Dirichlet series
| L(s) = 1 | + 2·2-s + 3-s + 6·4-s + 5-s + 2·6-s − 6·7-s + 10·8-s + 11·9-s + 2·10-s + 4·11-s + 6·12-s − 12·14-s + 15-s + 22·16-s + 5·17-s + 22·18-s − 19-s + 6·20-s − 6·21-s + 8·22-s − 23-s + 10·24-s + 19·25-s + 10·27-s − 36·28-s − 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3·4-s + 0.447·5-s + 0.816·6-s − 2.26·7-s + 3.53·8-s + 11/3·9-s + 0.632·10-s + 1.20·11-s + 1.73·12-s − 3.20·14-s + 0.258·15-s + 11/2·16-s + 1.21·17-s + 5.18·18-s − 0.229·19-s + 1.34·20-s − 1.30·21-s + 1.70·22-s − 0.208·23-s + 2.04·24-s + 19/5·25-s + 1.92·27-s − 6.80·28-s − 1.11·29-s + 0.365·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.04826\times 10^{11}\) |
| Root analytic conductor: | \(3.07348\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(516.7067971\) |
| \(L(\frac12)\) | \(\approx\) | \(516.7067971\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 + 6 T + 15 T^{2} + 54 T^{3} + 219 T^{4} + 516 T^{5} + 1069 T^{6} + 516 p T^{7} + 219 p^{2} T^{8} + 54 p^{3} T^{9} + 15 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T - p T^{2} + 3 p T^{3} - p T^{4} - p T^{5} + 7 T^{6} - p T^{7} - 7 p T^{8} + 3 p T^{9} - 5 p^{2} T^{10} - 5 p^{2} T^{11} + 153 T^{12} - 5 p^{3} T^{13} - 5 p^{4} T^{14} + 3 p^{4} T^{15} - 7 p^{5} T^{16} - p^{6} T^{17} + 7 p^{6} T^{18} - p^{8} T^{19} - p^{9} T^{20} + 3 p^{10} T^{21} - p^{11} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 - T - 10 T^{2} + 11 T^{3} + 53 T^{4} - 62 T^{5} - 167 T^{6} + 221 T^{7} + 98 p T^{8} - 535 T^{9} + 79 T^{10} + 604 T^{11} - 1559 T^{12} + 604 p T^{13} + 79 p^{2} T^{14} - 535 p^{3} T^{15} + 98 p^{5} T^{16} + 221 p^{5} T^{17} - 167 p^{6} T^{18} - 62 p^{7} T^{19} + 53 p^{8} T^{20} + 11 p^{9} T^{21} - 10 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 - T - 18 T^{2} - p T^{3} + 193 T^{4} + 192 T^{5} - 1181 T^{6} - 2139 T^{7} + 908 p T^{8} + 451 p^{2} T^{9} - 6679 T^{10} - 26266 T^{11} + 249 T^{12} - 26266 p T^{13} - 6679 p^{2} T^{14} + 451 p^{5} T^{15} + 908 p^{5} T^{16} - 2139 p^{5} T^{17} - 1181 p^{6} T^{18} + 192 p^{7} T^{19} + 193 p^{8} T^{20} - p^{10} T^{21} - 18 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 4 T - 29 T^{2} + 108 T^{3} + 477 T^{4} - 113 p T^{5} - 6686 T^{6} + 7665 T^{7} + 89323 T^{8} + 423 T^{9} - 1282040 T^{10} - 249219 T^{11} + 16505087 T^{12} - 249219 p T^{13} - 1282040 p^{2} T^{14} + 423 p^{3} T^{15} + 89323 p^{4} T^{16} + 7665 p^{5} T^{17} - 6686 p^{6} T^{18} - 113 p^{8} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} - 29 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 5 T - 65 T^{2} + 372 T^{3} + 2506 T^{4} - 15344 T^{5} - 62063 T^{6} + 395128 T^{7} + 1158376 T^{8} - 6526599 T^{9} - 17123414 T^{10} + 46016896 T^{11} + 268434807 T^{12} + 46016896 p T^{13} - 17123414 p^{2} T^{14} - 6526599 p^{3} T^{15} + 1158376 p^{4} T^{16} + 395128 p^{5} T^{17} - 62063 p^{6} T^{18} - 15344 p^{7} T^{19} + 2506 p^{8} T^{20} + 372 p^{9} T^{21} - 65 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + T - 49 T^{2} + 82 T^{3} + 1336 T^{4} - 4335 T^{5} - 10907 T^{6} + 99626 T^{7} - 263580 T^{8} - 1110690 T^{9} + 9684539 T^{10} + 2414194 T^{11} - 215227743 T^{12} + 2414194 p T^{13} + 9684539 p^{2} T^{14} - 1110690 p^{3} T^{15} - 263580 p^{4} T^{16} + 99626 p^{5} T^{17} - 10907 p^{6} T^{18} - 4335 p^{7} T^{19} + 1336 p^{8} T^{20} + 82 p^{9} T^{21} - 49 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + T - 31 T^{2} - 72 T^{3} - 242 T^{4} + 619 T^{5} + 6343 T^{6} + 28360 T^{7} + 121372 T^{8} - 9024 p T^{9} + 5143119 T^{10} - 4729028 T^{11} - 273608319 T^{12} - 4729028 p T^{13} + 5143119 p^{2} T^{14} - 9024 p^{4} T^{15} + 121372 p^{4} T^{16} + 28360 p^{5} T^{17} + 6343 