Dirichlet series
| L(s) = 1 | + 5·5-s − 13·7-s + 11-s − 10·13-s + 10·17-s − 5·19-s − 23-s − 5·25-s + 14·29-s − 21·31-s − 65·35-s − 21·37-s + 31·41-s − 18·43-s + 10·47-s + 57·49-s + 12·53-s + 5·55-s − 2·59-s − 32·61-s − 50·65-s − 25·67-s + 11·71-s − 14·73-s − 13·77-s − 9·79-s − 25·83-s + ⋯ |
| L(s) = 1 | + 2.23·5-s − 4.91·7-s + 0.301·11-s − 2.77·13-s + 2.42·17-s − 1.14·19-s − 0.208·23-s − 25-s + 2.59·29-s − 3.77·31-s − 10.9·35-s − 3.45·37-s + 4.84·41-s − 2.74·43-s + 1.45·47-s + 57/7·49-s + 1.64·53-s + 0.674·55-s − 0.260·59-s − 4.09·61-s − 6.20·65-s − 3.05·67-s + 1.30·71-s − 1.63·73-s − 1.48·77-s − 1.01·79-s − 2.74·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 223^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.85151\times 10^{14}\) |
| Root analytic conductor: | \(8.00649\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 223^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( 1 \) |
| 3 | \( 1 \) | |
| 223 | \( ( 1 + T )^{8} \) | |
| good | 5 | \( 1 - p T + 6 p T^{2} - 101 T^{3} + 359 T^{4} - 936 T^{5} + 2632 T^{6} - 5866 T^{7} + 14628 T^{8} - 5866 p T^{9} + 2632 p^{2} T^{10} - 936 p^{3} T^{11} + 359 p^{4} T^{12} - 101 p^{5} T^{13} + 6 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 13 T + 16 p T^{2} + 716 T^{3} + 528 p T^{4} + 16018 T^{5} + 59548 T^{6} + 192211 T^{7} + 543510 T^{8} + 192211 p T^{9} + 59548 p^{2} T^{10} + 16018 p^{3} T^{11} + 528 p^{5} T^{12} + 716 p^{5} T^{13} + 16 p^{7} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - T + 5 p T^{2} - 41 T^{3} + 1410 T^{4} - 909 T^{5} + 23505 T^{6} - 14397 T^{7} + 292986 T^{8} - 14397 p T^{9} + 23505 p^{2} T^{10} - 909 p^{3} T^{11} + 1410 p^{4} T^{12} - 41 p^{5} T^{13} + 5 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 10 T + 83 T^{2} + 456 T^{3} + 2598 T^{4} + 12232 T^{5} + 57413 T^{6} + 221918 T^{7} + 859826 T^{8} + 221918 p T^{9} + 57413 p^{2} T^{10} + 12232 p^{3} T^{11} + 2598 p^{4} T^{12} + 456 p^{5} T^{13} + 83 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 10 T + 108 T^{2} - 558 T^{3} + 3021 T^{4} - 6399 T^{5} + 762 p T^{6} + 118715 T^{7} - 402700 T^{8} + 118715 p T^{9} + 762 p^{3} T^{10} - 6399 p^{3} T^{11} + 3021 p^{4} T^{12} - 558 p^{5} T^{13} + 108 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 5 T + 83 T^{2} + 389 T^{3} + 206 p T^{4} + 16250 T^{5} + 122087 T^{6} + 441190 T^{7} + 2730658 T^{8} + 441190 p T^{9} + 122087 p^{2} T^{10} + 16250 p^{3} T^{11} + 206 p^{5} T^{12} + 389 p^{5} T^{13} + 83 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + T + 49 T^{2} + 237 T^{3} + 2226 T^{4} + 7181 T^{5} + 82855 T^{6} + 277089 T^{7} + 1786858 T^{8} + 277089 p T^{9} + 82855 p^{2} T^{10} + 7181 p^{3} T^{11} + 2226 p^{4} T^{12} + 237 p^{5} T^{13} + 49 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 14 T + 224 T^{2} - 1994 T^{3} + 19393 T^{4} - 134555 T^{5} + 1008624 T^{6} - 5792797 T^{7} + 35381532 T^{8} - 5792797 p T^{9} + 1008624 p^{2} T^{10} - 134555 p^{3} T^{11} + 19393 p^{4} T^{12} - 1994 p^{5} T^{13} + 224 