Dirichlet series
| L(s) = 1 | + 11·2-s + 11·3-s + 66·4-s + 5·5-s + 121·6-s + 4·7-s + 286·8-s + 66·9-s + 55·10-s − 7·11-s + 726·12-s + 11·13-s + 44·14-s + 55·15-s + 1.00e3·16-s + 10·17-s + 726·18-s − 7·19-s + 330·20-s + 44·21-s − 77·22-s + 18·23-s + 3.14e3·24-s + 25-s + 121·26-s + 286·27-s + 264·28-s + ⋯ |
| L(s) = 1 | + 7.77·2-s + 6.35·3-s + 33·4-s + 2.23·5-s + 49.3·6-s + 1.51·7-s + 101.·8-s + 22·9-s + 17.3·10-s − 2.11·11-s + 209.·12-s + 3.05·13-s + 11.7·14-s + 14.2·15-s + 250.·16-s + 2.42·17-s + 171.·18-s − 1.60·19-s + 73.7·20-s + 9.60·21-s − 16.4·22-s + 3.75·23-s + 642.·24-s + 1/5·25-s + 23.7·26-s + 55.0·27-s + 49.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.57352\times 10^{19}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{11} \cdot 3^{11} \cdot 13^{11} \cdot 103^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.972068375\times10^{6}\) |
| \(L(\frac12)\) | \(\approx\) | \(8.972068375\times10^{6}\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{11} \) |
| 3 | \( ( 1 - T )^{11} \) | |
| 13 | \( ( 1 - T )^{11} \) | |
| 103 | \( ( 1 + T )^{11} \) | |
| good | 5 | \( 1 - p T + 24 T^{2} - 82 T^{3} + 56 p T^{4} - 833 T^{5} + 2489 T^{6} - 6708 T^{7} + 3557 p T^{8} - 43104 T^{9} + 104049 T^{10} - 232064 T^{11} + 104049 p T^{12} - 43104 p^{2} T^{13} + 3557 p^{4} T^{14} - 6708 p^{4} T^{15} + 2489 p^{5} T^{16} - 833 p^{6} T^{17} + 56 p^{8} T^{18} - 82 p^{8} T^{19} + 24 p^{9} T^{20} - p^{11} T^{21} + p^{11} T^{22} \) |
| 7 | \( 1 - 4 T + 36 T^{2} - 100 T^{3} + 501 T^{4} - 18 p^{2} T^{5} + 3205 T^{6} - 1949 T^{7} + 1131 p T^{8} + 20049 T^{9} - 12995 T^{10} + 211660 T^{11} - 12995 p T^{12} + 20049 p^{2} T^{13} + 1131 p^{4} T^{14} - 1949 p^{4} T^{15} + 3205 p^{5} T^{16} - 18 p^{8} T^{17} + 501 p^{7} T^{18} - 100 p^{8} T^{19} + 36 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 7 T + 67 T^{2} + 345 T^{3} + 2048 T^{4} + 8386 T^{5} + 39070 T^{6} + 138238 T^{7} + 567086 T^{8} + 1833528 T^{9} + 7009003 T^{10} + 21381286 T^{11} + 7009003 p T^{12} + 1833528 p^{2} T^{13} + 567086 p^{3} T^{14} + 138238 p^{4} T^{15} + 39070 p^{5} T^{16} + 8386 p^{6} T^{17} + 2048 p^{7} T^{18} + 345 p^{8} T^{19} + 67 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 10 T + 93 T^{2} - 653 T^{3} + 4343 T^{4} - 24625 T^{5} + 136106 T^{6} - 684457 T^{7} + 3370952 T^{8} - 15454001 T^{9} + 69191945 T^{10} - 288127004 T^{11} + 69191945 p T^{12} - 15454001 p^{2} T^{13} + 3370952 p^{3} T^{14} - 684457 p^{4} T^{15} + 136106 p^{5} T^{16} - 24625 p^{6} T^{17} + 4343 p^{7} T^{18} - 653 p^{8} T^{19} + 93 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 + 7 T + 183 T^{2} + 1126 T^{3} + 15612 T^{4} + 84506 T^{5} + 820396 T^{6} + 3898083 T^{7} + 29522476 T^{8} + 122469749 T^{9} + 764144734 T^{10} + 2739865090 T^{11} + 764144734 p T^{12} + 122469749 p^{2} T^{13} + 29522476 p^{3} T^{14} + 3898083 p^{4} T^{15} + 820396 p^{5} T^{16} + 84506 p^{6} T^{17} + 15612 p^{7} T^{18} + 1126 p^{8} T^{19} + 183 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 18 T + 275 T^{2} - 2827 