p^{6} T^{18} + 619 p^{7} T^{19} - 242 p^{8} T^{20} - 72 p^{9} T^{21} - 31 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 3 T + 96 T^{2} + 191 T^{3} + 4061 T^{4} + 5126 T^{5} + 122643 T^{6} + 5126 p T^{7} + 4061 p^{2} T^{8} + 191 p^{3} T^{9} + 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 - 16 T + 20 T^{2} + 594 T^{3} + 2163 T^{4} - 43649 T^{5} - 56125 T^{6} + 1282696 T^{7} + 2984747 T^{8} - 22743273 T^{9} - 180497697 T^{10} + 655302586 T^{11} + 2182678017 T^{12} + 655302586 p T^{13} - 180497697 p^{2} T^{14} - 22743273 p^{3} T^{15} + 2984747 p^{4} T^{16} + 1282696 p^{5} T^{17} - 56125 p^{6} T^{18} - 43649 p^{7} T^{19} + 2163 p^{8} T^{20} + 594 p^{9} T^{21} + 20 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 13 T - 15 T^{2} + 284 T^{3} + 12996 T^{4} + 18401 T^{5} - 116147 T^{6} + 5523346 T^{7} + 19538810 T^{8} - 71463812 T^{9} + 1452640399 T^{10} + 7689412934 T^{11} - 18842100883 T^{12} + 7689412934 p T^{13} + 1452640399 p^{2} T^{14} - 71463812 p^{3} T^{15} + 19538810 p^{4} T^{16} + 5523346 p^{5} T^{17} - 116147 p^{6} T^{18} + 18401 p^{7} T^{19} + 12996 p^{8} T^{20} + 284 p^{9} T^{21} - 15 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 8 T + 225 T^{2} - 1362 T^{3} + 21488 T^{4} - 101725 T^{5} + 1145451 T^{6} - 101725 p T^{7} + 21488 p^{2} T^{8} - 1362 p^{3} T^{9} + 225 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 11 T + 259 T^{2} - 2099 T^{3} + 27622 T^{4} - 170696 T^{5} + 1576761 T^{6} - 170696 p T^{7} + 27622 p^{2} T^{8} - 2099 p^{3} T^{9} + 259 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + T - 104 T^{2} - 189 T^{3} + 5335 T^{4} + 164 p T^{5} - 69863 T^{6} + 514255 T^{7} - 7627520 T^{8} - 55687467 T^{9} + 662939941 T^{10} + 1686387922 T^{11} - 35399065407 T^{12} + 1686387922 p T^{13} + 662939941 p^{2} T^{14} - 55687467 p^{3} T^{15} - 7627520 p^{4} T^{16} + 514255 p^{5} T^{17} - 69863 p^{6} T^{18} + 164 p^{8} T^{19} + 5335 p^{8} T^{20} - 189 p^{9} T^{21} - 104 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T - 214 T^{2} - 252 T^{3} + 24796 T^{4} + 13772 T^{5} - 1921862 T^{6} + 82142 T^{7} + 113089342 T^{8} - 43114584 T^{9} - 5653831794 T^{10} + 1443208718 T^{11} + 285781391787 T^{12} + 1443208718 p T^{13} - 5653831794 p^{2} T^{14} - 43114584 p^{3} T^{15} + 113089342 p^{4} T^{16} + 82142 p^{5} T^{17} - 1921862 p^{6} T^{18} + 13772 p^{7} T^{19} + 24796 p^{8} T^{20} - 252 p^{9} T^{21} - 214 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 13 T - 126 T^{2} + 1843 T^{3} + 11161 T^{4} - 119322 T^{5} - 1337447 T^{6} + 7367025 T^{7} + 123366322 T^{8} - 382593671 T^{9} - 9090177085 T^{10} + 8156701016 T^{11} + 592237594305 T^{12} + 8156701016 p T^{13} - 9090177085 p^{2} T^{14} - 382593671 p^{3} T^{15} + 123366322 p^{4} T^{16} + 7367025 p^{5} T^{17} - 1337447 p^{6} T^{18} - 119322 p^{7} T^{19} + 11161 p^{8} T^{20} + 1843 p^{9} T^{21} - 126 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 5 T - 140 T^{2} - 373 T^{3} + 8487 T^{4} - 5202 T^{5} - 147441 T^{6} + 963135 T^{7} - 4711566 T^{8} - 13690661 T^{9} - 1296684385 T^{10} - 689962304 T^{11} + 162150963097 T^{12} - 689962304 p T^{13} - 1296684385 p^{2} T^{14} - 13690661 p^{3} T^{15} - 4711566 p^{4} T^{16} + 963135 p^{5} T^{17} - 147441 p^{6} T^{18} - 5202 p^{7} T^{19} + 8487 p^{8} T^{20} - 373 p^{9} T^{21} - 140 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 11 T - 175 T^{2} - 2336 T^{3} + 15663 T^{4} + 247450 T^{5} - 15954 p T^{6} - 18125445 T^{7} + 60512732 T^{8} + 977936543 T^{9} - 2490157221 T^{10} - 26393757979 T^{11} + 95373451231 T^{12} - 26393757979 p T^{13} - 2490157221 p^{2} T^{14} + 977936543 p^{3} T^{15} + 60512732 p^{4} T^{16} - 18125445 p^{5} T^{17} - 