Dirichlet series
| L(s) = 1 | + 4·2-s + 8·3-s + 32·6-s + 9·7-s − 20·8-s + 20·9-s − 3·11-s + 9·13-s + 36·14-s − 23·16-s + 4·17-s + 80·18-s − 33·19-s + 72·21-s − 12·22-s + 31·23-s − 160·24-s + 36·26-s − 8·27-s + 29-s + 6·31-s + 17·32-s − 24·33-s + 16·34-s + 23·37-s − 132·38-s + 72·39-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4.61·3-s + 13.0·6-s + 3.40·7-s − 7.07·8-s + 20/3·9-s − 0.904·11-s + 2.49·13-s + 9.62·14-s − 5.75·16-s + 0.970·17-s + 18.8·18-s − 7.57·19-s + 15.7·21-s − 2.55·22-s + 6.46·23-s − 32.6·24-s + 7.06·26-s − 1.53·27-s + 0.185·29-s + 1.07·31-s + 3.00·32-s − 4.17·33-s + 2.74·34-s + 3.78·37-s − 21.4·38-s + 11.5·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{22} \cdot 241^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{22} \cdot 241^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(5^{22} \cdot 241^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.19577\times 10^{18}\) |
| Root analytic conductor: | \(6.93612\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 5^{22} \cdot 241^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(457.0516208\) |
| \(L(\frac12)\) | \(\approx\) | \(457.0516208\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 241 | \( ( 1 - T )^{11} \) | |
| good | 2 | \( 1 - p^{2} T + p^{4} T^{2} - 11 p^{2} T^{3} + 119 T^{4} - 265 T^{5} + 569 T^{6} - 1075 T^{7} + 1955 T^{8} - 803 p^{2} T^{9} + 5073 T^{10} - 7317 T^{11} + 5073 p T^{12} - 803 p^{4} T^{13} + 1955 p^{3} T^{14} - 1075 p^{4} T^{15} + 569 p^{5} T^{16} - 265 p^{6} T^{17} + 119 p^{7} T^{18} - 11 p^{10} T^{19} + p^{13} T^{20} - p^{12} T^{21} + p^{11} T^{22} \) |
| 3 | \( 1 - 8 T + 44 T^{2} - 184 T^{3} + 644 T^{4} - 1954 T^{5} + 5282 T^{6} - 4307 p T^{7} + 28973 T^{8} - 2222 p^{3} T^{9} + 115445 T^{10} - 206986 T^{11} + 115445 p T^{12} - 2222 p^{5} T^{13} + 28973 p^{3} T^{14} - 4307 p^{5} T^{15} + 5282 p^{5} T^{16} - 1954 p^{6} T^{17} + 644 p^{7} T^{18} - 184 p^{8} T^{19} + 44 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 7 | \( 1 - 9 T + 11 p T^{2} - 443 T^{3} + 346 p T^{4} - 1544 p T^{5} + 45823 T^{6} - 168895 T^{7} + 593821 T^{8} - 1863877 T^{9} + 5589939 T^{10} - 15122740 T^{11} + 5589939 p T^{12} - 1863877 p^{2} T^{13} + 593821 p^{3} T^{14} - 168895 p^{4} T^{15} + 45823 p^{5} T^{16} - 1544 p^{7} T^{17} + 346 p^{8} T^{18} - 443 p^{8} T^{19} + 11 p^{10} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 3 T + 62 T^{2} + 14 p T^{3} + 1657 T^{4} + 2555 T^{5} + 22142 T^{6} - 7625 T^{7} + 93562 T^{8} - 937944 T^{9} - 1275223 T^{10} - 15528862 T^{11} - 1275223 p T^{12} - 937944 p^{2} T^{13} + 93562 p^{3} T^{14} - 7625 p^{4} T^{15} + 22142 p^{5} T^{16} + 2555 p^{6} T^{17} + 1657 p^{7} T^{18} + 14 p^{9} T^{19} + 62 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 - 9 T + 97 T^{2} - 587 T^{3} + 3646 T^{4} - 16217 T^{5} + 71982 T^{6} - 244837 T^{7} + 834242 T^{8} - 2260088 T^{9} + 6932271 T^{10} - 20417578 T^{11} + 6932271 p T^{12} - 2260088 p^{2} T^{13} + 834242 p^{3} T^{14} - 244837 p^{4} T^{15} + 71982 p^{5} T^{16} - 16217 p^{6} T^{17} + 3646 p^{7} T^{18} - 587 p^{8} T^{19} + 97 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 4 T + 106 T^{2} - 449 T^{3} + 6030 T^{4} - 25167 T^{5} + 232247 T^{6} - 915996 T^{7} + 6617860 T^{8} - 1405387 p T^{9} + 144618675 