Dirichlet series
| L(s) = 1 | + 6·3-s − 2·5-s + 3·7-s + 7·9-s + 13·11-s + 11·13-s − 12·15-s − 6·17-s + 12·19-s + 18·21-s + 13·23-s − 20·25-s − 28·27-s − 11·29-s + 11·31-s + 78·33-s − 6·35-s + 11·37-s + 66·39-s − 9·41-s + 30·43-s − 14·45-s + 47-s − 36·49-s − 36·51-s − 9·53-s − 26·55-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 0.894·5-s + 1.13·7-s + 7/3·9-s + 3.91·11-s + 3.05·13-s − 3.09·15-s − 1.45·17-s + 2.75·19-s + 3.92·21-s + 2.71·23-s − 4·25-s − 5.38·27-s − 2.04·29-s + 1.97·31-s + 13.5·33-s − 1.01·35-s + 1.80·37-s + 10.5·39-s − 1.40·41-s + 4.57·43-s − 2.08·45-s + 0.145·47-s − 5.14·49-s − 5.04·51-s − 1.23·53-s − 3.50·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{44} \cdot 13^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{44} \cdot 13^{11} \cdot 29^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{44} \cdot 13^{11} \cdot 29^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.23685\times 10^{18}\) |
| Root analytic conductor: | \(6.94015\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{44} \cdot 13^{11} \cdot 29^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(413.8202763\) |
| \(L(\frac12)\) | \(\approx\) | \(413.8202763\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 13 | \( ( 1 - T )^{11} \) | |
| 29 | \( ( 1 + T )^{11} \) | |
| good | 3 | \( 1 - 2 p T + 29 T^{2} - 104 T^{3} + 332 T^{4} - 917 T^{5} + 260 p^{2} T^{6} - 1801 p T^{7} + 11731 T^{8} - 7847 p T^{9} + 44815 T^{10} - 79594 T^{11} + 44815 p T^{12} - 7847 p^{3} T^{13} + 11731 p^{3} T^{14} - 1801 p^{5} T^{15} + 260 p^{7} T^{16} - 917 p^{6} T^{17} + 332 p^{7} T^{18} - 104 p^{8} T^{19} + 29 p^{9} T^{20} - 2 p^{11} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 + 2 T + 24 T^{2} + 49 T^{3} + 314 T^{4} + 598 T^{5} + 589 p T^{6} + 5189 T^{7} + 21686 T^{8} + 35652 T^{9} + 130874 T^{10} + 197836 T^{11} + 130874 p T^{12} + 35652 p^{2} T^{13} + 21686 p^{3} T^{14} + 5189 p^{4} T^{15} + 589 p^{6} T^{16} + 598 p^{6} T^{17} + 314 p^{7} T^{18} + 49 p^{8} T^{19} + 24 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 7 | \( 1 - 3 T + 45 T^{2} - 99 T^{3} + 947 T^{4} - 1564 T^{5} + 12820 T^{6} - 15783 T^{7} + 128609 T^{8} - 120097 T^{9} + 1047628 T^{10} - 831228 T^{11} + 1047628 p T^{12} - 120097 p^{2} T^{13} + 128609 p^{3} T^{14} - 15783 p^{4} T^{15} + 12820 p^{5} T^{16} - 1564 p^{6} T^{17} + 947 p^{7} T^{18} - 99 p^{8} T^{19} + 45 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 - 13 T + 127 T^{2} - 899 T^{3} + 5234 T^{4} - 25270 T^{5} + 856 p^{2} T^{6} - 360780 T^{7} + 1070587 T^{8} - 2740629 T^{9} + 6488127 T^{10} - 18109042 T^{11} + 6488127 p T^{12} - 2740629 p^{2} T^{13} + 1070587 p^{3} T^{14} - 360780 p^{4} T^{15} + 856 p^{7} T^{16} - 25270 p^{6} T^{17} + 5234 p^{7} T^{18} - 899 p^{8} T^{19} + 127 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 + 6 T + 122 T^{2} + 739 T^{3} + 7470 T^{4} + 43908 T^{5} + 300227 T^{6} + 1662881 T^{7} + 8738696 T^{8} + 44448546 T^{9} + 192609852 T^{10} + 875142240 T^{11} + 192609852 p T^{12} + 44448546 p^{2} T^{13} + 8738696 