T^{3} + 25995 T^{4} - 194675 T^{5} + 1346400 T^{6} - 8130329 T^{7} + 46639949 T^{8} - 244056006 T^{9} + 1249586870 T^{10} - 6012971762 T^{11} + 1249586870 p T^{12} - 244056006 p^{2} T^{13} + 46639949 p^{3} T^{14} - 8130329 p^{4} T^{15} + 1346400 p^{5} T^{16} - 194675 p^{6} T^{17} + 25995 p^{7} T^{18} - 2827 p^{8} T^{19} + 275 p^{9} T^{20} - 18 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 - p T + 531 T^{2} - 7059 T^{3} + 77145 T^{4} - 718400 T^{5} + 5956553 T^{6} - 44637532 T^{7} + 308605345 T^{8} - 1975416638 T^{9} + 11807273239 T^{10} - 65741457936 T^{11} + 11807273239 p T^{12} - 1975416638 p^{2} T^{13} + 308605345 p^{3} T^{14} - 44637532 p^{4} T^{15} + 5956553 p^{5} T^{16} - 718400 p^{6} T^{17} + 77145 p^{7} T^{18} - 7059 p^{8} T^{19} + 531 p^{9} T^{20} - p^{11} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 7 T + 315 T^{2} - 1966 T^{3} + 45804 T^{4} - 253922 T^{5} + 4059628 T^{6} - 19863291 T^{7} + 243726268 T^{8} - 1042376093 T^{9} + 10408598782 T^{10} - 38348771362 T^{11} + 10408598782 p T^{12} - 1042376093 p^{2} T^{13} + 243726268 p^{3} T^{14} - 19863291 p^{4} T^{15} + 4059628 p^{5} T^{16} - 253922 p^{6} T^{17} + 45804 p^{7} T^{18} - 1966 p^{8} T^{19} + 315 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 21 T + 413 T^{2} - 157 p T^{3} + 1933 p T^{4} - 765742 T^{5} + 7312171 T^{6} - 63629198 T^{7} + 505776257 T^{8} - 3708250550 T^{9} + 25244809269 T^{10} - 158906302224 T^{11} + 25244809269 p T^{12} - 3708250550 p^{2} T^{13} + 505776257 p^{3} T^{14} - 63629198 p^{4} T^{15} + 7312171 p^{5} T^{16} - 765742 p^{6} T^{17} + 1933 p^{8} T^{18} - 157 p^{9} T^{19} + 413 p^{9} T^{20} - 21 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 3 T + 272 T^{2} + 981 T^{3} + 36987 T^{4} + 138644 T^{5} + 3346829 T^{6} + 12014750 T^{7} + 223134441 T^{8} + 740908205 T^{9} + 11516674142 T^{10} + 34623256162 T^{11} + 11516674142 p T^{12} + 740908205 p^{2} T^{13} + 223134441 p^{3} T^{14} + 12014750 p^{4} T^{15} + 3346829 p^{5} T^{16} + 138644 p^{6} T^{17} + 36987 p^{7} T^{18} + 981 p^{8} T^{19} + 272 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 + 17 T + 273 T^{2} + 3734 T^{3} + 41419 T^{4} + 430199 T^{5} + 4079640 T^{6} + 35153006 T^{7} + 287832136 T^{8} + 2188617450 T^{9} + 15647130347 T^{10} + 106084328428 T^{11} + 15647130347 p T^{12} + 2188617450 p^{2} T^{13} + 287832136 p^{3} T^{14} + 35153006 p^{4} T^{15} + 4079640 p^{5} T^{16} + 430199 p^{6} T^{17} + 41419 p^{7} T^{18} + 3734 p^{8} T^{19} + 273 p^{9} T^{20} + 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 12 T + 484 T^{2} - 4902 T^{3} + 107103 T^{4} - 928970 T^{5} + 14384085 T^{6} - 107628357 T^{7} + 1306986508 T^{8} - 8444435427 T^{9} + 84518211201 T^{10} - 468835705808 T^{11} + 84518211201 p T^{12} - 8444435427 p^{2} T^{13} + 1306986508 p^{3} T^{14} - 107628357 p^{4} T^{15} + 14384085 p^{5} T^{16} - 928970 p^{6} T^{17} + 107103 p^{7} T^{18} - 4902 p^{8} T^{19} + 484 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 11 T + 285 T^{2} - 1907 T^{3} + 32259 T^{4} - 119589 T^{5} + 