Dirichlet series
| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·8-s + 5·9-s + 4·10-s − 2·11-s − 6·12-s + 2·13-s − 4·15-s + 4·16-s − 24·17-s + 10·18-s − 12·19-s + 6·20-s − 4·22-s + 4·23-s − 8·24-s + 11·25-s + 4·26-s − 6·27-s − 8·30-s − 6·31-s + 4·33-s − 48·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 5/3·9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s + 0.554·13-s − 1.03·15-s + 16-s − 5.82·17-s + 2.35·18-s − 2.75·19-s + 1.34·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s + 11/5·25-s + 0.784·26-s − 1.15·27-s − 1.46·30-s − 1.07·31-s + 0.696·33-s − 8.23·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(29^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.41157\times 10^{9}\) |
| Root analytic conductor: | \(2.59141\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 29^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(24.00934615\) |
| \(L(\frac12)\) | \(\approx\) | \(24.00934615\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - p T + T^{2} + T^{4} + p T^{5} - 11 T^{6} + 7 p T^{7} - 7 T^{8} + p^{3} T^{9} - 7 T^{10} - 19 p T^{11} + 93 T^{12} - 19 p^{2} T^{13} - 7 p^{2} T^{14} + p^{6} T^{15} - 7 p^{4} T^{16} + 7 p^{6} T^{17} - 11 p^{6} T^{18} + p^{8} T^{19} + p^{8} T^{20} + p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 + 2 T - T^{2} - 2 p T^{3} - 4 T^{4} - 2 T^{5} - 11 T^{6} - 124 T^{7} - 169 T^{8} + 26 p^{2} T^{9} + 668 T^{10} + 268 T^{11} + 121 T^{12} + 268 p T^{13} + 668 p^{2} T^{14} + 26 p^{5} T^{15} - 169 p^{4} T^{16} - 124 p^{5} T^{17} - 11 p^{6} T^{18} - 2 p^{7} T^{19} - 4 p^{8} T^{20} - 2 p^{10} T^{21} - p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( ( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} - 56 T^{5} + T^{6} - 56 p T^{7} + 11 p^{2} T^{8} + 9 p^{3} T^{9} - 4 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 7 | \( 1 - 6 T^{2} - 13 T^{4} + 372 T^{6} - 1595 T^{8} - 8658 T^{10} + 130103 T^{12} - 8658 p^{2} T^{14} - 1595 p^{4} T^{16} + 372 p^{6} T^{18} - 13 p^{8} T^{20} - 6 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 2 T - 17 T^{2} - 54 T^{3} + 172 T^{4} + 862 T^{5} - 1019 T^{6} - 9068 T^{7} + 1415 T^{8} + 66538 T^{9} + 27164 T^{10} - 214612 T^{11} + 292953 T^{12} - 214612 p T^{13} + 27164 p^{2} T^{14} + 66538 p^{3} T^{15} + 1415 p^{4} T^{16} - 9068 p^{5} T^{17} - 1019 p^{6} T^{18} + 862 p^{7} T^{19} + 172 p^{8} T^{20} - 54 p^{9} T^{21} - 17 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 2 T - 15 T^{2} + 42 T^{3} + 84 T^{4} - 238 T^{5} + 323 T^{6} + 21140 T^{7} - 56425 T^{8} - 256986 T^{9} + 981820 T^{10} + 813484 T^{11} - 4064087 T^{12} + 813484 p T^{13} + 981820 p^{2} T^{14} - 256986 p^{3} T^{15} - 56425 p^{4} T^{16} + 21140 p^{5} T^{17} + 323 p^{6} T^{18} - 238 p^{7} T^{19} + 84 p^{8} T^{20} + 42 p^{9} T^{21} - 15 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{6} \) | |
| 19 | \( ( 1 + 6 T + 17 T^{2} - 12 T^{3} - 395 T^{4} - 2142 T^{5} - 5347 T^{6} - 2142 p T^{7} - 395 p^{2} T^{8} - 12 p^{3} T^{9} + 17 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 23 | \( 1 - 4 T - 2 T^{2} - 12 T^{3} - 77 T^{4} + 2824 T^{5} - 7748 T^{6} + 17620 T^{7} - 150091 T^{8} - 497876 T^{9} + 7170794 T^{10} - 15234016 T^{11} + 57064503 T^{12} - 15234016 p T^{13} + 7170794 p^{2} T^{14} - 497876 p^{3} T^{15} - 150091 p^{4} T^{16} + 17620 p^{5} T^{17} - 7748 p^{6} T^{18} + 2824 p^{7} T^{19} - 77 p^{8} T^{20} - 12 p^{9} T^{21} - 2 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 