Dirichlet series
| L(s) = 1 | + 8·2-s + 36·4-s − 6·5-s − 6·7-s + 120·8-s − 48·10-s − 11·11-s − 13-s − 48·14-s + 330·16-s − 16·17-s − 19-s − 216·20-s − 88·22-s − 14·23-s + 3·25-s − 8·26-s − 216·28-s − 21·29-s − 6·31-s + 792·32-s − 128·34-s + 36·35-s − 14·37-s − 8·38-s − 720·40-s − 16·41-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s − 2.68·5-s − 2.26·7-s + 42.4·8-s − 15.1·10-s − 3.31·11-s − 0.277·13-s − 12.8·14-s + 82.5·16-s − 3.88·17-s − 0.229·19-s − 48.2·20-s − 18.7·22-s − 2.91·23-s + 3/5·25-s − 1.56·26-s − 40.8·28-s − 3.89·29-s − 1.07·31-s + 140.·32-s − 21.9·34-s + 6.08·35-s − 2.30·37-s − 1.29·38-s − 113.·40-s − 2.49·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 223^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11387\times 10^{12}\) |
| Root analytic conductor: | \(5.66144\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 223^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 \) | |
| 223 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + 6 T + 33 T^{2} + 128 T^{3} + 456 T^{4} + 1368 T^{5} + 3884 T^{6} + 9666 T^{7} + 22992 T^{8} + 9666 p T^{9} + 3884 p^{2} T^{10} + 1368 p^{3} T^{11} + 456 p^{4} T^{12} + 128 p^{5} T^{13} + 33 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 6 T + 47 T^{2} + 192 T^{3} + 134 p T^{4} + 3082 T^{5} + 11540 T^{6} + 4490 p T^{7} + 96232 T^{8} + 4490 p^{2} T^{9} + 11540 p^{2} T^{10} + 3082 p^{3} T^{11} + 134 p^{5} T^{12} + 192 p^{5} T^{13} + 47 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + p T + 101 T^{2} + 647 T^{3} + 3759 T^{4} + 17968 T^{5} + 79367 T^{6} + 302769 T^{7} + 1074995 T^{8} + 302769 p T^{9} + 79367 p^{2} T^{10} + 17968 p^{3} T^{11} + 3759 p^{4} T^{12} + 647 p^{5} T^{13} + 101 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + T + 30 T^{2} + 4 T^{3} + 565 T^{4} - 134 T^{5} + 10013 T^{6} + 1682 T^{7} + 150687 T^{8} + 1682 p T^{9} + 10013 p^{2} T^{10} - 134 p^{3} T^{11} + 565 p^{4} T^{12} + 4 p^{5} T^{13} + 30 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 16 T + 184 T^{2} + 1506 T^{3} + 10309 T^{4} + 59680 T^{5} + 307385 T^{6} + 1431328 T^{7} + 6133586 T^{8} + 1431328 p T^{9} + 307385 p^{2} T^{10} + 59680 p^{3} T^{11} + 10309 p^{4} T^{12} + 1506 p^{5} T^{13} + 184 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + T + 46 T^{2} + 54 T^{3} + 1665 T^{4} + 2430 T^{5} + 44311 T^{6} + 2950 p T^{7} + 957323 T^{8} + 2950 p^{2} T^{9} + 44311 p^{2} T^{10} + 2430 p^{3} T^{11} + 1665 p^{4} T^{12} + 54 p^{5} T^{13} + 46 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 14 T + 168 T^{2} + 1496 T^{3} + 11853 T^{4} + 79684 T^{5} + 493823 T^{6} + 2703978 T^{7} + 13676166 T^{8} + 2703978 p T^{9} + 493823 p^{2} T^{10} + 79684 p^{3} T^{11} + 11853 p^{4} T^{12} + 1496 p^{5} T^{13} + 168 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 21 T + 258 T^{2} + 2290 T^{3} + 16073 T^{4} + 95508 T^{5} + 522069 T^{6} + 2762546 T^{7} + 14727627 T^{8} + 2762546 p T^{9} + 522069 p^{2} T^{10} + 95508 p^{3} T^{11} + 16073 p^{4} T^{12} + 2290 p^{5} T^{13} + 