2261632 T^{6} - 4551641 T^{7} + 158350826 T^{8} - 384157832 T^{9} + 11433757765 T^{10} - 29678394008 T^{11} + 11433757765 p T^{12} - 384157832 p^{2} T^{13} + 158350826 p^{3} T^{14} - 4551641 p^{4} T^{15} + 2261632 p^{5} T^{16} - 119589 p^{6} T^{17} + 32259 p^{7} T^{18} - 1907 p^{8} T^{19} + 285 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + 48 T + 1321 T^{2} + 26714 T^{3} + 438328 T^{4} + 6117534 T^{5} + 74923400 T^{6} + 822254899 T^{7} + 8213126910 T^{8} + 75493098346 T^{9} + 643807800102 T^{10} + 5117654595910 T^{11} + 643807800102 p T^{12} + 75493098346 p^{2} T^{13} + 8213126910 p^{3} T^{14} + 822254899 p^{4} T^{15} + 74923400 p^{5} T^{16} + 6117534 p^{6} T^{17} + 438328 p^{7} T^{18} + 26714 p^{8} T^{19} + 1321 p^{9} T^{20} + 48 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + T + 121 T^{2} + 30 T^{3} + 16403 T^{4} + 6233 T^{5} + 1363610 T^{6} + 88270 T^{7} + 112494656 T^{8} + 54551114 T^{9} + 7512177509 T^{10} + 1117431328 T^{11} + 7512177509 p T^{12} + 54551114 p^{2} T^{13} + 112494656 p^{3} T^{14} + 88270 p^{4} T^{15} + 1363610 p^{5} T^{16} + 6233 p^{6} T^{17} + 16403 p^{7} T^{18} + 30 p^{8} T^{19} + 121 p^{9} T^{20} + p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + 9 T + 394 T^{2} + 57 p T^{3} + 80349 T^{4} + 781121 T^{5} + 11310962 T^{6} + 103101443 T^{7} + 1211982888 T^{8} + 9933981227 T^{9} + 102283799212 T^{10} + 745760608346 T^{11} + 102283799212 p T^{12} + 9933981227 p^{2} T^{13} + 1211982888 p^{3} T^{14} + 103101443 p^{4} T^{15} + 11310962 p^{5} T^{16} + 781121 p^{6} T^{17} + 80349 p^{7} T^{18} + 57 p^{9} T^{19} + 394 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 17 T + 410 T^{2} - 4719 T^{3} + 72231 T^{4} - 732838 T^{5} + 8966198 T^{6} - 83614697 T^{7} + 849560641 T^{8} - 7466122765 T^{9} + 67991804849 T^{10} - 572410773864 T^{11} + 67991804849 p T^{12} - 7466122765 p^{2} T^{13} + 849560641 p^{3} T^{14} - 83614697 p^{4} T^{15} + 8966198 p^{5} T^{16} - 732838 p^{6} T^{17} + 72231 p^{7} T^{18} - 4719 p^{8} T^{19} + 410 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 23 T + 549 T^{2} + 7902 T^{3} + 114786 T^{4} + 1299647 T^{5} + 15141597 T^{6} + 151246736 T^{7} + 1578703671 T^{8} + 14542339469 T^{9} + 139433976402 T^{10} + 1172680462970 T^{11} + 139433976402 p T^{12} + 14542339469 p^{2} T^{13} + 1578703671 p^{3} T^{14} + 151246736 p^{4} T^{15} + 15141597 p^{5} T^{16} + 1299647 p^{6} T^{17} + 114786 p^{7} T^{18} + 7902 p^{8} T^{19} + 549 p^{9} T^{20} + 23 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 41 T + 1364 T^{2} - 31828 T^{3} + 645877 T^{4} - 10916056 T^{5} + 165870441 T^{6} - 2217271855 T^{7} + 27117736202 T^{8} - 298641620511 T^{9} + 3035712014195 T^{10} - 28046383964666 T^{11} + 3035712014195 p T^{12} - 298641620511 p^{2} T^{13} + 27117736202 p^{3} T^{14} - 2217271855 p^{4} T^{15} + 165870441 p^{5} T^{16} - 10916056 p^{6} T^{17} + 645877 p^{7} T^{18} - 31828 p^{8} T^{19} + 1364 p^{9} T^{20} - 41 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 19 T + 672 T^{2} - 8894 T^{3} + 186956 T^{4} - 