Dirichlet series
| L(s) = 1 | − 12·2-s − 3·3-s + 78·4-s + 3·5-s + 36·6-s − 364·8-s − 5·9-s − 36·10-s − 2·11-s − 234·12-s − 13·13-s − 9·15-s + 1.36e3·16-s − 5·17-s + 60·18-s − 14·19-s + 234·20-s + 24·22-s + 2·23-s + 1.09e3·24-s − 15·25-s + 156·26-s + 25·27-s + 7·29-s + 108·30-s − 15·31-s − 4.36e3·32-s + ⋯ |
| L(s) = 1 | − 8.48·2-s − 1.73·3-s + 39·4-s + 1.34·5-s + 14.6·6-s − 128.·8-s − 5/3·9-s − 11.3·10-s − 0.603·11-s − 67.5·12-s − 3.60·13-s − 2.32·15-s + 341.·16-s − 1.21·17-s + 14.1·18-s − 3.21·19-s + 52.3·20-s + 5.11·22-s + 0.417·23-s + 222.·24-s − 3·25-s + 30.5·26-s + 4.81·27-s + 1.29·29-s + 19.7·30-s − 2.69·31-s − 772.·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{24} \cdot 79^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{24} \cdot 79^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{24} \cdot 79^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.11575\times 10^{21}\) |
| Root analytic conductor: | \(7.86258\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{24} \cdot 79^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | 2 | \( ( 1 + T )^{12} \) |
| 7 | \( 1 \) | |
| 79 | \( ( 1 + T )^{12} \) | |
| good | 3 | \( 1 + p T + 14 T^{2} + 32 T^{3} + 82 T^{4} + 49 p T^{5} + 92 p T^{6} + 431 T^{7} + 10 p^{4} T^{8} + 475 p T^{9} + 3197 T^{10} + 5704 T^{11} + 11561 T^{12} + 5704 p T^{13} + 3197 p^{2} T^{14} + 475 p^{4} T^{15} + 10 p^{8} T^{16} + 431 p^{5} T^{17} + 92 p^{7} T^{18} + 49 p^{8} T^{19} + 82 p^{8} T^{20} + 32 p^{9} T^{21} + 14 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 3 T + 24 T^{2} - 12 p T^{3} + 332 T^{4} - 788 T^{5} + 3444 T^{6} - 7541 T^{7} + 27799 T^{8} - 56253 T^{9} + 36699 p T^{10} - 342806 T^{11} + 1006757 T^{12} - 342806 p T^{13} + 36699 p^{3} T^{14} - 56253 p^{3} T^{15} + 27799 p^{4} T^{16} - 7541 p^{5} T^{17} + 3444 p^{6} T^{18} - 788 p^{7} T^{19} + 332 p^{8} T^{20} - 12 p^{10} T^{21} + 24 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 2 T + 84 T^{2} + 134 T^{3} + 3493 T^{4} + 4573 T^{5} + 95427 T^{6} + 104224 T^{7} + 173708 p T^{8} + 1776895 T^{9} + 2690674 p T^{10} + 24012856 T^{11} + 363953863 T^{12} + 24012856 p T^{13} + 2690674 p^{3} T^{14} + 1776895 p^{3} T^{15} + 173708 p^{5} T^{16} + 104224 p^{5} T^{17} + 95427 p^{6} T^{18} + 4573 p^{7} T^{19} + 3493 p^{8} T^{20} + 134 p^{9} T^{21} + 84 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + p T + 184 T^{2} + 1604 T^{3} + 13803 T^{4} + 92721 T^{5} + 602668 T^{6} + 3312721 T^{7} + 17530209 T^{8} + 81254737 T^{9} + 362153620 T^{10} + 1436431325 T^{11} + 5477423135 T^{12} + 1436431325 p T^{13} + 362153620 p^{2} T^{14} + 81254737 p^{3} T^{15} + 17530209 p^{4} T^{16} + 3312721 p^{5} T^{17} + 602668 p^{6} T^{18} + 92721 p^{7} T^{19} + 13803 p^{8} T^{20} + 1604 p^{9} T^{21} + 184 p^{10} T^{22} + p^{12} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 5 T + 107 T^{2} + 506 T^{3} + 6089 T^{4} + 27503 T^{5} + 235965 T^{6} + 1015360 T^{7} + 6894964 T^{8} + 27989556 T^{9} + 159560519 T^{10} + 600522553 T^{11} + 2998861955 T^{12} + 600522553 p T^{13} + 159560519 p^{2} T^{14} + 27989556 p^{3} T^{15} + 6894964 p^{4} T^{16} + 1015360 p^{5} T^{17} + 235965 p^{6} T^{18} + 27503 p^{7} T^{19} + 6089 p^{8} T^{20} + 506 p^{9} T^{21} + 107 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 14 T + 11 p T^{2} + 