6 T + 15 T^{2} + 150 T^{3} + 740 T^{4} - 1686 T^{5} - 12171 T^{6} + 121764 T^{7} - 38561 T^{8} - 3431970 T^{9} + 14562396 T^{10} + 35258196 T^{11} - 695550439 T^{12} + 35258196 p T^{13} + 14562396 p^{2} T^{14} - 3431970 p^{3} T^{15} - 38561 p^{4} T^{16} + 121764 p^{5} T^{17} - 12171 p^{6} T^{18} - 1686 p^{7} T^{19} + 740 p^{8} T^{20} + 150 p^{9} T^{21} + 15 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 - 4 T - 21 T^{2} + 232 T^{3} - 151 T^{4} - 7980 T^{5} + 37507 T^{6} - 7980 p T^{7} - 151 p^{2} T^{8} + 232 p^{3} T^{9} - 21 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{6} \) | |
| 43 | \( 1 + 10 T - 9 T^{2} - 750 T^{3} - 4068 T^{4} + 18710 T^{5} + 324653 T^{6} + 1377620 T^{7} - 6043945 T^{8} - 105594030 T^{9} - 405057092 T^{10} + 2313062620 T^{11} + 33051670633 T^{12} + 2313062620 p T^{13} - 405057092 p^{2} T^{14} - 105594030 p^{3} T^{15} - 6043945 p^{4} T^{16} + 1377620 p^{5} T^{17} + 324653 p^{6} T^{18} + 18710 p^{7} T^{19} - 4068 p^{8} T^{20} - 750 p^{9} T^{21} - 9 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 2 T - 73 T^{2} - 206 T^{3} + 3188 T^{4} + 10958 T^{5} - 81667 T^{6} - 1120740 T^{7} - 1965361 T^{8} + 50483338 T^{9} + 327352316 T^{10} - 941541668 T^{11} - 18002295447 T^{12} - 941541668 p T^{13} + 327352316 p^{2} T^{14} + 50483338 p^{3} T^{15} - 1965361 p^{4} T^{16} - 1120740 p^{5} T^{17} - 81667 p^{6} T^{18} + 10958 p^{7} T^{19} + 3188 p^{8} T^{20} - 206 p^{9} T^{21} - 73 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 2 T - 31 T^{2} - 26 T^{3} - 1564 T^{4} - 11122 T^{5} + 116819 T^{6} - 276804 T^{7} - 1427929 T^{8} + 50456794 T^{9} - 206594692 T^{10} - 1552995212 T^{11} + 13484296521 T^{12} - 1552995212 p T^{13} - 206594692 p^{2} T^{14} + 50456794 p^{3} T^{15} - 1427929 p^{4} T^{16} - 276804 p^{5} T^{17} + 116819 p^{6} T^{18} - 11122 p^{7} T^{19} - 1564 p^{8} T^{20} - 26 p^{9} T^{21} - 31 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{6} \) | |
| 61 | \( 1 - 4 T - 102 T^{2} + 636 T^{3} + 6747 T^{4} - 62264 T^{5} - 322732 T^{6} + 3045052 T^{7} + 15988997 T^{8} - 112841172 T^{9} - 977443874 T^{10} + 1993080032 T^{11} + 75404245135 T^{12} + 1993080032 p T^{13} - 977443874 p^{2} T^{14} - 112841172 p^{3} T^{15} + 15988997 p^{4} T^{16} + 3045052 p^{5} T^{17} - 322732 p^{6} T^{18} - 62264 p^{7} T^{19} + 6747 p^{8} T^{20} + 636 p^{9} T^{21} - 102 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 102 T^{2} + 5915 T^{4} - 145452 T^{6} - 11716331 T^{8} + 1847999790 T^{10} - 135901368721 T^{12} + 1847999790 p^{2} T^{14} - 11716331 p^{4} T^{16} - 145452 p^{6} T^{18} + 5915 p^{8} T^{20} - 102 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 12 T - 26 T^{2} + 1500 T^{3} - 8397 T^{4} - 71592 T^{5} + 1139660 T^{6} - 3794628 T^{7} - 44881003 T^{8} + 573563748 T^{9} - 1764997470 T^{10} - 18958533216 T^{11} + 265120791319 T^{12} - 18958533216 p T^{13} - 1764997470 p^{2} T^{14} + 573563748 p^{3} T^{15} - 44881003 p^{4} T^{16} - 3794628 p^{5} T^{17} + 1139660 p^{6} T^{18} - 71592 p^{7} T^{19} - 8397 p^{8} T^{20} + 1500 p^{9} T^{21} - 26 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 + 4 T - 57 T^{2} - 520 T^{3} + 2081 T^{4} + 46284 T^{5} + 33223 T^{6} + 46284 p T^{7} + 2081 p^{2} T^{8} - 520 p^{3} T^{9} - 57 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 - 2 T - 153 T^{2} + 462 T^{3} + 17172 T^{4} - 70222 T^{5} - 1673683 T^{6} + 3051524 T^{7} + 160589807 T^{8} - 97571466 