Dirichlet series
| L(s) = 1 | − 6·3-s + 3·5-s − 5·7-s + 5·9-s + 11·11-s − 13·13-s − 18·15-s − 7·17-s − 21·19-s + 30·21-s + 11·23-s − 21·25-s + 46·27-s − 5·31-s − 66·33-s − 15·35-s − 4·37-s + 78·39-s − 11·41-s + 15·45-s − 17·47-s − 27·49-s + 42·51-s + 11·53-s + 33·55-s + 126·57-s − 19·59-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 1.34·5-s − 1.88·7-s + 5/3·9-s + 3.31·11-s − 3.60·13-s − 4.64·15-s − 1.69·17-s − 4.81·19-s + 6.54·21-s + 2.29·23-s − 4.19·25-s + 8.85·27-s − 0.898·31-s − 11.4·33-s − 2.53·35-s − 0.657·37-s + 12.4·39-s − 1.71·41-s + 2.23·45-s − 2.47·47-s − 3.85·49-s + 5.88·51-s + 1.51·53-s + 4.44·55-s + 16.6·57-s − 2.47·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 11^{11} \cdot 53^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 11^{11} \cdot 53^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{33} \cdot 11^{11} \cdot 53^{11}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.91157\times 10^{17}\) |
| Root analytic conductor: | \(6.10264\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(11\) |
| Selberg data: | \((22,\ 2^{33} \cdot 11^{11} \cdot 53^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\) |
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 | \( 1 \) |
| 11 | \( ( 1 - T )^{11} \) | |
| 53 | \( ( 1 - T )^{11} \) | |
| good | 3 | \( 1 + 2 p T + 31 T^{2} + 110 T^{3} + 356 T^{4} + 320 p T^{5} + 811 p T^{6} + 5488 T^{7} + 146 p^{4} T^{8} + 23344 T^{9} + 44398 T^{10} + 78248 T^{11} + 44398 p T^{12} + 23344 p^{2} T^{13} + 146 p^{7} T^{14} + 5488 p^{4} T^{15} + 811 p^{6} T^{16} + 320 p^{7} T^{17} + 356 p^{7} T^{18} + 110 p^{8} T^{19} + 31 p^{9} T^{20} + 2 p^{11} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 - 3 T + 6 p T^{2} - 17 p T^{3} + 447 T^{4} - 1179 T^{5} + 4421 T^{6} - 10724 T^{7} + 32868 T^{8} - 2914 p^{2} T^{9} + 196929 T^{10} - 399566 T^{11} + 196929 p T^{12} - 2914 p^{4} T^{13} + 32868 p^{3} T^{14} - 10724 p^{4} T^{15} + 4421 p^{5} T^{16} - 1179 p^{6} T^{17} + 447 p^{7} T^{18} - 17 p^{9} T^{19} + 6 p^{10} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 7 | \( 1 + 5 T + 52 T^{2} + 227 T^{3} + 1373 T^{4} + 5219 T^{5} + 23582 T^{6} + 78352 T^{7} + 289683 T^{8} + 844824 T^{9} + 2660911 T^{10} + 6803194 T^{11} + 2660911 p T^{12} + 844824 p^{2} T^{13} + 289683 p^{3} T^{14} + 78352 p^{4} T^{15} + 23582 p^{5} T^{16} + 5219 p^{6} T^{17} + 1373 p^{7} T^{18} + 227 p^{8} T^{19} + 52 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 + p T + 171 T^{2} + 1504 T^{3} + 11931 T^{4} + 6122 p T^{5} + 476603 T^{6} + 196472 p T^{7} + 12450245 T^{8} + 55350571 T^{9} + 226445649 T^{10} + 849259924 T^{11} + 226445649 p T^{12} + 55350571 p^{2} T^{13} + 12450245 p^{3} T^{14} + 196472 p^{5} T^{15} + 476603 p^{5} T^{16} + 6122 p^{7} T^{17} + 11931 p^{7} T^{18} + 1504 p^{8} T^{19} + 171 p^{9} T^{20} + p^{11} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 + 7 T + 165 T^{2} + 940 T^{3} + 12355 T^{4} + 59318 T^{5} + 565373 T^{6} + 2331880 T^{7} + 17776793 T^{8} + 63548821 T^{9} + 406149933 T^{10} + 1258555100 T^{11} + 406149933 p T^{12} + 63548821 p^{2} T^{13} + 