258 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 6 T + 3 p T^{2} + 502 T^{3} + 6586 T^{4} + 30218 T^{5} + 288086 T^{6} + 1229152 T^{7} + 10731368 T^{8} + 1229152 p T^{9} + 288086 p^{2} T^{10} + 30218 p^{3} T^{11} + 6586 p^{4} T^{12} + 502 p^{5} T^{13} + 3 p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 14 T + 248 T^{2} + 2542 T^{3} + 27627 T^{4} + 224696 T^{5} + 1859201 T^{6} + 12366486 T^{7} + 83220526 T^{8} + 12366486 p T^{9} + 1859201 p^{2} T^{10} + 224696 p^{3} T^{11} + 27627 p^{4} T^{12} + 2542 p^{5} T^{13} + 248 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 16 T + 358 T^{2} + 4364 T^{3} + 53943 T^{4} + 518694 T^{5} + 4533887 T^{6} + 34838962 T^{7} + 234118138 T^{8} + 34838962 p T^{9} + 4533887 p^{2} T^{10} + 518694 p^{3} T^{11} + 53943 p^{4} T^{12} + 4364 p^{5} T^{13} + 358 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 29 T + 587 T^{2} + 8661 T^{3} + 106007 T^{4} + 1089814 T^{5} + 9793353 T^{6} + 76899377 T^{7} + 536926075 T^{8} + 76899377 p T^{9} + 9793353 p^{2} T^{10} + 1089814 p^{3} T^{11} + 106007 p^{4} T^{12} + 8661 p^{5} T^{13} + 587 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 9 T + 192 T^{2} + 856 T^{3} + 13037 T^{4} + 5994 T^{5} + 396723 T^{6} - 2875252 T^{7} + 8189679 T^{8} - 2875252 p T^{9} + 396723 p^{2} T^{10} + 5994 p^{3} T^{11} + 13037 p^{4} T^{12} + 856 p^{5} T^{13} + 192 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 284 T^{2} + 1808 T^{3} + 30979 T^{4} + 128438 T^{5} + 2235793 T^{6} + 7349734 T^{7} + 132132131 T^{8} + 7349734 p T^{9} + 2235793 p^{2} T^{10} + 128438 p^{3} T^{11} + 30979 p^{4} T^{12} + 1808 p^{5} T^{13} + 284 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 21 T + 471 T^{2} + 6327 T^{3} + 82527 T^{4} + 835620 T^{5} + 8145533 T^{6} + 68273733 T^{7} + 555976677 T^{8} + 68273733 p T^{9} + 8145533 p^{2} T^{10} + 835620 p^{3} T^{11} + 82527 p^{4} T^{12} + 6327 p^{5} T^{13} + 471 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 3 T + 198 T^{2} + 222 T^{3} + 15669 T^{4} + 110866 T^{5} + 795949 T^{6} + 11947206 T^{7} + 43070391 T^{8} + 11947206 p T^{9} + 795949 p^{2} T^{10} + 110866 p^{3} T^{11} + 15669 p^{4} T^{12} + 222 p^{5} T^{13} + 198 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 20 T + 443 T^{2} + 5246 T^{3} + 70214 T^{4} + 662146 T^{5} + 7410764 T^{6} + 62681818 T^{7} + 594236152 T^{8} + 62681818 p T^{9} + 7410764 p^{2} T^{10} + 662146 p^{3} T^{11} + 70214 p^{4} T^{12} + 5246 p^{5} T^{13} + 443 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 32 T + 641 T^{2} + 9004 T^{3} + 110069 T^{4} + 1213992 T^{5} + 12786174 T^{6} + 122742572 T^{7} + 1089839354 T^{8} + 122742572 p T^{9} + 12786174 p^{2} T^{10} + 1213992 p^{3} T^{11} + 110069 p^{4} T^{12} + 9004 p^{5} T^{13} + 641 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 13 T + 373 T^{2} - 3779 T^{3} + 59871 T^{4} - 487622 T^{5} + 5786257 T^{6} - 41105185 T^{7} + 440500479 T^{8} - 41105185 p T^{9} + 5786257 p^{2} T^{10} - 487622 p^{3} T^{11} + 59871 p^{4} T^{12} - 3779 