T^{9} - 15054141956 T^{10} + 1009253284 T^{11} + 1374657086665 T^{12} + 1009253284 p T^{13} - 15054141956 p^{2} T^{14} - 97571466 p^{3} T^{15} + 160589807 p^{4} T^{16} + 3051524 p^{5} T^{17} - 1673683 p^{6} T^{18} - 70222 p^{7} T^{19} + 17172 p^{8} T^{20} + 462 p^{9} T^{21} - 153 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 4 T - 122 T^{2} - 708 T^{3} + 8443 T^{4} + 63416 T^{5} - 306068 T^{6} - 11486980 T^{7} - 40820251 T^{8} + 883434836 T^{9} + 7461759074 T^{10} - 26485489184 T^{11} - 561153634257 T^{12} - 26485489184 p T^{13} + 7461759074 p^{2} T^{14} + 883434836 p^{3} T^{15} - 40820251 p^{4} T^{16} - 11486980 p^{5} T^{17} - 306068 p^{6} T^{18} + 63416 p^{7} T^{19} + 8443 p^{8} T^{20} - 708 p^{9} T^{21} - 122 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 8 T - 58 T^{2} + 728 T^{3} - 973 T^{4} + 23632 T^{5} - 129268 T^{6} + 18797016 T^{7} - 143724283 T^{8} - 1238591944 T^{9} + 15083554514 T^{10} - 16119693184 T^{11} + 309081404535 T^{12} - 16119693184 p T^{13} + 15083554514 p^{2} T^{14} - 1238591944 p^{3} T^{15} - 143724283 p^{4} T^{16} + 18797016 p^{5} T^{17} - 129268 p^{6} T^{18} + 23632 p^{7} T^{19} - 973 p^{8} T^{20} + 728 p^{9} T^{21} - 58 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 8 T - 74 T^{2} + 920 T^{3} - 349 T^{4} + 8528 T^{5} - 37716 T^{6} + 26604056 T^{7} - 210961627 T^{8} - 2034639048 T^{9} + 25099939298 T^{10} - 6630666112 T^{11} + 153071555463 T^{12} - 6630666112 p T^{13} + 25099939298 p^{2} T^{14} - 2034639048 p^{3} T^{15} - 210961627 p^{4} T^{16} + 26604056 p^{5} T^{17} - 37716 p^{6} T^{18} + 8528 p^{7} T^{19} - 349 p^{8} T^{20} + 920 p^{9} T^{21} - 74 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.20712174088636956492492552182, −3.08809539918140425934697908220, −3.02964510995801303773866080692, −3.02672991394293779209960180779, −2.79757472611951062168404157234, −2.79530424399108927083643204803, −2.65832785419631649050235224406, −2.56996292500046844146149735898, −2.45331475648531914849479260377, −2.36580081380291123328599448126, −2.29504996597483172484529119019, −2.02103877084818031845182020650, −1.97417881063653645072917151907, −1.96905803319852487910584569694, −1.96607414579134336926880466710, −1.80912186519909524031343718907, −1.80488654398499356800401042462, −1.58835952916660353698575374839, −1.28218722185610260080672308533, −0.966415505681229254605213773632, −0.792948669759499669336840440010, −0.78844514174138766580760551541, −0.64211670183940325012502179823, −0.56054857004110585455592721097, −0.31125417577498656784524517216, 0.31125417577498656784524517216, 0.56054857004110585455592721097, 0.64211670183940325012502179823, 0.78844514174138766580760551541, 0.792948669759499669336840440010, 0.966415505681229254605213773632, 1.28218722185610260080672308533, 1.58835952916660353698575374839, 1.80488654398499356800401042462, 1.80912186519909524031343718907, 1.96607414579134336926880466710, 1.96905803319852487910584569694, 1.97417881063653645072917151907, 2.02103877084818031845182020650, 2.29504996597483172484529119019, 2.36580081380291123328599448126, 2.45331475648531914849479260377, 2.56996292500046844146149735898, 2.65832785419631649050235224406, 2.79530424399108927083643204803, 2.79757472611951062168404157234, 3.02672991394293779209960180779, 3.02964510995801303773866080692, 3.08809539918140425934697908220, 3.20712174088636956492492552182
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.