Dirichlet series
| L(s) = 1 | − 8·3-s + 4·4-s − 6·5-s + 38·9-s − 4·11-s − 32·12-s + 12·13-s + 48·15-s + 10·16-s − 16·17-s − 2·19-s − 24·20-s + 6·23-s + 35·25-s − 136·27-s − 12·29-s − 6·31-s + 4·32-s + 32·33-s + 152·36-s − 96·39-s − 16·41-s + 4·43-s − 16·44-s − 228·45-s − 30·47-s − 80·48-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 2·4-s − 2.68·5-s + 38/3·9-s − 1.20·11-s − 9.23·12-s + 3.32·13-s + 12.3·15-s + 5/2·16-s − 3.88·17-s − 0.458·19-s − 5.36·20-s + 1.25·23-s + 7·25-s − 26.1·27-s − 2.22·29-s − 1.07·31-s + 0.707·32-s + 5.57·33-s + 76/3·36-s − 15.3·39-s − 2.49·41-s + 0.609·43-s − 2.41·44-s − 33.9·45-s − 4.37·47-s − 11.5·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.99915\times 10^{8}\) |
| Root analytic conductor: | \(2.25532\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.4891684374\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4891684374\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( ( 1 - T )^{12} \) | |
| good | 2 | \( 1 - p^{2} T^{2} + 3 p T^{4} - p^{2} T^{5} + p^{2} T^{6} + 3 p^{3} T^{7} - 7 p^{2} T^{8} - 9 p^{3} T^{9} + 13 p^{2} T^{10} + 9 p^{3} T^{11} - 19 p^{2} T^{12} + 9 p^{4} T^{13} + 13 p^{4} T^{14} - 9 p^{6} T^{15} - 7 p^{6} T^{16} + 3 p^{8} T^{17} + p^{8} T^{18} - p^{9} T^{19} + 3 p^{9} T^{20} - p^{12} T^{22} + p^{12} T^{24} \) |
| 3 | \( 1 + 8 T + 26 T^{2} + 40 T^{3} + 25 T^{4} + 28 T^{5} + 170 T^{6} + 152 p T^{7} + 722 T^{8} + 316 p T^{9} + 1732 T^{10} + 4412 T^{11} + 9217 T^{12} + 4412 p T^{13} + 1732 p^{2} T^{14} + 316 p^{4} T^{15} + 722 p^{4} T^{16} + 152 p^{6} T^{17} + 170 p^{6} T^{18} + 28 p^{7} T^{19} + 25 p^{8} T^{20} + 40 p^{9} T^{21} + 26 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + 6 T + T^{2} - 42 T^{3} + p^{2} T^{4} + 248 T^{5} - 718 T^{6} - 404 p T^{7} + 1019 p T^{8} + 8162 T^{9} - 33523 T^{10} - 6754 T^{11} + 222786 T^{12} - 6754 p T^{13} - 33523 p^{2} T^{14} + 8162 p^{3} T^{15} + 1019 p^{5} T^{16} - 404 p^{6} T^{17} - 718 p^{6} T^{18} + 248 p^{7} T^{19} + p^{10} T^{20} - 42 p^{9} T^{21} + p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T - 12 T^{2} - 16 T^{3} + 199 T^{4} - 456 T^{5} - 1528 T^{6} + 14024 T^{7} + 5912 T^{8} - 94460 T^{9} + 46622 p T^{10} + 501964 T^{11} - 7764019 T^{12} + 501964 p T^{13} + 46622 p^{3} T^{14} - 94460 p^{3} T^{15} + 5912 p^{4} T^{16} + 14024 p^{5} T^{17} - 1528 p^{6} T^{18} - 456 p^{7} T^{19} + 199 p^{8} T^{20} - 16 p^{9} T^{21} - 12 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 16 T + 98 T^{2} + 456 T^{3} + 3037 T^{4} + 15380 T^{5} + 51338 T^{6} + 211632 T^{7} + 775718 T^{8} + 1669252 T^{9} + 6707964 T^{10} + 17125676 T^{11} - 10681119 T^{12} + 17125676 p T^{13} + 6707964 p^{2} T^{14} + 1669252 p^{3} T^{15} + 775718 p^{4} T^{16} + 211632 p^{5} T^{17} + 51338 p^{6} T^{18} + 15380 p^{7} T^{19} + 3037 