Dirichlet series
| L(s) = 1 | − 4·2-s + 2·4-s + 12·8-s − 20·11-s − 17·16-s + 80·22-s − 32·23-s − 24·25-s − 16·29-s − 12·32-s + 12·37-s − 40·44-s + 128·46-s + 96·50-s − 32·53-s + 64·58-s + 50·64-s + 12·67-s − 56·71-s − 48·74-s − 12·79-s − 240·88-s − 64·92-s − 48·100-s + 128·106-s − 60·107-s − 12·109-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4-s + 4.24·8-s − 6.03·11-s − 4.25·16-s + 17.0·22-s − 6.67·23-s − 4.79·25-s − 2.97·29-s − 2.12·32-s + 1.97·37-s − 6.03·44-s + 18.8·46-s + 13.5·50-s − 4.39·53-s + 8.40·58-s + 25/4·64-s + 1.46·67-s − 6.64·71-s − 5.57·74-s − 1.35·79-s − 25.5·88-s − 6.67·92-s − 4.79·100-s + 12.4·106-s − 5.80·107-s − 1.14·109-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{48} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02682\times 10^{18}\) |
| Root analytic conductor: | \(5.62962\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 3^{48} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | \( 1 \) |
| 7 | \( 1 \) | |
| good | 2 | \( ( 1 + p T + 5 T^{2} + p^{3} T^{3} + 7 p T^{4} + 11 p T^{5} + 31 T^{6} + 11 p^{2} T^{7} + 7 p^{3} T^{8} + p^{6} T^{9} + 5 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( 1 + 24 T^{2} + 63 p T^{4} + 604 p T^{6} + 933 p^{2} T^{8} + 30006 p T^{10} + 814099 T^{12} + 30006 p^{3} T^{14} + 933 p^{6} T^{16} + 604 p^{7} T^{18} + 63 p^{9} T^{20} + 24 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( ( 1 + 10 T + 74 T^{2} + 416 T^{3} + 1956 T^{4} + 7838 T^{5} + 28069 T^{6} + 7838 p T^{7} + 1956 p^{2} T^{8} + 416 p^{3} T^{9} + 74 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 13 | \( 1 + 6 p T^{2} + 3297 T^{4} + 96082 T^{6} + 2131710 T^{8} + 37655910 T^{10} + 541159525 T^{12} + 37655910 p^{2} T^{14} + 2131710 p^{4} T^{16} + 96082 p^{6} T^{18} + 3297 p^{8} T^{20} + 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 + 6 p T^{2} + 5424 T^{4} + 198330 T^{6} + 5544384 T^{8} + 124741560 T^{10} + 2321396831 T^{12} + 124741560 p^{2} T^{14} + 5544384 p^{4} T^{16} + 198330 p^{6} T^{18} + 5424 p^{8} T^{20} + 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 84 T^{2} + 3543 T^{4} + 110022 T^{6} + 2924529 T^{8} + 67365570 T^{10} + 1356656807 T^{12} + 67365570 p^{2} T^{14} + 2924529 p^{4} T^{16} + 110022 p^{6} T^{18} + 3543 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 16 T + 236 T^{2} + 2128 T^{3} + 17366 T^{4} + 104774 T^{5} + 573529 T^{6} + 104774 p T^{7} + 17366 p^{2} T^{8} + 2128 p^{3} T^{9} + 236 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 8 T + 101 T^{2} + 566 T^{3} + 4919 T^{4} + 27100 T^{5} + 182593 T^{6} + 27100 p T^{7} + 4919 p^{2} T^{8} + 566 p^{3} T^{9} + 101 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 180 T^{2} + 16107 T^{4} + 996376 T^{6} + 48243561 T^{8} + 1915110246 T^{10} + 64160027371 T^{12} + 1915110246 p^{2} T^{14} + 48243561 p^{4} T^{16} + 996376 p^{6} T^{18} + 16107 p^{8} T^{20} + 180 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 6 T + 141 T^{2} - 454 T^{3} + 7695 T^{4} - 11826 T^{5} + 289185 T^{6} - 11826 p T^{7} + 7695 p^{2} T^{8} - 454 p^{3} T^{9} + 141 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 210 T^{2} + 25812 T^{4} + 2226030 T^{6} + 149265072 T^{8} + 8078895858 T^{10} + 8856283387 p T^{12} + 8078895858 p^{2} T^{14} + 149265072 p^{4} T^{16} + 2226030 p^{6} T^{18} + 25812 p^{8} T^{20} + 210 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 87 T^{2} + 522 T^{3} + 5457 T^{4} + 24174 T^{5} + 371595 T^{6} + 24174 p T^{7} + 5457 p^{2} T^{8} + 522 p^{3} T^{9} + 87 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 372 T^{2} + 65523 T^{4} + 7295818 T^{6} + 583265421 T^{8} + 36319137786 T^{10} + 1861693717767 T^{12} + 36319137786 p^{2} T^{14} + 583265421 p^{4} T^{16} + 7295818 p^{6} T^{18} + 65523 p^{8} T^{20} + 372 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 + 16 T + 290 T^{2} + 2834 T^{3} + 31722 T^{4} + 236978 T^{5} + 2063893 T^{6} + 236978 p T^{7} + 31722 p^{2} T^{8} + 2834 p^{3} T^{9} + 290 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 