196507401585094190844429, 3.19867428819935466716384318663, 3.33858123952618020793268587630
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.