p^{8} T^{20} + 456 p^{9} T^{21} + 98 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 2 T - 93 T^{2} - 154 T^{3} + 4855 T^{4} + 6368 T^{5} - 183456 T^{6} - 166228 T^{7} + 5519715 T^{8} + 2846414 T^{9} - 136536987 T^{10} - 21690610 T^{11} + 2829237246 T^{12} - 21690610 p T^{13} - 136536987 p^{2} T^{14} + 2846414 p^{3} T^{15} + 5519715 p^{4} T^{16} - 166228 p^{5} T^{17} - 183456 p^{6} T^{18} + 6368 p^{7} T^{19} + 4855 p^{8} T^{20} - 154 p^{9} T^{21} - 93 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 6 T - 65 T^{2} + 10 p T^{3} + 3331 T^{4} - 3720 T^{5} - 111600 T^{6} - 56824 T^{7} + 2769057 T^{8} + 128842 p T^{9} - 2245641 p T^{10} - 80054 p^{2} T^{11} + 2015590 p^{2} T^{12} - 80054 p^{3} T^{13} - 2245641 p^{3} T^{14} + 128842 p^{4} T^{15} + 2769057 p^{4} T^{16} - 56824 p^{5} T^{17} - 111600 p^{6} T^{18} - 3720 p^{7} T^{19} + 3331 p^{8} T^{20} + 10 p^{10} T^{21} - 65 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( ( 1 + 6 T + 141 T^{2} + 602 T^{3} + 8386 T^{4} + 27374 T^{5} + 298533 T^{6} + 27374 p T^{7} + 8386 p^{2} T^{8} + 602 p^{3} T^{9} + 141 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 + 6 T - 35 T^{2} - 418 T^{3} - 1827 T^{4} - 1608 T^{5} + 42862 T^{6} + 444728 T^{7} + 3123987 T^{8} + 8956534 T^{9} - 38732219 T^{10} - 442578530 T^{11} - 2531492158 T^{12} - 442578530 p T^{13} - 38732219 p^{2} T^{14} + 8956534 p^{3} T^{15} + 3123987 p^{4} T^{16} + 444728 p^{5} T^{17} + 42862 p^{6} T^{18} - 1608 p^{7} T^{19} - 1827 p^{8} T^{20} - 418 p^{9} T^{21} - 35 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 140 T^{2} + 472 T^{3} + 10823 T^{4} - 58520 T^{5} - 437608 T^{6} + 4120268 T^{7} + 7523416 T^{8} - 158143808 T^{9} + 301119714 T^{10} + 2669961412 T^{11} - 20211964963 T^{12} + 2669961412 p T^{13} + 301119714 p^{2} T^{14} - 158143808 p^{3} T^{15} + 7523416 p^{4} T^{16} + 4120268 p^{5} T^{17} - 437608 p^{6} T^{18} - 58520 p^{7} T^{19} + 10823 p^{8} T^{20} + 472 p^{9} T^{21} - 140 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( ( 1 + 8 T + 126 T^{2} + 672 T^{3} + 7255 T^{4} + 35160 T^{5} + 337924 T^{6} + 35160 p T^{7} + 7255 p^{2} T^{8} + 672 p^{3} T^{9} + 126 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 2 T + 97 T^{2} - 618 T^{3} + 6618 T^{4} - 32018 T^{5} + 404609 T^{6} - 32018 p T^{7} + 6618 p^{2} T^{8} - 618 p^{3} T^{9} + 97 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 30 T + 403 T^{2} + 3770 T^{3} + 31975 T^{4} + 238920 T^{5} + 1565760 T^{6} + 10158260 T^{7} + 54546387 T^{8} + 198782210 T^{9} + 719638821 T^{10} + 2337630490 T^{11} + 2305971166 T^{12} + 2337630490 p T^{13} + 719638821 p^{2} T^{14} + 198782210 p^{3} T^{15} + 54546387 p^{4} T^{16} + 10158260 p^{5} T^{17} + 1565760 p^{6} T^{18} + 238920 p^{7} T^{19} + 31975 p^{8} T^{20} + 3770 p^{9} T^{21} + 403 