Dirichlet series
| L(s) = 1 | + 3·2-s + 8·3-s + 4-s + 7·5-s + 24·6-s − 4·7-s − 7·8-s + 36·9-s + 21·10-s + 13·11-s + 8·12-s − 12·14-s + 56·15-s − 12·16-s + 11·17-s + 108·18-s + 12·19-s + 7·20-s − 32·21-s + 39·22-s + 7·23-s − 56·24-s + 2·25-s + 120·27-s − 4·28-s + 29-s + 168·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4.61·3-s + 1/2·4-s + 3.13·5-s + 9.79·6-s − 1.51·7-s − 2.47·8-s + 12·9-s + 6.64·10-s + 3.91·11-s + 2.30·12-s − 3.20·14-s + 14.4·15-s − 3·16-s + 2.66·17-s + 25.4·18-s + 2.75·19-s + 1.56·20-s − 6.98·21-s + 8.31·22-s + 1.45·23-s − 11.4·24-s + 2/5·25-s + 23.0·27-s − 0.755·28-s + 0.185·29-s + 30.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 167^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(65602.2\) |
| Root analytic conductor: | \(2.00012\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 167^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(231.0827109\) |
| \(L(\frac12)\) | \(\approx\) | \(231.0827109\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{8} \) |
| 167 | \( ( 1 + T )^{8} \) | |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} - 7 p T^{3} + 25 T^{4} - 9 p^{2} T^{5} + 57 T^{6} - 81 T^{7} + 125 T^{8} - 81 p T^{9} + 57 p^{2} T^{10} - 9 p^{5} T^{11} + 25 p^{4} T^{12} - 7 p^{6} T^{13} + p^{9} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - 7 T + 47 T^{2} - 39 p T^{3} + 157 p T^{4} - 2416 T^{5} + 7321 T^{6} - 18114 T^{7} + 44442 T^{8} - 18114 p T^{9} + 7321 p^{2} T^{10} - 2416 p^{3} T^{11} + 157 p^{5} T^{12} - 39 p^{6} T^{13} + 47 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 4 T + 37 T^{2} + 131 T^{3} + 13 p^{2} T^{4} + 2096 T^{5} + 1013 p T^{6} + 21477 T^{7} + 57412 T^{8} + 21477 p T^{9} + 1013 p^{3} T^{10} + 2096 p^{3} T^{11} + 13 p^{6} T^{12} + 131 p^{5} T^{13} + 37 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 13 T + 129 T^{2} - 897 T^{3} + 5403 T^{4} - 26878 T^{5} + 10941 p T^{6} - 466726 T^{7} + 1652716 T^{8} - 466726 p T^{9} + 10941 p^{3} T^{10} - 26878 p^{3} T^{11} + 5403 p^{4} T^{12} - 897 p^{5} T^{13} + 129 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 67 T^{2} - 4 p T^{3} + 2166 T^{4} - 2407 T^{5} + 46871 T^{6} - 49521 T^{7} + 723870 T^{8} - 49521 p T^{9} + 46871 p^{2} T^{10} - 2407 p^{3} T^{11} + 2166 p^{4} T^{12} - 4 p^{6} T^{13} + 67 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 11 T + 116 T^{2} - 683 T^{3} + 4251 T^{4} - 18412 T^{5} + 96214 T^{6} - 371378 T^{7} + 1794794 T^{8} - 371378 p T^{9} + 96214 p^{2} T^{10} - 18412 p^{3} T^{11} + 4251 p^{4} T^{12} - 683 p^{5} T^{13} + 116 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 12 T + 157 T^{2} - 1298 T^{3} + 10390 T^{4} - 65307 T^{5} + 390291 T^{6} - 1947317 T^{7} + 9226946 T^{8} - 1947317 p T^{9} + 390291 p^{2} T^{10} - 65307 p^{3} T^{11} + 10390 p^{4} T^{12} - 1298 p^{5} T^{13} + 157 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 7 T + 165 T^{2} - 1013 T^{3} + 12309 T^{4} - 65238 T^{5} + 541223 T^{6} - 