1917416 T^{5} + 31354219 T^{6} - 264708379 T^{7} + 3775625321 T^{8} - 27532638971 T^{9} + 364635969903 T^{10} - 2421478738370 T^{11} + 364635969903 p T^{12} - 27532638971 p^{2} T^{13} + 3775625321 p^{3} T^{14} - 264708379 p^{4} T^{15} + 31354219 p^{5} T^{16} - 1917416 p^{6} T^{17} + 186956 p^{7} T^{18} - 8894 p^{8} T^{19} + 672 p^{9} T^{20} - 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 32 T + 737 T^{2} - 12110 T^{3} + 178130 T^{4} - 2224893 T^{5} + 26830182 T^{6} - 294866004 T^{7} + 3198934300 T^{8} - 31904252221 T^{9} + 318517941246 T^{10} - 2985282062736 T^{11} + 318517941246 p T^{12} - 31904252221 p^{2} T^{13} + 3198934300 p^{3} T^{14} - 294866004 p^{4} T^{15} + 26830182 p^{5} T^{16} - 2224893 p^{6} T^{17} + 178130 p^{7} T^{18} - 12110 p^{8} T^{19} + 737 p^{9} T^{20} - 32 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 16 T + 714 T^{2} + 7608 T^{3} + 216879 T^{4} + 1742645 T^{5} + 43101471 T^{6} + 288336086 T^{7} + 6566757567 T^{8} + 37997410219 T^{9} + 790655981656 T^{10} + 4051916832740 T^{11} + 790655981656 p T^{12} + 37997410219 p^{2} T^{13} + 6566757567 p^{3} T^{14} + 288336086 p^{4} T^{15} + 43101471 p^{5} T^{16} + 1742645 p^{6} T^{17} + 216879 p^{7} T^{18} + 7608 p^{8} T^{19} + 714 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.71300244377780879630124393056, −2.66268821206349287379230103051, −2.48211759515138022517340585796, −2.27819681231267296188458153631, −2.17578498671790579823692030074, −2.11946970445257171129143412609, −2.09862324609326522058792227748, −1.93559769420035452853972063146, −1.91810035279241648192200165134, −1.90140802492242869210670232971, −1.88384989744958886768245026672, −1.84966587879183069005583756925, −1.84307120409156599799924718698, −1.64284479308998410421338711793, −1.45979792308207253906949000103, −1.32001503075844620774958763263, −1.27817084898797039268920571187, −1.21549216585654728629726439143, −1.15650588154129988583079303193, −1.00269298909760475608504662762, −0.888228473045301792293649017106, −0.866303103549487719582428372551, −0.75405137002776566803773516498, −0.64622012925829370038428128537, −0.44485692871559063761591385617, 0.44485692871559063761591385617, 0.64622012925829370038428128537, 0.75405137002776566803773516498, 0.866303103549487719582428372551, 0.888228473045301792293649017106, 1.00269298909760475608504662762, 1.15650588154129988583079303193, 1.21549216585654728629726439143, 1.27817084898797039268920571187, 1.32001503075844620774958763263, 1.45979792308207253906949000103, 1.64284479308998410421338711793, 1.84307120409156599799924718698, 1.84966587879183069005583756925, 1.88384989744958886768245026672, 1.90140802492242869210670232971, 1.91810035279241648192200165134, 1.93559769420035452853972063146, 2.09862324609326522058792227748, 2.11946970445257171129143412609, 2.17578498671790579823692030074, 2.27819681231267296188458153631, 2.48211759515138022517340585796, 2.66268821206349287379230103051, 2.71300244377780879630124393056
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.