17776793 p^{3} T^{14} + 2331880 p^{4} T^{15} + 565373 p^{5} T^{16} + 59318 p^{6} T^{17} + 12355 p^{7} T^{18} + 940 p^{8} T^{19} + 165 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + 21 T + 334 T^{2} + 3793 T^{3} + 36912 T^{4} + 302242 T^{5} + 2217848 T^{6} + 14420073 T^{7} + 85720394 T^{8} + 461220585 T^{9} + 2289804949 T^{10} + 10376330396 T^{11} + 2289804949 p T^{12} + 461220585 p^{2} T^{13} + 85720394 p^{3} T^{14} + 14420073 p^{4} T^{15} + 2217848 p^{5} T^{16} + 302242 p^{6} T^{17} + 36912 p^{7} T^{18} + 3793 p^{8} T^{19} + 334 p^{9} T^{20} + 21 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 11 T + 191 T^{2} - 1670 T^{3} + 16832 T^{4} - 123939 T^{5} + 930001 T^{6} - 5942552 T^{7} + 36462742 T^{8} - 205449835 T^{9} + 1079856674 T^{10} - 5385602918 T^{11} + 1079856674 p T^{12} - 205449835 p^{2} T^{13} + 36462742 p^{3} T^{14} - 5942552 p^{4} T^{15} + 930001 p^{5} T^{16} - 123939 p^{6} T^{17} + 16832 p^{7} T^{18} - 1670 p^{8} T^{19} + 191 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 207 T^{2} + 53 T^{3} + 20419 T^{4} + 12730 T^{5} + 1284695 T^{6} + 1299663 T^{7} + 58570709 T^{8} + 75219432 T^{9} + 2087303525 T^{10} + 2722514036 T^{11} + 2087303525 p T^{12} + 75219432 p^{2} T^{13} + 58570709 p^{3} T^{14} + 1299663 p^{4} T^{15} + 1284695 p^{5} T^{16} + 12730 p^{6} T^{17} + 20419 p^{7} T^{18} + 53 p^{8} T^{19} + 207 p^{9} T^{20} + p^{11} T^{22} \) | |
| 31 | \( 1 + 5 T + 226 T^{2} + 799 T^{3} + 23279 T^{4} + 54077 T^{5} + 1486255 T^{6} + 1913376 T^{7} + 67996728 T^{8} + 35287746 T^{9} + 2473960867 T^{10} + 542833738 T^{11} + 2473960867 p T^{12} + 35287746 p^{2} T^{13} + 67996728 p^{3} T^{14} + 1913376 p^{4} T^{15} + 1486255 p^{5} T^{16} + 54077 p^{6} T^{17} + 23279 p^{7} T^{18} + 799 p^{8} T^{19} + 226 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 + 4 T + 201 T^{2} + 1178 T^{3} + 21604 T^{4} + 144144 T^{5} + 1673737 T^{6} + 295090 p T^{7} + 99636080 T^{8} + 599793822 T^{9} + 4639733084 T^{10} + 25234152970 T^{11} + 4639733084 p T^{12} + 599793822 p^{2} T^{13} + 99636080 p^{3} T^{14} + 295090 p^{5} T^{15} + 1673737 p^{5} T^{16} + 144144 p^{6} T^{17} + 21604 p^{7} T^{18} + 1178 p^{8} T^{19} + 201 p^{9} T^{20} + 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 11 T + 365 T^{2} + 3158 T^{3} + 59498 T^{4} + 428699 T^{5} + 5990490 T^{6} + 37310786 T^{7} + 426478223 T^{8} + 2337461150 T^{9} + 22803112847 T^{10} + 110027543784 T^{11} + 22803112847 p T^{12} + 2337461150 p^{2} T^{13} + 426478223 p^{3} T^{14} + 37310786 p^{4} T^{15} + 5990490 p^{5} T^{16} + 428699 p^{6} T^{17} + 59498 p^{7} T^{18} + 3158 p^{8} T^{19} + 365 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 + 177 T^{2} + 533 T^{3} + 13578 T^{4} + 78870 T^{5} + 738636 T^{6} + 4755984 T^{7} + 37590589 T^{8} + 171534474 T^{9} + 1764463867 T^{10} + 6066619926 T^{11} + 1764463867 p T^{12} + 171534474 p^{2} T^{13} + 37590589 p^{3} T^{14} + 4755984 p^{4} T^{15} + 738636 p^{5} T^{16} + 78870 p^{6} T^{17} + 13578 p^{7} T^{18} + 533 p^{8} T^{19} + 177 p^{9} T^{20} + p^{11} T^{22} \) | |