6 p T^{2} + 65751 T^{4} + 8465138 T^{6} + 836843829 T^{8} + 66423207186 T^{10} + 4317343305043 T^{12} + 66423207186 p^{2} T^{14} + 836843829 p^{4} T^{16} + 8465138 p^{6} T^{18} + 65751 p^{8} T^{20} + 6 p^{11} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 348 T^{2} + 64080 T^{4} + 8309234 T^{6} + 834479412 T^{8} + 67660436166 T^{10} + 4526819688495 T^{12} + 67660436166 p^{2} T^{14} + 834479412 p^{4} T^{16} + 8309234 p^{6} T^{18} + 64080 p^{8} T^{20} + 348 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 6 T + 306 T^{2} - 1590 T^{3} + 43680 T^{4} - 189840 T^{5} + 3691485 T^{6} - 189840 p T^{7} + 43680 p^{2} T^{8} - 1590 p^{3} T^{9} + 306 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 28 T + 653 T^{2} + 10222 T^{3} + 137939 T^{4} + 1465208 T^{5} + 13714957 T^{6} + 1465208 p T^{7} + 137939 p^{2} T^{8} + 10222 p^{3} T^{9} + 653 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 708 T^{2} + 236331 T^{4} + 49425010 T^{6} + 7249853901 T^{8} + 789454108674 T^{10} + 65655726835543 T^{12} + 789454108674 p^{2} T^{14} + 7249853901 p^{4} T^{16} + 49425010 p^{6} T^{18} + 236331 p^{8} T^{20} + 708 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 6 T + 255 T^{2} + 2108 T^{3} + 35457 T^{4} + 293580 T^{5} + 3413581 T^{6} + 293580 p T^{7} + 35457 p^{2} T^{8} + 2108 p^{3} T^{9} + 255 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 + 594 T^{2} + 174828 T^{4} + 33977966 T^{6} + 4890872196 T^{8} + 553427422020 T^{10} + 50770554002527 T^{12} + 553427422020 p^{2} T^{14} + 4890872196 p^{4} T^{16} + 33977966 p^{6} T^{18} + 174828 p^{8} T^{20} + 594 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 + 822 T^{2} + 322068 T^{4} + 79812294 T^{6} + 13991580948 T^{8} + 1835673681090 T^{10} + 185397377514059 T^{12} + 1835673681090 p^{2} T^{14} + 13991580948 p^{4} T^{16} + 79812294 p^{6} T^{18} + 322068 p^{8} T^{20} + 822 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 + 498 T^{2} + 130071 T^{4} + 24012862 T^{6} + 3539984577 T^{8} + 435867825654 T^{10} + 45695934843163 T^{12} + 435867825654 p^{2} T^{14} + 3539984577 p^{4} T^{16} + 24012862 p^{6} T^{18} + 130071 p^{8} T^{20} + 498 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−2.89244950549249768782577459890, −2.88589082853573816651176782329, −2.85262618705959849714569813588, −2.69534568578854636691310326055, −2.48156367126328371268297274766, −2.47594853371543896608838667043, −2.39089257075746576244522763454, −2.36048574822390884800763465527, −2.25873217048147784925008443487, −2.25536224171240505012866003128, −2.18006143330922281298907267907, −2.17024384709953281110334063299, −2.12137811989962538402511966050, −2.09196809690710769288713025677, −1.92408150059858839455149186004, −1.84263466215454483253148744055, −1.43731319965590403919249904623, −1.36017552595118290642188682528, −1.32100027347833208105545051225, −1.31128314169352260675628879870, −1.29943702587730951221302110698, −1.25129122661590616766574193921, −1.10162603060270831930553861157, −1.04222528772957338004201130543, −0.899922344633579401438742184780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.899922344633579401438742184780, 1.04222528772957338004201130543, 1.10162603060270831930553861157, 1.25129122661590616766574193921, 1.29943702587730951221302110698, 1.31128314169352260675628879870, 1.32100027347833208105545051225, 1.36017552595118290642188682528, 1.43731319965590403919249904623, 1.84263466215454483253148744055, 1.92408150059858839455149186004, 2.09196809690710769288713025677, 2.12137811989962538402511966050, 2.17024384709953281110334063299, 2.18006143330922281298907267907, 2.25536224171240505012866003128, 2.25873217048147784925008443487, 2.36048574822390884800763465527, 2.39089257075746576244522763454, 2.47594853371543896608838667043, 2.48156367126328371268297274766, 2.69534568578854636691310326055, 2.85262618705959849714569813588, 2.88589082853573816651176782329, 2.89244950549249768782577459890
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.