Dirichlet series
| L(s) = 1 | − 11·2-s − 11·3-s + 66·4-s + 2·5-s + 121·6-s − 2·7-s − 286·8-s + 66·9-s − 22·10-s − 6·11-s − 726·12-s − 7·13-s + 22·14-s − 22·15-s + 1.00e3·16-s − 16·17-s − 726·18-s + 11·19-s + 132·20-s + 22·21-s + 66·22-s − 11·23-s + 3.14e3·24-s − 16·25-s + 77·26-s − 286·27-s − 132·28-s + ⋯ |
| L(s) = 1 | − 7.77·2-s − 6.35·3-s + 33·4-s + 0.894·5-s + 49.3·6-s − 0.755·7-s − 101.·8-s + 22·9-s − 6.95·10-s − 1.80·11-s − 209.·12-s − 1.94·13-s + 5.87·14-s − 5.68·15-s + 250.·16-s − 3.88·17-s − 171.·18-s + 2.52·19-s + 29.5·20-s + 4.80·21-s + 14.0·22-s − 2.29·23-s + 642.·24-s − 3.19·25-s + 15.1·26-s − 55.0·27-s − 24.9·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 19^{11} \cdot 53^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{11} \cdot 3^{11} \cdot 19^{11} \cdot 53^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{11} \cdot 3^{11} \cdot 19^{11} \cdot 53^{11}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.29637\times 10^{18}\) |
| Root analytic conductor: | \(6.94590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(11\) |
| Selberg data: | \((22,\ 2^{11} \cdot 3^{11} \cdot 19^{11} \cdot 53^{11} ,\ ( \ : [1/2]^{11} ),\ -1 )\) |
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 | \( ( 1 + T )^{11} \) |
| 3 | \( ( 1 + T )^{11} \) | |
| 19 | \( ( 1 - T )^{11} \) | |
| 53 | \( ( 1 - T )^{11} \) | |
| good | 5 | \( 1 - 2 T + 4 p T^{2} - 23 T^{3} + 194 T^{4} - 164 T^{5} + 1438 T^{6} - 1237 T^{7} + 9238 T^{8} - 8784 T^{9} + 51703 T^{10} - 49206 T^{11} + 51703 p T^{12} - 8784 p^{2} T^{13} + 9238 p^{3} T^{14} - 1237 p^{4} T^{15} + 1438 p^{5} T^{16} - 164 p^{6} T^{17} + 194 p^{7} T^{18} - 23 p^{8} T^{19} + 4 p^{10} T^{20} - 2 p^{10} T^{21} + p^{11} T^{22} \) |
| 7 | \( 1 + 2 T + 29 T^{2} + 54 T^{3} + 475 T^{4} + 940 T^{5} + 849 p T^{6} + 11549 T^{7} + 59778 T^{8} + 109110 T^{9} + 504906 T^{10} + 846114 T^{11} + 504906 p T^{12} + 109110 p^{2} T^{13} + 59778 p^{3} T^{14} + 11549 p^{4} T^{15} + 849 p^{6} T^{16} + 940 p^{6} T^{17} + 475 p^{7} T^{18} + 54 p^{8} T^{19} + 29 p^{9} T^{20} + 2 p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 6 T + 60 T^{2} + 219 T^{3} + 1401 T^{4} + 3360 T^{5} + 1660 p T^{6} + 14422 T^{7} + 107359 T^{8} - 386534 T^{9} - 114283 T^{10} - 7259618 T^{11} - 114283 p T^{12} - 386534 p^{2} T^{13} + 107359 p^{3} T^{14} + 14422 p^{4} T^{15} + 1660 p^{6} T^{16} + 3360 p^{6} T^{17} + 1401 p^{7} T^{18} + 219 p^{8} T^{19} + 60 p^{9} T^{20} + 6 p^{10} T^{21} + p^{11} T^{22} \) | |
| 13 | \( 1 + 7 T + 6 p T^{2} + 393 T^{3} + 2835 T^{4} + 12608 T^{5} + 73565 T^{6} + 297340 T^{7} + 1467640 T^{8} + 5385045 T^{9} + 23374393 T^{10} + 5995894 p T^{11} + 23374393 p T^{12} + 5385045 p^{2} T^{13} + 1467640 p^{3} T^{14} + 297340 p^{4} T^{15} + 73565 p^{5} T^{16} + 12608 p^{6} T^{17} + 2835 p^{7} T^{18} + 393 p^{8} T^{19} + 6 p^{10} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 + 16 T + 183 T^{2} + 1334 T^{3} + 7533 T^{4} + 1720 p T^{5} + 84251 T^{6} + 169239 T^{7} + 941322 T^{8} + 8789472 T^{9} + 64280790 