| 47 | \( 1 + 17 T + 358 T^{2} + 4115 T^{3} + 53143 T^{4} + 469949 T^{5} + 4609332 T^{6} + 33119748 T^{7} + 274650665 T^{8} + 1712702098 T^{9} + 13316665007 T^{10} + 79584703442 T^{11} + 13316665007 p T^{12} + 1712702098 p^{2} T^{13} + 274650665 p^{3} T^{14} + 33119748 p^{4} T^{15} + 4609332 p^{5} T^{16} + 469949 p^{6} T^{17} + 53143 p^{7} T^{18} + 4115 p^{8} T^{19} + 358 p^{9} T^{20} + 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + 19 T + 569 T^{2} + 7978 T^{3} + 140547 T^{4} + 1612518 T^{5} + 21333742 T^{6} + 209238914 T^{7} + 2261354523 T^{8} + 19267692059 T^{9} + 176789563214 T^{10} + 1313080051928 T^{11} + 176789563214 p T^{12} + 19267692059 p^{2} T^{13} + 2261354523 p^{3} T^{14} + 209238914 p^{4} T^{15} + 21333742 p^{5} T^{16} + 1612518 p^{6} T^{17} + 140547 p^{7} T^{18} + 7978 p^{8} T^{19} + 569 p^{9} T^{20} + 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 2 T + 439 T^{2} + 99 T^{3} + 91248 T^{4} - 121136 T^{5} + 12087284 T^{6} - 31418280 T^{7} + 1160674279 T^{8} - 3922060370 T^{9} + 87103457725 T^{10} - 299065792662 T^{11} + 87103457725 p T^{12} - 3922060370 p^{2} T^{13} + 1160674279 p^{3} T^{14} - 31418280 p^{4} T^{15} + 12087284 p^{5} T^{16} - 121136 p^{6} T^{17} + 91248 p^{7} T^{18} + 99 p^{8} T^{19} + 439 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 25 T + 703 T^{2} - 12928 T^{3} + 223271 T^{4} - 48188 p T^{5} + 42751734 T^{6} - 507222130 T^{7} + 5527238407 T^{8} - 55171841827 T^{9} + 508801047958 T^{10} - 4327586634044 T^{11} + 508801047958 p T^{12} - 55171841827 p^{2} T^{13} + 5527238407 p^{3} T^{14} - 507222130 p^{4} T^{15} + 42751734 p^{5} T^{16} - 48188 p^{7} T^{17} + 223271 p^{7} T^{18} - 12928 p^{8} T^{19} + 703 p^{9} T^{20} - 25 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 + 30 T + 874 T^{2} + 16603 T^{3} + 294570 T^{4} + 4222543 T^{5} + 56795201 T^{6} + 663239779 T^{7} + 7333585531 T^{8} + 72763634296 T^{9} + 688627511064 T^{10} + 5932929461838 T^{11} + 688627511064 p T^{12} + 72763634296 p^{2} T^{13} + 7333585531 p^{3} T^{14} + 663239779 p^{4} T^{15} + 56795201 p^{5} T^{16} + 4222543 p^{6} T^{17} + 294570 p^{7} T^{18} + 16603 p^{8} T^{19} + 874 p^{9} T^{20} + 30 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 5 T + 388 T^{2} - 1859 T^{3} + 73931 T^{4} - 365733 T^{5} + 9394598 T^{6} - 49683676 T^{7} + 909498541 T^{8} - 5108156278 T^{9} + 73847811557 T^{10} - 415946394338 T^{11} + 73847811557 p T^{12} - 5108156278 p^{2} T^{13} + 909498541 p^{3} T^{14} - 49683676 p^{4} T^{15} + 9394598 p^{5} T^{16} - 365733 p^{6} T^{17} + 73931 p^{7} T^{18} - 1859 p^{8} T^{19} + 388 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 + 23 T + 820 T^{2} + 13243 T^{3} + 276612 T^{4} + 3506466 T^{5} + 54938346 T^{6} + 579324361 T^{7} + 7488237368 T^{8} + 68091941043 T^{9} + 761488932815 T^{10} + 6095282713472 T^{11} + 761488932815 p T^{12} + 68091941043 p^{2} T^{13} + 7488237368 p^{3} T^{14} + 579324361 p^{4} T^{15} + 54938346 p^{5} T^{16} + 3506466 p^{6} T^{17} + 276612 p^{7} T^{18} + 13243 p^{8} T^{19} + 820 p^{9} T^{20} + 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 + 19 T + 409 T^{2} + 3656 T^{3} + 57069 