T^{10} - 466504364 T^{11} + 144618675 p T^{12} - 1405387 p^{3} T^{13} + 6617860 p^{3} T^{14} - 915996 p^{4} T^{15} + 232247 p^{5} T^{16} - 25167 p^{6} T^{17} + 6030 p^{7} T^{18} - 449 p^{8} T^{19} + 106 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + 33 T + 583 T^{2} + 376 p T^{3} + 68237 T^{4} + 542625 T^{5} + 3756768 T^{6} + 23329258 T^{7} + 132524391 T^{8} + 695866631 T^{9} + 3391438531 T^{10} + 15344291290 T^{11} + 3391438531 p T^{12} + 695866631 p^{2} T^{13} + 132524391 p^{3} T^{14} + 23329258 p^{4} T^{15} + 3756768 p^{5} T^{16} + 542625 p^{6} T^{17} + 68237 p^{7} T^{18} + 376 p^{9} T^{19} + 583 p^{9} T^{20} + 33 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 31 T + 645 T^{2} - 9693 T^{3} + 119003 T^{4} - 1220599 T^{5} + 10860307 T^{6} - 84638686 T^{7} + 587470399 T^{8} - 3645456586 T^{9} + 20391619600 T^{10} - 102819393318 T^{11} + 20391619600 p T^{12} - 3645456586 p^{2} T^{13} + 587470399 p^{3} T^{14} - 84638686 p^{4} T^{15} + 10860307 p^{5} T^{16} - 1220599 p^{6} T^{17} + 119003 p^{7} T^{18} - 9693 p^{8} T^{19} + 645 p^{9} T^{20} - 31 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - T + 165 T^{2} - 481 T^{3} + 13537 T^{4} - 58070 T^{5} + 790904 T^{6} - 3714933 T^{7} + 36720806 T^{8} - 160589230 T^{9} + 1356513220 T^{10} - 5258726656 T^{11} + 1356513220 p T^{12} - 160589230 p^{2} T^{13} + 36720806 p^{3} T^{14} - 3714933 p^{4} T^{15} + 790904 p^{5} T^{16} - 58070 p^{6} T^{17} + 13537 p^{7} T^{18} - 481 p^{8} T^{19} + 165 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 6 T + 167 T^{2} - 640 T^{3} + 12503 T^{4} - 33462 T^{5} + 660282 T^{6} - 1410885 T^{7} + 28528780 T^{8} - 49501876 T^{9} + 1017551600 T^{10} - 1515235802 T^{11} + 1017551600 p T^{12} - 49501876 p^{2} T^{13} + 28528780 p^{3} T^{14} - 1410885 p^{4} T^{15} + 660282 p^{5} T^{16} - 33462 p^{6} T^{17} + 12503 p^{7} T^{18} - 640 p^{8} T^{19} + 167 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 23 T + 452 T^{2} - 6195 T^{3} + 76439 T^{4} - 786870 T^{5} + 7479142 T^{6} - 62919285 T^{7} + 496649263 T^{8} - 3562483986 T^{9} + 24196899884 T^{10} - 151017785580 T^{11} + 24196899884 p T^{12} - 3562483986 p^{2} T^{13} + 496649263 p^{3} T^{14} - 62919285 p^{4} T^{15} + 7479142 p^{5} T^{16} - 786870 p^{6} T^{17} + 76439 p^{7} T^{18} - 6195 p^{8} T^{19} + 452 p^{9} T^{20} - 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 8 T + 248 T^{2} - 1581 T^{3} + 27398 T^{4} - 146428 T^{5} + 1949997 T^{6} - 9676579 T^{7} + 112075607 T^{8} - 551199103 T^{9} + 5581442406 T^{10} - 25768211940 T^{11} + 5581442406 p T^{12} - 551199103 p^{2} T^{13} + 112075607 p^{3} T^{14} - 9676579 p^{4} T^{15} + 1949997 p^{5} T^{16} - 146428 p^{6} T^{17} + 27398 p^{7} T^{18} - 1581 p^{8} T^{19} + 248 p^{9} T^{20} - 8 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 19 T + 467 T^{2} - 6416 T^{3} + 94986 T^{4} - 1036284 T^{5} + 11574715 T^{6} - 104788953 T^{7} + 953247821 T^{8} - 7321184820 T^{9} + 56082791347 T^{10} - 368610934200 T^{11} + 56082791347 p T^{12} - 7321184820 p^{2} T^{13} + 953247821 p^{3} T^{14} - 104788953 p^{4} T^{15} + 11574715 p^{5} T^{16} - 1036284 p^{6} T^{17} + 94986 p^{7} T^{18} - 6416 p^{8} T^{19} + 467 p^{9} T^{20} - 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 35 T + 880 T^{2} - 15376 