p^{3} T^{14} + 1662881 p^{4} T^{15} + 300227 p^{5} T^{16} + 43908 p^{6} T^{17} + 7470 p^{7} T^{18} + 739 p^{8} T^{19} + 122 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - 12 T + 148 T^{2} - 1103 T^{3} + 8425 T^{4} - 49643 T^{5} + 303180 T^{6} - 1554284 T^{7} + 8275303 T^{8} - 38521945 T^{9} + 186259797 T^{10} - 798260026 T^{11} + 186259797 p T^{12} - 38521945 p^{2} T^{13} + 8275303 p^{3} T^{14} - 1554284 p^{4} T^{15} + 303180 p^{5} T^{16} - 49643 p^{6} T^{17} + 8425 p^{7} T^{18} - 1103 p^{8} T^{19} + 148 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 - 13 T + 210 T^{2} - 1797 T^{3} + 17170 T^{4} - 111242 T^{5} + 797393 T^{6} - 4185168 T^{7} + 25083604 T^{8} - 113832625 T^{9} + 627260204 T^{10} - 2678358662 T^{11} + 627260204 p T^{12} - 113832625 p^{2} T^{13} + 25083604 p^{3} T^{14} - 4185168 p^{4} T^{15} + 797393 p^{5} T^{16} - 111242 p^{6} T^{17} + 17170 p^{7} T^{18} - 1797 p^{8} T^{19} + 210 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 11 T + 256 T^{2} - 2229 T^{3} + 29696 T^{4} - 7062 p T^{5} + 69901 p T^{6} - 14018958 T^{7} + 113431882 T^{8} - 652551355 T^{9} + 4516901528 T^{10} - 23070551898 T^{11} + 4516901528 p T^{12} - 652551355 p^{2} T^{13} + 113431882 p^{3} T^{14} - 14018958 p^{4} T^{15} + 69901 p^{6} T^{16} - 7062 p^{7} T^{17} + 29696 p^{7} T^{18} - 2229 p^{8} T^{19} + 256 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 11 T + 186 T^{2} - 36 p T^{3} + 13381 T^{4} - 71849 T^{5} + 574101 T^{6} - 2983567 T^{7} + 23083132 T^{8} - 140130896 T^{9} + 997620599 T^{10} - 6012388578 T^{11} + 997620599 p T^{12} - 140130896 p^{2} T^{13} + 23083132 p^{3} T^{14} - 2983567 p^{4} T^{15} + 574101 p^{5} T^{16} - 71849 p^{6} T^{17} + 13381 p^{7} T^{18} - 36 p^{9} T^{19} + 186 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 + 9 T + 232 T^{2} + 1906 T^{3} + 26571 T^{4} + 196584 T^{5} + 2032638 T^{6} + 13416832 T^{7} + 118170853 T^{8} + 17135807 p T^{9} + 5649585169 T^{10} + 30915013740 T^{11} + 5649585169 p T^{12} + 17135807 p^{3} T^{13} + 118170853 p^{3} T^{14} + 13416832 p^{4} T^{15} + 2032638 p^{5} T^{16} + 196584 p^{6} T^{17} + 26571 p^{7} T^{18} + 1906 p^{8} T^{19} + 232 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 30 T + 701 T^{2} - 12004 T^{3} + 174762 T^{4} - 2172787 T^{5} + 23980560 T^{6} - 236435049 T^{7} + 2114920243 T^{8} - 17213040751 T^{9} + 128434946185 T^{10} - 878230734934 T^{11} + 128434946185 p T^{12} - 17213040751 p^{2} T^{13} + 2114920243 p^{3} T^{14} - 236435049 p^{4} T^{15} + 23980560 p^{5} T^{16} - 2172787 p^{6} T^{17} + 174762 p^{7} T^{18} - 12004 p^{8} T^{19} + 701 p^{9} T^{20} - 30 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - T + 271 T^{2} - 674 T^{3} + 39800 T^{4} - 119127 T^{5} + 86968 p T^{6} - 12596583 T^{7} + 313345903 T^{8} - 932979252 T^{9} + 18684612817 T^{10} - 50630223558 T^{11} + 18684612817 p T^{12} - 932979252 p^{2} T^{13} + 313345903 p^{3} T^{14} - 12596583 p^{4} T^{15} + 86968 p^{6} T^{16} - 119127 p^{6} T^{17} + 39800 p^{7} T^{18} - 674 p^{8} T^{19} + 271 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 + 9 T + 287 T^{2} + 2917 T^{3} + 43696 T^{4} + 438472 T^{5} + 4810016 T^{6} + 43137862 T^{7} + 405382115 T^{8} + 3232709151 T^{9} + 26655294941 T^{10} + 192369944938 T^{11} + 26655294941 p T^{12} + 3232709151 p^{2} T^{13} + 405382115 p^{3} T^{14} + 43137862 p^{4} T^{15} + 4810016 p^{5} T^{16} + 438472 p^{6} T^{17} + 43696 p^{7} T^{18} + 2917 p^{8} T^{19} + 287 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 9 T + 309 T^{2} - 2995 T^{3} + 56736 T^{4} - 513344 T^{5} + 7196512 T^{6} - 60377200 T^{7} + 687127067 T^{8} - 5246460695 T^{9} + 51196691607 T^{10} - 351357052378 T^{11} + 51196691607 p T^{12} - 5246460695 p^{2} T^{13} + 687127067 p^{3} T^{14} - 60377200 p^{4} T^{15} + 7196512 p^{5} T^{16} - 513344 p^{6} T^{17} + 56736 p^{7} T^{18} - 2995 p^{8} T^{19} + 309 p^{9} T^{20} - 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 5 T + 274 T^{2} + 1510 T^{3} + 45189 T^{4} + 254438 T^{5} + 5301274 T^{6} + 29854408 T^{7} + 488468649 T^{8} + 2626010061 T^{9} + 36309296409 T^{10} + 180584213812 T^{11} + 36309296409 p T^{12} + 2626010061 p^{2} T^{13} + 488468649 p^{3} T^{14} + 29854408 p^{4} T^{15} + 5301274 p^{5} T^{16} + 254438 p^{6} T^{17} + 45189 p^{7} T^{18} + 1510 p^{8} T^{19} + 274 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 25 T + 824 T^{2} - 15079 T^{3} + 287560 T^{4} - 4149810 T^{5} + 58243735 T^{6} - 690458918 T^{7} + 7766040850 T^{8} - 77355289541 T^{9} + 724516206712 T^{10} - 6122675971974 T^{11} + 724516206712 p T^{12} - 77355289541 p^{2} T^{13} + 7766040850 p^{3} T^{14} - 690458918 p^{4} T^{15} + 58243735 p^{5} T^{16} - 4149810 p^{6} T^{17} + 287560 p^{7} T^{18} - 15079 p^{8} T^{19} + 824 p^{9} T^{20} - 25 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 26 T + 669 T^{2} - 11598 T^{3} + 186880 T^{4} - 2491515 T^{5} + 31168688 T^{6} - 344609759 T^{7} + 3628703739 T^{8} - 34828198139 T^{9} + 322511347151 T^{10} - 2759152405878 T^{11} + 322511347151 p T^{12} - 34828198139 p^{2} T^{13} + 3628703739 p^{3} T^{14} - 344609759 p^{4} T^{15} + 31168688 p^{5} T^{16} - 2491515 p^{6} T^{17} + 186880 p^{7} T^{18} - 11598 p^{8} T^{19} + 669 p^{9} T^{20} - 26 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 - 10 T + 429 T^{2} - 2845 T^{3} + 72849 T^{4} - 227119 T^{5} + 6099955 T^{6} + 16215180 T^{7} + 213779544 T^{8} + 5010043569 T^{9} - 4085593914 T^{10} + 501981744242 T^{11} - 4085593914 p T^{12} + 5010043569 p^{2} T^{13} + 213779544 p^{3} T^{14} + 16215180 p^{4} T^{15} + 6099955 p^{5} T^{16} - 227119 p^{6} T^{17} + 72849 p^{7} T^{18} - 2845 p^{8} T^{19} + 429 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 14 T + 432 T^{2} - 5213 T^{3} + 91391 T^{4} - 971355 T^{5} + 13186754 T^{6} - 1580888 p T^{7} + 1466806133 T^{8} - 12562722519 T^{9} + 133877096947 T^{10} - 1063877575318 T^{11} + 133877096947 p T^{12} - 12562722519 p^{2} T^{13} + 1466806133 p^{3} T^{14} - 1580888 p^{5} T^{15} + 13186754 p^{5} T^{16} - 971355 p^{6} T^{17} + 91391 p^{7} T^{18} - 5213 p^{8} T^{19} + 432 p^{9} T^{20} - 14 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 