15954 p^{7} T^{18} + 247450 p^{7} T^{19} + 15663 p^{8} T^{20} - 2336 p^{9} T^{21} - 175 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 + 6 T + 285 T^{2} + 14 p T^{3} + 35468 T^{4} + 74185 T^{5} + 2901951 T^{6} + 74185 p T^{7} + 35468 p^{2} T^{8} + 14 p^{4} T^{9} + 285 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 + 30 T + 224 T^{2} - 1118 T^{3} - 5021 T^{4} + 290169 T^{5} + 1854677 T^{6} - 9817892 T^{7} - 27971653 T^{8} + 688598777 T^{9} - 1819010273 T^{10} - 20701972840 T^{11} + 235631264151 T^{12} - 20701972840 p T^{13} - 1819010273 p^{2} T^{14} + 688598777 p^{3} T^{15} - 27971653 p^{4} T^{16} - 9817892 p^{5} T^{17} + 1854677 p^{6} T^{18} + 290169 p^{7} T^{19} - 5021 p^{8} T^{20} - 1118 p^{9} T^{21} + 224 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 7 T - 277 T^{2} + 2628 T^{3} + 34995 T^{4} - 387429 T^{5} - 3070086 T^{6} + 26237658 T^{7} + 339376855 T^{8} - 565746882 T^{9} - 45365142063 T^{10} - 8895648284 T^{11} + 4474615429807 T^{12} - 8895648284 p T^{13} - 45365142063 p^{2} T^{14} - 565746882 p^{3} T^{15} + 339376855 p^{4} T^{16} + 26237658 p^{5} T^{17} - 3070086 p^{6} T^{18} - 387429 p^{7} T^{19} + 34995 p^{8} T^{20} + 2628 p^{9} T^{21} - 277 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 4 T - 151 T^{2} + 2628 T^{3} + 262 T^{4} - 309046 T^{5} + 2802769 T^{6} + 6034970 T^{7} - 281402495 T^{8} + 1666391304 T^{9} + 5367237150 T^{10} - 95837476354 T^{11} + 701675320941 T^{12} - 95837476354 p T^{13} + 5367237150 p^{2} T^{14} + 1666391304 p^{3} T^{15} - 281402495 p^{4} T^{16} + 6034970 p^{5} T^{17} + 2802769 p^{6} T^{18} - 309046 p^{7} T^{19} + 262 p^{8} T^{20} + 2628 p^{9} T^{21} - 151 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 35 T + 947 T^{2} - 18161 T^{3} + 281670 T^{4} - 3629766 T^{5} + 38644781 T^{6} - 3629766 p T^{7} + 281670 p^{2} T^{8} - 18161 p^{3} T^{9} + 947 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.13679713754667639444242213842, −3.04542252205656983029221781959, −2.85949635619482549788466658570, −2.85446628604575294612051890057, −2.78283243420235661771960741806, −2.65223387478996395545338192992, −2.61421934187211140168286093510, −2.58637762954337047507637185959, −2.39674073322770788286633990806, −2.18393441743156440932815716528, −1.98717490099525154133052795015, −1.87410683825105462642809504180, −1.86498835157656590535755585547, −1.79404808333237189492533285548, −1.77495561398805324496363258538, −1.76556814732911963227987015914, −1.62481084358282326141045090172, −1.29710065117532689406433266524, −1.16527599217888866878367137103, −0.882986977365847155704218535426, −0.875041436693927283282975098719, −0.826931056324815880384631235460, −0.822757818520953141613265308000, −0.67078506398929520254109184985, −0.46661322384760203045745736940, 0.46661322384760203045745736940, 0.67078506398929520254109184985, 0.822757818520953141613265308000, 0.826931056324815880384631235460, 0.875041436693927283282975098719, 0.882986977365847155704218535426, 1.16527599217888866878367137103, 1.29710065117532689406433266524, 1.62481084358282326141045090172, 1.76556814732911963227987015914, 1.77495561398805324496363258538, 1.79404808333237189492533285548, 1.86498835157656590535755585547, 1.87410683825105462642809504180, 1.98717490099525154133052795015, 2.18393441743156440932815716528, 2.39674073322770788286633990806, 2.58637762954337047507637185959, 2.61421934187211140168286093510, 2.65223387478996395545338192992, 2.78283243420235661771960741806, 2.85446628604575294612051890057, 2.85949635619482549788466658570, 3.04542252205656983029221781959, 3.13679713754667639444242213842
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.