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 21 T + 289 T^{2} + 2861 T^{3} + 25850 T^{4} + 205006 T^{5} + 47925 p T^{6} + 9336450 T^{7} + 54588270 T^{8} + 9336450 p T^{9} + 47925 p^{3} T^{10} + 205006 p^{3} T^{11} + 25850 p^{4} T^{12} + 2861 p^{5} T^{13} + 289 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 21 T + 317 T^{2} + 3724 T^{3} + 35983 T^{4} + 308260 T^{5} + 2355447 T^{6} + 16331091 T^{7} + 104208025 T^{8} + 16331091 p T^{9} + 2355447 p^{2} T^{10} + 308260 p^{3} T^{11} + 35983 p^{4} T^{12} + 3724 p^{5} T^{13} + 317 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 31 T + 529 T^{2} - 6543 T^{3} + 65990 T^{4} - 576298 T^{5} + 4523223 T^{6} - 32547204 T^{7} + 216517746 T^{8} - 32547204 p T^{9} + 4523223 p^{2} T^{10} - 576298 p^{3} T^{11} + 65990 p^{4} T^{12} - 6543 p^{5} T^{13} + 529 p^{6} T^{14} - 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 18 T + 290 T^{2} + 3392 T^{3} + 34087 T^{4} + 306841 T^{5} + 2498008 T^{6} + 18492099 T^{7} + 128152936 T^{8} + 18492099 p T^{9} + 2498008 p^{2} T^{10} + 306841 p^{3} T^{11} + 34087 p^{4} T^{12} + 3392 p^{5} T^{13} + 290 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 226 T^{2} - 1979 T^{3} + 24748 T^{4} - 202646 T^{5} + 1815079 T^{6} - 13538111 T^{7} + 97818600 T^{8} - 13538111 p T^{9} + 1815079 p^{2} T^{10} - 202646 p^{3} T^{11} + 24748 p^{4} T^{12} - 1979 p^{5} T^{13} + 226 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 12 T + 230 T^{2} - 1960 T^{3} + 24687 T^{4} - 192703 T^{5} + 1926168 T^{6} - 13529797 T^{7} + 113704456 T^{8} - 13529797 p T^{9} + 1926168 p^{2} T^{10} - 192703 p^{3} T^{11} + 24687 p^{4} T^{12} - 1960 p^{5} T^{13} + 230 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 2 T + 265 T^{2} + 342 T^{3} + 34642 T^{4} + 29142 T^{5} + 3061055 T^{6} + 1769250 T^{7} + 204478938 T^{8} + 1769250 p T^{9} + 3061055 p^{2} T^{10} + 29142 p^{3} T^{11} + 34642 p^{4} T^{12} + 342 p^{5} T^{13} + 265 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 32 T + 749 T^{2} + 12362 T^{3} + 174494 T^{4} + 2051410 T^{5} + 21570827 T^{6} + 197747644 T^{7} + 1644692610 T^{8} + 197747644 p T^{9} + 21570827 p^{2} T^{10} + 2051410 p^{3} T^{11} + 174494 p^{4} T^{12} + 12362 p^{5} T^{13} + 749 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 25 T + 525 T^{2} + 6780 T^{3} + 76820 T^{4} + 636082 T^{5} + 4955955 T^{6} + 31715563 T^{7} + 253871590 T^{8} + 31715563 p T^{9} + 4955955 p^{2} T^{10} + 636082 p^{3} T^{11} + 76820 p^{4} T^{12} + 6780 p^{5} T^{13} + 525 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 11 T + 373 T^{2} - 3085 T^{3} + 60638 T^{4} - 381893 T^{5} + 6068747 T^{6} - 30759971 T^{7} + 466104290 T^{8} - 30759971 p T^{9} + 6068747 p^{2} T^{10} - 381893 p^{3} T^{11} + 60638 p^{4} T^{12} - 3085 p^{5} T^{13} + 373 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 14 T + 434 T^{2} + 5525 T^{3} + 95216 T^{4} + 1011912 T^{5} + 12819531 T^{6} + 113320225 T^{7} + 1137285260 T^{8} + 113320225 p T^{9} + 12819531 p^{2} T^{10} + 1011912 p^{3} T^{11} + 