T^{3} + 224009 T^{4} - 2672909 T^{5} + 28152005 T^{6} - 258452490 T^{7} + 2180451303 T^{8} - 16759642682 T^{9} + 123192609095 T^{10} - 854553467168 T^{11} + 123192609095 p T^{12} - 16759642682 p^{2} T^{13} + 2180451303 p^{3} T^{14} - 258452490 p^{4} T^{15} + 28152005 p^{5} T^{16} - 2672909 p^{6} T^{17} + 224009 p^{7} T^{18} - 15376 p^{8} T^{19} + 880 p^{9} T^{20} - 35 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 14 T + 282 T^{2} + 2766 T^{3} + 33849 T^{4} + 298024 T^{5} + 3100616 T^{6} + 26287378 T^{7} + 243443677 T^{8} + 1835387119 T^{9} + 15238389718 T^{10} + 103865986416 T^{11} + 15238389718 p T^{12} + 1835387119 p^{2} T^{13} + 243443677 p^{3} T^{14} + 26287378 p^{4} T^{15} + 3100616 p^{5} T^{16} + 298024 p^{6} T^{17} + 33849 p^{7} T^{18} + 2766 p^{8} T^{19} + 282 p^{9} T^{20} + 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + 6 T + 348 T^{2} + 1596 T^{3} + 61335 T^{4} + 232670 T^{5} + 7399972 T^{6} + 24581802 T^{7} + 680201535 T^{8} + 2028486741 T^{9} + 49647828250 T^{10} + 133465942558 T^{11} + 49647828250 p T^{12} + 2028486741 p^{2} T^{13} + 680201535 p^{3} T^{14} + 24581802 p^{4} T^{15} + 7399972 p^{5} T^{16} + 232670 p^{6} T^{17} + 61335 p^{7} T^{18} + 1596 p^{8} T^{19} + 348 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 9 T + 523 T^{2} - 4167 T^{3} + 130382 T^{4} - 923100 T^{5} + 20475161 T^{6} - 128835705 T^{7} + 2253750195 T^{8} - 12545722717 T^{9} + 182770561509 T^{10} - 890638865574 T^{11} + 182770561509 p T^{12} - 12545722717 p^{2} T^{13} + 2253750195 p^{3} T^{14} - 128835705 p^{4} T^{15} + 20475161 p^{5} T^{16} - 923100 p^{6} T^{17} + 130382 p^{7} T^{18} - 4167 p^{8} T^{19} + 523 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 54 T + 1793 T^{2} - 43376 T^{3} + 848860 T^{4} - 14000247 T^{5} + 200942236 T^{6} - 2552265686 T^{7} + 29093829595 T^{8} - 299932159848 T^{9} + 2814333707634 T^{10} - 24095922393102 T^{11} + 2814333707634 p T^{12} - 299932159848 p^{2} T^{13} + 29093829595 p^{3} T^{14} - 2552265686 p^{4} T^{15} + 200942236 p^{5} T^{16} - 14000247 p^{6} T^{17} + 848860 p^{7} T^{18} - 43376 p^{8} T^{19} + 1793 p^{9} T^{20} - 54 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 + 5 T + 369 T^{2} + 2726 T^{3} + 1097 p T^{4} + 632327 T^{5} + 11819152 T^{6} + 93566238 T^{7} + 1360132324 T^{8} + 10053683216 T^{9} + 121983570726 T^{10} + 816591365980 T^{11} + 121983570726 p T^{12} + 10053683216 p^{2} T^{13} + 1360132324 p^{3} T^{14} + 93566238 p^{4} T^{15} + 11819152 p^{5} T^{16} + 632327 p^{6} T^{17} + 1097 p^{8} T^{18} + 2726 p^{8} T^{19} + 369 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 17 T + 578 T^{2} + 8389 T^{3} + 169543 T^{4} + 2083597 T^{5} + 31678731 T^{6} + 336054025 T^{7} + 4171190164 T^{8} + 38460101207 T^{9} + 405026306682 T^{10} + 3250225364632 T^{11} + 405026306682 p T^{12} + 38460101207 p^{2} T^{13} + 4171190164 p^{3} T^{14} + 336054025 p^{4} T^{15} + 31678731 p^{5} T^{16} + 2083597 p^{6} T^{17} + 169543 p^{7} T^{18} + 8389 p^{8} T^{19} + 578 p^{9} T^{20} + 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 + 16 T + 643 T^{2} + 8618 T^{3} + 185961 T^{4} + 2117687 T^{5} + 32353783 T^{6} + 320137040 T^{7} + 3904045847 T^{8} + 34623954136 T^{9} + 365890566412 T^{10} + 2989968768694 T^{11} + 365890566412 p T^{12} + 