T^{10} + 311087510 T^{11} + 64280790 p T^{12} + 8789472 p^{2} T^{13} + 941322 p^{3} T^{14} + 169239 p^{4} T^{15} + 84251 p^{5} T^{16} + 1720 p^{7} T^{17} + 7533 p^{7} T^{18} + 1334 p^{8} T^{19} + 183 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 23 | \( 1 + 11 T + 194 T^{2} + 1716 T^{3} + 17508 T^{4} + 130270 T^{5} + 989465 T^{6} + 6361430 T^{7} + 39447060 T^{8} + 9674353 p T^{9} + 1178474742 T^{10} + 5856349228 T^{11} + 1178474742 p T^{12} + 9674353 p^{3} T^{13} + 39447060 p^{3} T^{14} + 6361430 p^{4} T^{15} + 989465 p^{5} T^{16} + 130270 p^{6} T^{17} + 17508 p^{7} T^{18} + 1716 p^{8} T^{19} + 194 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 7 T + 163 T^{2} + 40 p T^{3} + 14128 T^{4} + 96020 T^{5} + 832765 T^{6} + 5264185 T^{7} + 37177784 T^{8} + 214662965 T^{9} + 1322471407 T^{10} + 6929329166 T^{11} + 1322471407 p T^{12} + 214662965 p^{2} T^{13} + 37177784 p^{3} T^{14} + 5264185 p^{4} T^{15} + 832765 p^{5} T^{16} + 96020 p^{6} T^{17} + 14128 p^{7} T^{18} + 40 p^{9} T^{19} + 163 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 12 T + 207 T^{2} - 1927 T^{3} + 20712 T^{4} - 167111 T^{5} + 1390266 T^{6} - 10015042 T^{7} + 69660249 T^{8} - 450767853 T^{9} + 2718340377 T^{10} - 15758444366 T^{11} + 2718340377 p T^{12} - 450767853 p^{2} T^{13} + 69660249 p^{3} T^{14} - 10015042 p^{4} T^{15} + 1390266 p^{5} T^{16} - 167111 p^{6} T^{17} + 20712 p^{7} T^{18} - 1927 p^{8} T^{19} + 207 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 3 T + 98 T^{2} - 283 T^{3} + 6431 T^{4} - 9606 T^{5} + 319403 T^{6} - 268054 T^{7} + 15530426 T^{8} - 14199723 T^{9} + 679980329 T^{10} - 789107974 T^{11} + 679980329 p T^{12} - 14199723 p^{2} T^{13} + 15530426 p^{3} T^{14} - 268054 p^{4} T^{15} + 319403 p^{5} T^{16} - 9606 p^{6} T^{17} + 6431 p^{7} T^{18} - 283 p^{8} T^{19} + 98 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 11 T + 311 T^{2} - 2705 T^{3} + 45635 T^{4} - 343947 T^{5} + 4370628 T^{6} - 29198142 T^{7} + 303092829 T^{8} - 1804913570 T^{9} + 16003133244 T^{10} - 84591371122 T^{11} + 16003133244 p T^{12} - 1804913570 p^{2} T^{13} + 303092829 p^{3} T^{14} - 29198142 p^{4} T^{15} + 4370628 p^{5} T^{16} - 343947 p^{6} T^{17} + 45635 p^{7} T^{18} - 2705 p^{8} T^{19} + 311 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 - 7 T + 254 T^{2} - 1828 T^{3} + 28016 T^{4} - 189850 T^{5} + 1708741 T^{6} - 9323822 T^{7} + 60537376 T^{8} - 170629807 T^{9} + 1406280154 T^{10} + 57487268 T^{11} + 1406280154 p T^{12} - 170629807 p^{2} T^{13} + 60537376 p^{3} T^{14} - 9323822 p^{4} T^{15} + 1708741 p^{5} T^{16} - 189850 p^{6} T^{17} + 28016 p^{7} T^{18} - 1828 p^{8} T^{19} + 254 p^{9} T^{20} - 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 + 37 T + 961 T^{2} + 18079 T^{3} + 283801 T^{4} + 3757391 T^{5} + 43918102 T^{6} + 455166870 T^{7} + 4274135363 T^{8} + 36384289722 T^{9} + 284133579722 T^{10} + 2029934153562 T^{11} + 284133579722 p T^{12} + 36384289722 p^{2} T^{13} + 4274135363 p^{3} T^{14} + 455166870 p^{4} T^{15} + 43918102 