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 14 T - 85 T^{2} + 2126 T^{3} + 1051 T^{4} - 146236 T^{5} + 128280 T^{6} + 4608996 T^{7} + 16987017 T^{8} - 3205558 T^{9} - 3598309075 T^{10} - 2641075506 T^{11} + 271815701350 T^{12} - 2641075506 p T^{13} - 3598309075 p^{2} T^{14} - 3205558 p^{3} T^{15} + 16987017 p^{4} T^{16} + 4608996 p^{5} T^{17} + 128280 p^{6} T^{18} - 146236 p^{7} T^{19} + 1051 p^{8} T^{20} + 2126 p^{9} T^{21} - 85 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 24 T + 76 T^{2} - 2072 T^{3} - 4299 T^{4} + 244012 T^{5} + 967628 T^{6} - 8681832 T^{7} + 19176022 T^{8} + 604731732 T^{9} - 1976089600 T^{10} + 3874109316 T^{11} + 405184017425 T^{12} + 3874109316 p T^{13} - 1976089600 p^{2} T^{14} + 604731732 p^{3} T^{15} + 19176022 p^{4} T^{16} - 8681832 p^{5} T^{17} + 967628 p^{6} T^{18} + 244012 p^{7} T^{19} - 4299 p^{8} T^{20} - 2072 p^{9} T^{21} + 76 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 120 T^{2} + 224 T^{3} + 2357 T^{4} - 13168 T^{5} + 194552 T^{6} - 1397504 T^{7} + 2622230 T^{8} + 238353296 T^{9} - 702680024 T^{10} - 8247909040 T^{11} + 31598597553 T^{12} - 8247909040 p T^{13} - 702680024 p^{2} T^{14} + 238353296 p^{3} T^{15} + 2622230 p^{4} T^{16} - 1397504 p^{5} T^{17} + 194552 p^{6} T^{18} - 13168 p^{7} T^{19} + 2357 p^{8} T^{20} + 224 p^{9} T^{21} - 120 p^{10} T^{22} + p^{12} T^{24} \) | |
| 67 | \( 1 + 16 T - 146 T^{2} - 2688 T^{3} + 23293 T^{4} + 303040 T^{5} - 2944902 T^{6} - 23480208 T^{7} + 307931898 T^{8} + 1246418064 T^{9} - 26862783418 T^{10} - 32870663168 T^{11} + 1943492323461 T^{12} - 32870663168 p T^{13} - 26862783418 p^{2} T^{14} + 1246418064 p^{3} T^{15} + 307931898 p^{4} T^{16} - 23480208 p^{5} T^{17} - 2944902 p^{6} T^{18} + 303040 p^{7} T^{19} + 23293 p^{8} T^{20} - 2688 p^{9} T^{21} - 146 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( ( 1 - 8 T + 110 T^{2} - 984 T^{3} + 19145 T^{4} - 111980 T^{5} + 1119230 T^{6} - 111980 p T^{7} + 19145 p^{2} T^{8} - 984 p^{3} T^{9} + 110 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 73 | \( 1 - 6 T - 161 T^{2} + 1582 T^{3} + 8943 T^{4} - 178388 T^{5} + 40824 T^{6} + 13043464 T^{7} - 65085837 T^{8} - 889056078 T^{9} + 10696836305 T^{10} + 31531313154 T^{11} - 1015259317234 T^{12} + 31531313154 p T^{13} + 10696836305 p^{2} T^{14} - 889056078 p^{3} T^{15} - 65085837 p^{4} T^{16} + 13043464 p^{5} T^{17} + 40824 p^{6} T^{18} - 178388 p^{7} T^{19} + 8943 p^{8} T^{20} + 1582 p^{9} T^{21} - 161 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 22 T + 75 T^{2} + 2038 T^{3} - 16061 T^{4} - 161980 T^{5} + 2588728 T^{6} - 578752 T^{7} - 158888523 T^{8} + 127898574 T^{9} + 11715108749 T^{10} - 19683798394 T^{11} - 651320994834 T^{12} - 19683798394 p T^{13} + 11715108749 p^{2} T^{14} + 127898574 p^{3} T^{15} - 158888523 p^{4} T^{16} - 