p^{5} T^{13} + 373 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 21 T + 521 T^{2} - 8025 T^{3} + 126065 T^{4} - 1520426 T^{5} + 18280415 T^{6} - 180049441 T^{7} + 1753921105 T^{8} - 180049441 p T^{9} + 18280415 p^{2} T^{10} - 1520426 p^{3} T^{11} + 126065 p^{4} T^{12} - 8025 p^{5} T^{13} + 521 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 28 T + 617 T^{2} + 9216 T^{3} + 127226 T^{4} + 1485538 T^{5} + 17135118 T^{6} + 174639662 T^{7} + 1708967588 T^{8} + 174639662 p T^{9} + 17135118 p^{2} T^{10} + 1485538 p^{3} T^{11} + 127226 p^{4} T^{12} + 9216 p^{5} T^{13} + 617 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 54 T + 1663 T^{2} + 36914 T^{3} + 650736 T^{4} + 9564914 T^{5} + 121318598 T^{6} + 1354932734 T^{7} + 13504663096 T^{8} + 1354932734 p T^{9} + 121318598 p^{2} T^{10} + 9564914 p^{3} T^{11} + 650736 p^{4} T^{12} + 36914 p^{5} T^{13} + 1663 p^{6} T^{14} + 54 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 10 T + 597 T^{2} - 5938 T^{3} + 165132 T^{4} - 1584880 T^{5} + 28137136 T^{6} - 245736968 T^{7} + 3263822916 T^{8} - 245736968 p T^{9} + 28137136 p^{2} T^{10} - 1584880 p^{3} T^{11} + 165132 p^{4} T^{12} - 5938 p^{5} T^{13} + 597 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.01772580204873048800756837227, −3.74051695919514805976531210060, −3.62060785966131343324896850634, −3.56283817323602495162687783565, −3.54676018036572953022021747453, −3.51034644409883099468811528026, −3.50159091068440440888520601613, −3.37049355893131588760479551436, −3.00442306590501460282799691495, −2.97802395340530693473728023186, −2.97632590128300952942077164640, −2.79276263709000285597088477881, −2.77411492682191424164672791509, −2.54477504426462935487448769559, −2.47967846041226867238508449190, −2.42181778614752998939332761321, −2.35370718607057698071628219632, −1.85624589239993067755664404811, −1.79622243480650012687949248033, −1.76383512147384008864342261446, −1.76212715796657133982678193782, −1.74931299121668475312376553564, −1.55695054665038699785426099275, −1.38205868161981650961221270366, −1.37764553331378980800886629173, 0, 0, 0, 0, 0, 0, 0, 0, 1.37764553331378980800886629173, 1.38205868161981650961221270366, 1.55695054665038699785426099275, 1.74931299121668475312376553564, 1.76212715796657133982678193782, 1.76383512147384008864342261446, 1.79622243480650012687949248033, 1.85624589239993067755664404811, 2.35370718607057698071628219632, 2.42181778614752998939332761321, 2.47967846041226867238508449190, 2.54477504426462935487448769559, 2.77411492682191424164672791509, 2.79276263709000285597088477881, 2.97632590128300952942077164640, 2.97802395340530693473728023186, 3.00442306590501460282799691495, 3.37049355893131588760479551436, 3.50159091068440440888520601613, 3.51034644409883099468811528026, 3.54676018036572953022021747453, 3.56283817323602495162687783565, 3.62060785966131343324896850634, 3.74051695919514805976531210060, 4.01772580204873048800756837227
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.