p^{5} T^{16} + 3757391 p^{6} T^{17} + 283801 p^{7} T^{18} + 18079 p^{8} T^{19} + 961 p^{9} T^{20} + 37 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 + 16 T + 506 T^{2} + 6032 T^{3} + 103517 T^{4} + 951448 T^{5} + 11403524 T^{6} + 81585588 T^{7} + 768371536 T^{8} + 4414384312 T^{9} + 39019869652 T^{10} + 220404658440 T^{11} + 39019869652 p T^{12} + 4414384312 p^{2} T^{13} + 768371536 p^{3} T^{14} + 81585588 p^{4} T^{15} + 11403524 p^{5} T^{16} + 951448 p^{6} T^{17} + 103517 p^{7} T^{18} + 6032 p^{8} T^{19} + 506 p^{9} T^{20} + 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 - 24 T + 661 T^{2} - 10794 T^{3} + 178276 T^{4} - 2258942 T^{5} + 28235013 T^{6} - 296228966 T^{7} + 3054104252 T^{8} - 27508310034 T^{9} + 243772105121 T^{10} - 1918589268944 T^{11} + 243772105121 p T^{12} - 27508310034 p^{2} T^{13} + 3054104252 p^{3} T^{14} - 296228966 p^{4} T^{15} + 28235013 p^{5} T^{16} - 2258942 p^{6} T^{17} + 178276 p^{7} T^{18} - 10794 p^{8} T^{19} + 661 p^{9} T^{20} - 24 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 + 28 T + 766 T^{2} + 12685 T^{3} + 199539 T^{4} + 2291678 T^{5} + 25195860 T^{6} + 207387576 T^{7} + 1695219223 T^{8} + 9914344278 T^{9} + 71263761365 T^{10} + 409476997078 T^{11} + 71263761365 p T^{12} + 9914344278 p^{2} T^{13} + 1695219223 p^{3} T^{14} + 207387576 p^{4} T^{15} + 25195860 p^{5} T^{16} + 2291678 p^{6} T^{17} + 199539 p^{7} T^{18} + 12685 p^{8} T^{19} + 766 p^{9} T^{20} + 28 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 - 4 T + 396 T^{2} - 1631 T^{3} + 75168 T^{4} - 311736 T^{5} + 8876715 T^{6} - 37764052 T^{7} + 746510386 T^{8} - 3382350308 T^{9} + 52530128596 T^{10} - 254577659802 T^{11} + 52530128596 p T^{12} - 3382350308 p^{2} T^{13} + 746510386 p^{3} T^{14} - 37764052 p^{4} T^{15} + 8876715 p^{5} T^{16} - 311736 p^{6} T^{17} + 75168 p^{7} T^{18} - 1631 p^{8} T^{19} + 396 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 7 T + 263 T^{2} + 1437 T^{3} + 33449 T^{4} + 124687 T^{5} + 2748471 T^{6} + 9384756 T^{7} + 211304504 T^{8} + 1024533594 T^{9} + 17686597072 T^{10} + 95260794270 T^{11} + 17686597072 p T^{12} + 1024533594 p^{2} T^{13} + 211304504 p^{3} T^{14} + 9384756 p^{4} T^{15} + 2748471 p^{5} T^{16} + 124687 p^{6} T^{17} + 33449 p^{7} T^{18} + 1437 p^{8} T^{19} + 263 p^{9} T^{20} + 7 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 - 5 T + 461 T^{2} - 3121 T^{3} + 96102 T^{4} - 774834 T^{5} + 12572030 T^{6} - 104861325 T^{7} + 1243408761 T^{8} - 9358180061 T^{9} + 106300984409 T^{10} - 721408297564 T^{11} + 106300984409 p T^{12} - 9358180061 p^{2} T^{13} + 1243408761 p^{3} T^{14} - 104861325 p^{4} T^{15} + 12572030 p^{5} T^{16} - 774834 p^{6} T^{17} + 96102 p^{7} T^{18} - 3121 p^{8} T^{19} + 461 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 + 11 T + 458 T^{2} + 5971 T^{3} + 121471 T^{4} + 1515064 T^{5} + 23086287 T^{6} + 253101862 T^{7} + 3233145964 T^{8} + 31458378941 T^{9} + 343494397731 T^{10} + 2987492467822 T^{11} + 343494397731 p T^{12} + 31458378941 