91 p T^{3} + 14536 T^{4} + 82090 T^{5} + 25701 p T^{6} + 1968607 T^{7} + 9480104 T^{8} + 28057900 T^{9} + 137429411 T^{10} + 340679307 T^{11} + 2228652507 T^{12} + 340679307 p T^{13} + 137429411 p^{2} T^{14} + 28057900 p^{3} T^{15} + 9480104 p^{4} T^{16} + 1968607 p^{5} T^{17} + 25701 p^{7} T^{18} + 82090 p^{7} T^{19} + 14536 p^{8} T^{20} + 91 p^{10} T^{21} + 11 p^{11} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 2 T + 134 T^{2} - 146 T^{3} + 8284 T^{4} - 1696 T^{5} + 336287 T^{6} + 161593 T^{7} + 11121779 T^{8} + 7933992 T^{9} + 328590378 T^{10} + 202359882 T^{11} + 8324446997 T^{12} + 202359882 p T^{13} + 328590378 p^{2} T^{14} + 7933992 p^{3} T^{15} + 11121779 p^{4} T^{16} + 161593 p^{5} T^{17} + 336287 p^{6} T^{18} - 1696 p^{7} T^{19} + 8284 p^{8} T^{20} - 146 p^{9} T^{21} + 134 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 7 T + 217 T^{2} - 1482 T^{3} + 23286 T^{4} - 150615 T^{5} + 1644847 T^{6} - 9853442 T^{7} + 85740195 T^{8} - 467669851 T^{9} + 3481356304 T^{10} - 17115670215 T^{11} + 112707704300 T^{12} - 17115670215 p T^{13} + 3481356304 p^{2} T^{14} - 467669851 p^{3} T^{15} + 85740195 p^{4} T^{16} - 9853442 p^{5} T^{17} + 1644847 p^{6} T^{18} - 150615 p^{7} T^{19} + 23286 p^{8} T^{20} - 1482 p^{9} T^{21} + 217 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 15 T + 303 T^{2} + 3306 T^{3} + 40812 T^{4} + 362691 T^{5} + 3444430 T^{6} + 26131755 T^{7} + 206638463 T^{8} + 1369084181 T^{9} + 9347147995 T^{10} + 54657374364 T^{11} + 327784892059 T^{12} + 54657374364 p T^{13} + 9347147995 p^{2} T^{14} + 1369084181 p^{3} T^{15} + 206638463 p^{4} T^{16} + 26131755 p^{5} T^{17} + 3444430 p^{6} T^{18} + 362691 p^{7} T^{19} + 40812 p^{8} T^{20} + 3306 p^{9} T^{21} + 303 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 15 T + 339 T^{2} + 3765 T^{3} + 52657 T^{4} + 483406 T^{5} + 5207176 T^{6} + 41070442 T^{7} + 368305231 T^{8} + 2548971342 T^{9} + 19745011217 T^{10} + 120977210879 T^{11} + 824287977749 T^{12} + 120977210879 p T^{13} + 19745011217 p^{2} T^{14} + 2548971342 p^{3} T^{15} + 368305231 p^{4} T^{16} + 41070442 p^{5} T^{17} + 5207176 p^{6} T^{18} + 483406 p^{7} T^{19} + 52657 p^{8} T^{20} + 3765 p^{9} T^{21} + 339 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + 10 T + 263 T^{2} + 2388 T^{3} + 35880 T^{4} + 6989 p T^{5} + 3291002 T^{6} + 23429442 T^{7} + 227031460 T^{8} + 35756612 p T^{9} + 12509338252 T^{10} + 73613145090 T^{11} + 564952615995 T^{12} + 73613145090 p T^{13} + 12509338252 p^{2} T^{14} + 35756612 p^{4} T^{15} + 227031460 p^{4} T^{16} + 23429442 p^{5} T^{17} + 3291002 p^{6} T^{18} + 6989 p^{8} T^{19} + 35880 p^{8} T^{20} + 2388 p^{9} T^{21} + 263 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 8 T + 315 T^{2} - 2503 T^{3} + 50558 T^{4} - 386499 T^{5} + 5399927 T^{6} - 38946595 T^{7} + 425266124 T^{8} - 2843444083 T^{9} + 25911681947 T^{10} - 157784054598 T^{11} + 1248776590211 T^{12} - 157784054598 p T^{13} + 25911681947 p^{2} T^{14} - 2843444083 p^{3} T^{15} + 425266124 p^{4} T^{16} - 38946595 p^{5} T^{17} + 5399927 p^{6} T^{18} - 386499 p^{7} T^{19} + 50558 p^{8} T^{20} - 2503 p^{9} T^{21} + 315 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 13 T + 365 T^{2} - 3651 T^{3} + 58537 T^{4} - 473486 T^{5} + 5784820 T^{6} - 40175006 T^{7} + 425164083 