T^{4} + 460920 T^{5} + 8494871 T^{6} + 68366260 T^{7} + 1012999441 T^{8} + 6707919453 T^{9} + 93177170435 T^{10} + 578506314888 T^{11} + 93177170435 p T^{12} + 6707919453 p^{2} T^{13} + 1012999441 p^{3} T^{14} + 68366260 p^{4} T^{15} + 8494871 p^{5} T^{16} + 460920 p^{6} T^{17} + 57069 p^{7} T^{18} + 3656 p^{8} T^{19} + 409 p^{9} T^{20} + 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 6 T + 716 T^{2} - 4715 T^{3} + 243155 T^{4} - 1737073 T^{5} + 52184011 T^{6} - 390656372 T^{7} + 7951958538 T^{8} - 59025238001 T^{9} + 913470294359 T^{10} - 6242229821994 T^{11} + 913470294359 p T^{12} - 59025238001 p^{2} T^{13} + 7951958538 p^{3} T^{14} - 390656372 p^{4} T^{15} + 52184011 p^{5} T^{16} - 1737073 p^{6} T^{17} + 243155 p^{7} T^{18} - 4715 p^{8} T^{19} + 716 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 35 T + 1128 T^{2} + 24031 T^{3} + 473534 T^{4} + 7513326 T^{5} + 112420507 T^{6} + 1450496203 T^{7} + 18009016705 T^{8} + 199902011561 T^{9} + 2173868209792 T^{10} + 21545816108594 T^{11} + 2173868209792 p T^{12} + 199902011561 p^{2} T^{13} + 18009016705 p^{3} T^{14} + 1450496203 p^{4} T^{15} + 112420507 p^{5} T^{16} + 7513326 p^{6} T^{17} + 473534 p^{7} T^{18} + 24031 p^{8} T^{19} + 1128 p^{9} T^{20} + 35 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−3.17192694697037740253330269922, −3.13479263907067278894530513939, −3.00204138020397601731135265631, −2.66422936752188328173098216429, −2.63200625387647523877340324450, −2.62905305761675970209061201176, −2.45499795690449180678170496626, −2.43869259810281278217473515302, −2.42394811565367632503121736157, −2.29441873891986210534033046769, −2.28182970126039901965750207333, −2.27484192388759051347877661295, −2.24727551427127587254731763552, −2.05449762586867988418354985150, −1.90423509786434676964396975702, −1.71471391236150442177110025179, −1.63004044294217811563473394270, −1.55893714002353409632701844106, −1.35224825282459073603149447926, −1.32747074739325918742946017171, −1.27735250769986516686292058434, −1.19838822831932522814725042549, −1.17072705952569252208597561482, −1.04262464132129469220234510920, −0.909863567867264072244213546422, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.909863567867264072244213546422, 1.04262464132129469220234510920, 1.17072705952569252208597561482, 1.19838822831932522814725042549, 1.27735250769986516686292058434, 1.32747074739325918742946017171, 1.35224825282459073603149447926, 1.55893714002353409632701844106, 1.63004044294217811563473394270, 1.71471391236150442177110025179, 1.90423509786434676964396975702, 2.05449762586867988418354985150, 2.24727551427127587254731763552, 2.27484192388759051347877661295, 2.28182970126039901965750207333, 2.29441873891986210534033046769, 2.42394811565367632503121736157, 2.43869259810281278217473515302, 2.45499795690449180678170496626, 2.62905305761675970209061201176, 2.63200625387647523877340324450, 2.66422936752188328173098216429, 3.00204138020397601731135265631, 3.13479263907067278894530513939, 3.17192694697037740253330269922
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.