578752 p^{5} T^{17} + 2588728 p^{6} T^{18} - 161980 p^{7} T^{19} - 16061 p^{8} T^{20} + 2038 p^{9} T^{21} + 75 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( ( 1 - 50 T + 1439 T^{2} - 28742 T^{3} + 5324 p T^{4} - 5421474 T^{5} + 54504063 T^{6} - 5421474 p T^{7} + 5324 p^{3} T^{8} - 28742 p^{3} T^{9} + 1439 p^{4} T^{10} - 50 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 26 T + 11 T^{2} - 3474 T^{3} + 15895 T^{4} + 513624 T^{5} - 4070664 T^{6} - 51638768 T^{7} + 745034871 T^{8} + 4398459658 T^{9} - 87039924331 T^{10} - 176826779450 T^{11} + 8109085005622 T^{12} - 176826779450 p T^{13} - 87039924331 p^{2} T^{14} + 4398459658 p^{3} T^{15} + 745034871 p^{4} T^{16} - 51638768 p^{5} T^{17} - 4070664 p^{6} T^{18} + 513624 p^{7} T^{19} + 15895 p^{8} T^{20} - 3474 p^{9} T^{21} + 11 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 14 T + 321 T^{2} + 3170 T^{3} + 53038 T^{4} + 457742 T^{5} + 6291427 T^{6} + 457742 p T^{7} + 53038 p^{2} T^{8} + 3170 p^{3} T^{9} + 321 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.58802049802836212621727640326, −3.47258557122457648877907316510, −3.41146337650272776619838796079, −3.27526661618185708618707421708, −3.17837679678233812712082468460, −2.76446926278301755068372521098, −2.76326822632018647138816443641, −2.69861898932120978669218945956, −2.68141291035781260462684965015, −2.58697800231998390392696015595, −2.54336117366787452554649954586, −2.13477671766083700704479983465, −2.00341341663886294148050333056, −1.98144921099396773280485559571, −1.66341684139127248045674183310, −1.63037130036487967972601267287, −1.50231218827177621438038059862, −1.41336570396533091983992863286, −1.41226558446146016876854709612, −1.30152600034902563566890766782, −0.911221143111126059948230595038, −0.71832641340275595539265413317, −0.42196071163705207331399610011, −0.40944254555089205145594104064, −0.24882245791168718362197934149, 0.24882245791168718362197934149, 0.40944254555089205145594104064, 0.42196071163705207331399610011, 0.71832641340275595539265413317, 0.911221143111126059948230595038, 1.30152600034902563566890766782, 1.41226558446146016876854709612, 1.41336570396533091983992863286, 1.50231218827177621438038059862, 1.63037130036487967972601267287, 1.66341684139127248045674183310, 1.98144921099396773280485559571, 2.00341341663886294148050333056, 2.13477671766083700704479983465, 2.54336117366787452554649954586, 2.58697800231998390392696015595, 2.68141291035781260462684965015, 2.69861898932120978669218945956, 2.76326822632018647138816443641, 2.76446926278301755068372521098, 3.17837679678233812712082468460, 3.27526661618185708618707421708, 3.41146337650272776619838796079, 3.47258557122457648877907316510, 3.58802049802836212621727640326
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.