34623954136 p^{2} T^{13} + 3904045847 p^{3} T^{14} + 320137040 p^{4} T^{15} + 32353783 p^{5} T^{16} + 2117687 p^{6} T^{17} + 185961 p^{7} T^{18} + 8618 p^{8} T^{19} + 643 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 29 T + 828 T^{2} - 15713 T^{3} + 281415 T^{4} - 4136872 T^{5} + 57745871 T^{6} - 709662065 T^{7} + 8324128788 T^{8} - 88511841411 T^{9} + 899942775938 T^{10} - 8382930168216 T^{11} + 899942775938 p T^{12} - 88511841411 p^{2} T^{13} + 8324128788 p^{3} T^{14} - 709662065 p^{4} T^{15} + 57745871 p^{5} T^{16} - 4136872 p^{6} T^{17} + 281415 p^{7} T^{18} - 15713 p^{8} T^{19} + 828 p^{9} T^{20} - 29 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 5 T + 195 T^{2} + 1230 T^{3} + 33329 T^{4} + 250880 T^{5} + 4285411 T^{6} + 36248750 T^{7} + 498318379 T^{8} + 4149499905 T^{9} + 49565915508 T^{10} + 406334801638 T^{11} + 49565915508 p T^{12} + 4149499905 p^{2} T^{13} + 498318379 p^{3} T^{14} + 36248750 p^{4} T^{15} + 4285411 p^{5} T^{16} + 250880 p^{6} T^{17} + 33329 p^{7} T^{18} + 1230 p^{8} T^{19} + 195 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 6 T + 496 T^{2} + 4042 T^{3} + 136231 T^{4} + 1232695 T^{5} + 26746490 T^{6} + 239578498 T^{7} + 4080758420 T^{8} + 33830250458 T^{9} + 495002175583 T^{10} + 3696666740452 T^{11} + 495002175583 p T^{12} + 33830250458 p^{2} T^{13} + 4080758420 p^{3} T^{14} + 239578498 p^{4} T^{15} + 26746490 p^{5} T^{16} + 1232695 p^{6} T^{17} + 136231 p^{7} T^{18} + 4042 p^{8} T^{19} + 496 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.56377905308509374207196040532, −2.53671291047589837134664633642, −2.48717231197133441019022180486, −2.45059241257496034737692182004, −2.44039343476055881785143385568, −2.22291013784660246650533798188, −2.19237292763430353616042842299, −2.17649099115214769400981464857, −2.01668992833030025965173153743, −1.76391569211927330198134229541, −1.73258210071733254534733563847, −1.71874695986199069007698875852, −1.60876189026174703810930735390, −1.59360334225434433281041113139, −1.31179924581610809062780948026, −1.19650731321265511077735403147, −1.13587521934836189871290245792, −0.968411903908015575324512215082, −0.846438957358517436873463481500, −0.815381120948519156225727014144, −0.69609156815159149759178558064, −0.49329374012665182470860357654, −0.47073302569378394007493285080, −0.41327615005422044072505719951, −0.17005510871581806933953770573, 0.17005510871581806933953770573, 0.41327615005422044072505719951, 0.47073302569378394007493285080, 0.49329374012665182470860357654, 0.69609156815159149759178558064, 0.815381120948519156225727014144, 0.846438957358517436873463481500, 0.968411903908015575324512215082, 1.13587521934836189871290245792, 1.19650731321265511077735403147, 1.31179924581610809062780948026, 1.59360334225434433281041113139, 1.60876189026174703810930735390, 1.71874695986199069007698875852, 1.73258210071733254534733563847, 1.76391569211927330198134229541, 2.01668992833030025965173153743, 2.17649099115214769400981464857, 2.19237292763430353616042842299, 2.22291013784660246650533798188, 2.44039343476055881785143385568, 2.45059241257496034737692182004, 2.48717231197133441019022180486, 2.53671291047589837134664633642, 2.56377905308509374207196040532
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.