p^{2} T^{13} + 3233145964 p^{3} T^{14} + 253101862 p^{4} T^{15} + 23086287 p^{5} T^{16} + 1515064 p^{6} T^{17} + 121471 p^{7} T^{18} + 5971 p^{8} T^{19} + 458 p^{9} T^{20} + 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 + 10 T + 577 T^{2} + 4523 T^{3} + 146272 T^{4} + 828871 T^{5} + 21565126 T^{6} + 69737170 T^{7} + 2123459053 T^{8} + 1110902383 T^{9} + 170312612667 T^{10} - 194269945114 T^{11} + 170312612667 p T^{12} + 1110902383 p^{2} T^{13} + 2123459053 p^{3} T^{14} + 69737170 p^{4} T^{15} + 21565126 p^{5} T^{16} + 828871 p^{6} T^{17} + 146272 p^{7} T^{18} + 4523 p^{8} T^{19} + 577 p^{9} T^{20} + 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 + 24 T + 814 T^{2} + 15031 T^{3} + 315107 T^{4} + 4768796 T^{5} + 76720544 T^{6} + 988481484 T^{7} + 13176065299 T^{8} + 147130773772 T^{9} + 1681670485851 T^{10} + 16398179468410 T^{11} + 1681670485851 p T^{12} + 147130773772 p^{2} T^{13} + 13176065299 p^{3} T^{14} + 988481484 p^{4} T^{15} + 76720544 p^{5} T^{16} + 4768796 p^{6} T^{17} + 315107 p^{7} T^{18} + 15031 p^{8} T^{19} + 814 p^{9} T^{20} + 24 p^{10} T^{21} + p^{11} T^{22} \) | |
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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
−2.99651240724733272094320045991, −2.85693802623287592158184860950, −2.56182353352266568243968723013, −2.41178151286988743705303476872, −2.38743635461352556591395959426, −2.29319148153548192945352714789, −2.24721824297029967339236645445, −2.18673018687192013088088596458, −2.15595299699098995613376651395, −2.05263287008315425424756124801, −2.01687412561450706298057131132, −2.01327034598780239047156986925, −1.93969769661782771733266286315, −1.88942815868145833889116243813, −1.47879161371297053853265391678, −1.40047933535407613697192247132, −1.38019595959850813839538452364, −1.35837345284929925121007389616, −1.18990905784933293822388014541, −1.06461193325320060009758215797, −1.06309181521412593648405122066, −1.03300299019970064501218649114, −1.02741572464397157273230332400, −0.999121747361389695177722647443, −0.72775166550274418711206467849, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.72775166550274418711206467849, 0.999121747361389695177722647443, 1.02741572464397157273230332400, 1.03300299019970064501218649114, 1.06309181521412593648405122066, 1.06461193325320060009758215797, 1.18990905784933293822388014541, 1.35837345284929925121007389616, 1.38019595959850813839538452364, 1.40047933535407613697192247132, 1.47879161371297053853265391678, 1.88942815868145833889116243813, 1.93969769661782771733266286315, 2.01327034598780239047156986925, 2.01687412561450706298057131132, 2.05263287008315425424756124801, 2.15595299699098995613376651395, 2.18673018687192013088088596458, 2.24721824297029967339236645445, 2.29319148153548192945352714789, 2.38743635461352556591395959426, 2.41178151286988743705303476872, 2.56182353352266568243968723013, 2.85693802623287592158184860950, 2.99651240724733272094320045991
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.