T^{8} - 2684406378 T^{9} + 25895698331 T^{10} - 151334224015 T^{11} + 1331595048367 T^{12} - 151334224015 p T^{13} + 25895698331 p^{2} T^{14} - 2684406378 p^{3} T^{15} + 425164083 p^{4} T^{16} - 40175006 p^{5} T^{17} + 5784820 p^{6} T^{18} - 473486 p^{7} T^{19} + 58537 p^{8} T^{20} - 3651 p^{9} T^{21} + 365 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 8 T + 290 T^{2} - 1679 T^{3} + 38316 T^{4} - 180391 T^{5} + 3520405 T^{6} - 15469634 T^{7} + 267429470 T^{8} - 1114096108 T^{9} + 17011165264 T^{10} - 65294203929 T^{11} + 942357390937 T^{12} - 65294203929 p T^{13} + 17011165264 p^{2} T^{14} - 1114096108 p^{3} T^{15} + 267429470 p^{4} T^{16} - 15469634 p^{5} T^{17} + 3520405 p^{6} T^{18} - 180391 p^{7} T^{19} + 38316 p^{8} T^{20} - 1679 p^{9} T^{21} + 290 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 10 T + 356 T^{2} + 2711 T^{3} + 62131 T^{4} + 405184 T^{5} + 7627036 T^{6} + 45007975 T^{7} + 735066215 T^{8} + 3961326628 T^{9} + 57374523121 T^{10} + 281933410970 T^{11} + 3701183400765 T^{12} + 281933410970 p T^{13} + 57374523121 p^{2} T^{14} + 3961326628 p^{3} T^{15} + 735066215 p^{4} T^{16} + 45007975 p^{5} T^{17} + 7627036 p^{6} T^{18} + 405184 p^{7} T^{19} + 62131 p^{8} T^{20} + 2711 p^{9} T^{21} + 356 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 6 T + 527 T^{2} - 3366 T^{3} + 133045 T^{4} - 889337 T^{5} + 21438825 T^{6} - 146147804 T^{7} + 2474336027 T^{8} - 16599176192 T^{9} + 216937814268 T^{10} - 1367905022224 T^{11} + 14887740436577 T^{12} - 1367905022224 p T^{13} + 216937814268 p^{2} T^{14} - 16599176192 p^{3} T^{15} + 2474336027 p^{4} T^{16} - 146147804 p^{5} T^{17} + 21438825 p^{6} T^{18} - 889337 p^{7} T^{19} + 133045 p^{8} T^{20} - 3366 p^{9} T^{21} + 527 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 18 T + 346 T^{2} - 3285 T^{3} + 42366 T^{4} - 345292 T^{5} + 4761694 T^{6} - 40074031 T^{7} + 490554359 T^{8} - 3598474301 T^{9} + 38973653880 T^{10} - 263086845667 T^{11} + 2736733664820 T^{12} - 263086845667 p T^{13} + 38973653880 p^{2} T^{14} - 3598474301 p^{3} T^{15} + 490554359 p^{4} T^{16} - 40074031 p^{5} T^{17} + 4761694 p^{6} T^{18} - 345292 p^{7} T^{19} + 42366 p^{8} T^{20} - 3285 p^{9} T^{21} + 346 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 21 T + 602 T^{2} + 9406 T^{3} + 167887 T^{4} + 2154666 T^{5} + 29893581 T^{6} + 328997880 T^{7} + 3841642558 T^{8} + 37257435386 T^{9} + 380649693354 T^{10} + 3300656241124 T^{11} + 30086577979573 T^{12} + 3300656241124 p T^{13} + 380649693354 p^{2} T^{14} + 37257435386 p^{3} T^{15} + 3841642558 p^{4} T^{16} + 328997880 p^{5} T^{17} + 29893581 p^{6} T^{18} + 2154666 p^{7} T^{19} + 167887 p^{8} T^{20} + 9406 p^{9} T^{21} + 602 p^{10} T^{22} + 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 45 T + 1476 T^{2} + 35041 T^{3} + 701715 T^{4} + 11907723 T^{5} + 179666638 T^{6} + 2419899214 T^{7} + 29722801717 T^{8} + 333523977607 T^{9} + 47343921572 p T^{10} + 33059751406354 T^{11} + 293555059233443 T^{12} + 33059751406354 p T^{13} + 47343921572 p^{3} T^{14} + 333523977607 p^{3} T^{15} + 29722801717 p^{4} T^{16} + 2419899214 p^{5} T^{17} + 179666638 p^{6} T^{18} + 11907723 p^{7} T^{19} + 701715 p^{8} T^{20} + 35041 p^{9} T^{21} + 1476 p^{10} T^{22} + 45 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 15 T + 758 