95216 p^{4} T^{12} + 5525 p^{5} T^{13} + 434 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 9 T + 189 T^{2} + 2128 T^{3} + 34396 T^{4} + 337898 T^{5} + 3743563 T^{6} + 35902435 T^{7} + 356457174 T^{8} + 35902435 p T^{9} + 3743563 p^{2} T^{10} + 337898 p^{3} T^{11} + 34396 p^{4} T^{12} + 2128 p^{5} T^{13} + 189 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 25 T + 573 T^{2} + 9002 T^{3} + 134789 T^{4} + 1650666 T^{5} + 19348375 T^{6} + 195648613 T^{7} + 1902337721 T^{8} + 195648613 p T^{9} + 19348375 p^{2} T^{10} + 1650666 p^{3} T^{11} + 134789 p^{4} T^{12} + 9002 p^{5} T^{13} + 573 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 23 T + 637 T^{2} - 10127 T^{3} + 171614 T^{4} - 2116722 T^{5} + 27197943 T^{6} - 275440868 T^{7} + 2897150346 T^{8} - 275440868 p T^{9} + 27197943 p^{2} T^{10} - 2116722 p^{3} T^{11} + 171614 p^{4} T^{12} - 10127 p^{5} T^{13} + 637 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 11 T + 541 T^{2} + 3845 T^{3} + 127598 T^{4} + 634135 T^{5} + 19241763 T^{6} + 73630137 T^{7} + 2136662818 T^{8} + 73630137 p T^{9} + 19241763 p^{2} T^{10} + 634135 p^{3} T^{11} + 127598 p^{4} T^{12} + 3845 p^{5} T^{13} + 541 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.55998614819301280404937054786, −3.29586498185932232566040721445, −3.25236906388521915940163842715, −3.19385618262827298042522618292, −3.17499516321317910209631775917, −3.09501068912829013777205529533, −3.07542493992310473814056742665, −2.77455951603469209409628691368, −2.73056448711501088359337002849, −2.64063994950186727056907389144, −2.43843850689705863751597001796, −2.34653255636167153481533104890, −2.33767743713912157137207456365, −2.28383693997229581283228803117, −2.25784847763664836605854528824, −2.08703678289345527511040491371, −2.02331204872724167279418955934, −1.50563718876197762467711548543, −1.42138412160761671443274756683, −1.41789583320954440897848984125, −1.33633923364068867274071046461, −1.31202940752388541539646079464, −1.23859110407115269486435830442, −1.13519312446881800325304916984, −0.857981749622557558994417405738, 0, 0, 0, 0, 0, 0, 0, 0, 0.857981749622557558994417405738, 1.13519312446881800325304916984, 1.23859110407115269486435830442, 1.31202940752388541539646079464, 1.33633923364068867274071046461, 1.41789583320954440897848984125, 1.42138412160761671443274756683, 1.50563718876197762467711548543, 2.02331204872724167279418955934, 2.08703678289345527511040491371, 2.25784847763664836605854528824, 2.28383693997229581283228803117, 2.33767743713912157137207456365, 2.34653255636167153481533104890, 2.43843850689705863751597001796, 2.64063994950186727056907389144, 2.73056448711501088359337002849, 2.77455951603469209409628691368, 3.07542493992310473814056742665, 3.09501068912829013777205529533, 3.17499516321317910209631775917, 3.19385618262827298042522618292, 3.25236906388521915940163842715, 3.29586498185932232566040721445, 3.55998614819301280404937054786
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.