6 T + 278 T^{2} - 1653 T^{3} + 52205 T^{4} - 184765 T^{5} + 5883822 T^{6} - 9780072 T^{7} + 515116641 T^{8} + 766604571 T^{9} + 37658186759 T^{10} + 106229769626 T^{11} + 37658186759 p T^{12} + 766604571 p^{2} T^{13} + 515116641 p^{3} T^{14} - 9780072 p^{4} T^{15} + 5883822 p^{5} T^{16} - 184765 p^{6} T^{17} + 52205 p^{7} T^{18} - 1653 p^{8} T^{19} + 278 p^{9} T^{20} - 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 11 T + 709 T^{2} + 8200 T^{3} + 251742 T^{4} + 2803998 T^{5} + 57930728 T^{6} + 590971558 T^{7} + 9417171155 T^{8} + 85658750135 T^{9} + 1123475782353 T^{10} + 8932444105380 T^{11} + 1123475782353 p T^{12} + 85658750135 p^{2} T^{13} + 9417171155 p^{3} T^{14} + 590971558 p^{4} T^{15} + 57930728 p^{5} T^{16} + 2803998 p^{6} T^{17} + 251742 p^{7} T^{18} + 8200 p^{8} T^{19} + 709 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 12 T + 540 T^{2} - 5305 T^{3} + 149041 T^{4} - 1290625 T^{5} + 28993800 T^{6} - 228615070 T^{7} + 4355384651 T^{8} - 31079080275 T^{9} + 519218750463 T^{10} - 3350009169554 T^{11} + 519218750463 p T^{12} - 31079080275 p^{2} T^{13} + 4355384651 p^{3} T^{14} - 228615070 p^{4} T^{15} + 28993800 p^{5} T^{16} - 1290625 p^{6} T^{17} + 149041 p^{7} T^{18} - 5305 p^{8} T^{19} + 540 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.63021615208896224349118631513, −2.61105117683753290056393823228, −2.55854071038650597342798180191, −2.36669826209070313131493401152, −2.06646663265961951105010872116, −1.97468511620853357539780444473, −1.96718873855967561249491849954, −1.90966070490207513431132980905, −1.89128175344365772275890696884, −1.82866755090597671809648122203, −1.73680188707531252485018085109, −1.71593763960218407290257354477, −1.52976062631517212578287766371, −1.52097783129383622451409166016, −1.49490662147377805926723337073, −1.14122650042037715829717721434, −1.03410229667270366405955682200, −0.887084715520281318181415636793, −0.886274717986117585879001315477, −0.850506737254918407583931777681, −0.67684563429216991243980847609, −0.57299766300372686642609737005, −0.54896692008450585153477403731, −0.41411692505650964292306899140, −0.22030260114921519763969893299, 0.22030260114921519763969893299, 0.41411692505650964292306899140, 0.54896692008450585153477403731, 0.57299766300372686642609737005, 0.67684563429216991243980847609, 0.850506737254918407583931777681, 0.886274717986117585879001315477, 0.887084715520281318181415636793, 1.03410229667270366405955682200, 1.14122650042037715829717721434, 1.49490662147377805926723337073, 1.52097783129383622451409166016, 1.52976062631517212578287766371, 1.71593763960218407290257354477, 1.73680188707531252485018085109, 1.82866755090597671809648122203, 1.89128175344365772275890696884, 1.90966070490207513431132980905, 1.96718873855967561249491849954, 1.97468511620853357539780444473, 2.06646663265961951105010872116, 2.36669826209070313131493401152, 2.55854071038650597342798180191, 2.61105117683753290056393823228, 2.63021615208896224349118631513
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.