T^{2} - 8977 T^{3} + 258589 T^{4} - 2497428 T^{5} + 53937556 T^{6} - 433240708 T^{7} + 7880713636 T^{8} - 53805123113 T^{9} + 880851747894 T^{10} - 5286752750986 T^{11} + 80095990246935 T^{12} - 5286752750986 p T^{13} + 880851747894 p^{2} T^{14} - 53805123113 p^{3} T^{15} + 7880713636 p^{4} T^{16} - 433240708 p^{5} T^{17} + 53937556 p^{6} T^{18} - 2497428 p^{7} T^{19} + 258589 p^{8} T^{20} - 8977 p^{9} T^{21} + 758 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 5 T + 776 T^{2} + 3968 T^{3} + 287695 T^{4} + 1478921 T^{5} + 67726712 T^{6} + 343860072 T^{7} + 11370817200 T^{8} + 55662078506 T^{9} + 1448619091945 T^{10} + 6608085526158 T^{11} + 144779248618483 T^{12} + 6608085526158 p T^{13} + 1448619091945 p^{2} T^{14} + 55662078506 p^{3} T^{15} + 11370817200 p^{4} T^{16} + 343860072 p^{5} T^{17} + 67726712 p^{6} T^{18} + 1478921 p^{7} T^{19} + 287695 p^{8} T^{20} + 3968 p^{9} T^{21} + 776 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 53 T + 2010 T^{2} + 55908 T^{3} + 1302489 T^{4} + 25848178 T^{5} + 453945205 T^{6} + 7127359826 T^{7} + 101642353050 T^{8} + 1324153199679 T^{9} + 15875695746956 T^{10} + 175620042521928 T^{11} + 1798027330581971 T^{12} + 175620042521928 p T^{13} + 15875695746956 p^{2} T^{14} + 1324153199679 p^{3} T^{15} + 101642353050 p^{4} T^{16} + 7127359826 p^{5} T^{17} + 453945205 p^{6} T^{18} + 25848178 p^{7} T^{19} + 1302489 p^{8} T^{20} + 55908 p^{9} T^{21} + 2010 p^{10} T^{22} + 53 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.57782714618249783348851277711, −2.40023630808505815814532311822, −2.39313998560082084892444423204, −2.33331441340426901182928536568, −2.23122740309967489820902553611, −2.19211730698867265852192720730, −2.17455288103241665273567100385, −2.16668948095668245894690772369, −2.16097412528749528623328631692, −2.10811677139191812930155798698, −2.03231015204319745801612064952, −1.74504797418826922029857827383, −1.74201849987462320297088875518, −1.60933579726463611858345571965, −1.57355389809959897159758882018, −1.40511260430193582269969198589, −1.39731844137197690285978112674, −1.31665324063410444815622197508, −1.27529928174913941658220442149, −1.09542666112129922573040265214, −1.08924725019230489289766047421, −0.987558036687841028183369764113, −0.868137451330149699162455961633, −0.845290081325202067232276940264, −0.78477738506659022325976445336, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.78477738506659022325976445336, 0.845290081325202067232276940264, 0.868137451330149699162455961633, 0.987558036687841028183369764113, 1.08924725019230489289766047421, 1.09542666112129922573040265214, 1.27529928174913941658220442149, 1.31665324063410444815622197508, 1.39731844137197690285978112674, 1.40511260430193582269969198589, 1.57355389809959897159758882018, 1.60933579726463611858345571965, 1.74201849987462320297088875518, 1.74504797418826922029857827383, 2.03231015204319745801612064952, 2.10811677139191812930155798698, 2.16097412528749528623328631692, 2.16668948095668245894690772369, 2.17455288103241665273567100385, 2.19211730698867265852192720730, 2.23122740309967489820902553611, 2.33331441340426901182928536568, 2.39313998560082084892444423204, 2.40023630808505815814532311822, 2.57782714618249783348851277711
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.