2417292 T^{7} + 15329148 T^{8} - 2417292 p T^{9} + 541223 p^{2} T^{10} - 65238 p^{3} T^{11} + 12309 p^{4} T^{12} - 1013 p^{5} T^{13} + 165 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - T + 129 T^{2} - 195 T^{3} + 6633 T^{4} - 16732 T^{5} + 181393 T^{6} - 805780 T^{7} + 4241976 T^{8} - 805780 p T^{9} + 181393 p^{2} T^{10} - 16732 p^{3} T^{11} + 6633 p^{4} T^{12} - 195 p^{5} T^{13} + 129 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 2 T + 42 T^{2} - 147 T^{3} + 2272 T^{4} + 556 T^{5} + 120062 T^{6} - 147459 T^{7} + 2714766 T^{8} - 147459 p T^{9} + 120062 p^{2} T^{10} + 556 p^{3} T^{11} + 2272 p^{4} T^{12} - 147 p^{5} T^{13} + 42 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 9 T + 189 T^{2} + 1437 T^{3} + 17491 T^{4} + 112148 T^{5} + 1044441 T^{6} + 5719578 T^{7} + 44753916 T^{8} + 5719578 p T^{9} + 1044441 p^{2} T^{10} + 112148 p^{3} T^{11} + 17491 p^{4} T^{12} + 1437 p^{5} T^{13} + 189 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4 T + 140 T^{2} - 355 T^{3} + 11350 T^{4} - 28966 T^{5} + 691586 T^{6} - 1546337 T^{7} + 31338264 T^{8} - 1546337 p T^{9} + 691586 p^{2} T^{10} - 28966 p^{3} T^{11} + 11350 p^{4} T^{12} - 355 p^{5} T^{13} + 140 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 2 T + 306 T^{2} - 625 T^{3} + 42368 T^{4} - 81466 T^{5} + 3468520 T^{6} - 5843465 T^{7} + 183436652 T^{8} - 5843465 p T^{9} + 3468520 p^{2} T^{10} - 81466 p^{3} T^{11} + 42368 p^{4} T^{12} - 625 p^{5} T^{13} + 306 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 17 T + 343 T^{2} - 3782 T^{3} + 46162 T^{4} - 389283 T^{5} + 3631945 T^{6} - 25382322 T^{7} + 198853598 T^{8} - 25382322 p T^{9} + 3631945 p^{2} T^{10} - 389283 p^{3} T^{11} + 46162 p^{4} T^{12} - 3782 p^{5} T^{13} + 343 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 9 T + 7 p T^{2} - 2932 T^{3} + 62840 T^{4} - 427509 T^{5} + 6314527 T^{6} - 36165056 T^{7} + 411010840 T^{8} - 36165056 p T^{9} + 6314527 p^{2} T^{10} - 427509 p^{3} T^{11} + 62840 p^{4} T^{12} - 2932 p^{5} T^{13} + 7 p^{7} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 29 T + 741 T^{2} - 12436 T^{3} + 187204 T^{4} - 2234733 T^{5} + 24274943 T^{6} - 220913290 T^{7} + 1841244926 T^{8} - 220913290 p T^{9} + 24274943 p^{2} T^{10} - 2234733 p^{3} T^{11} + 187204 p^{4} T^{12} - 12436 p^{5} T^{13} + 741 p^{6} T^{14} - 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 12 T + 389 T^{2} + 4096 T^{3} + 72802 T^{4} + 647219 T^{5} + 8262029 T^{6} + 61325593 T^{7} + 615125834 T^{8} + 61325593 p T^{9} + 8262029 p^{2} T^{10} + 647219 p^{3} T^{11} + 72802 p^{4} T^{12} + 4096 p^{5} T^{13} + 389 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 359 T^{2} - 320 T^{3} + 64940 T^{4} - 71061 T^{5} + 7533739 T^{6} - 8111583 T^{7} + 600908932 T^{8} - 8111583 p T^{9} + 7533739 p^{2} T^{10} - 71061 p^{3} T^{11} + 64940 p^{4} T^{12} - 320 p^{5} T^{13} + 359 p^{6} T^{14} + p^{8} T^{16} \) | |
| 71 | \( 1 - 13 T + 534 T^{2} - 5116 T^{3} + 118293 T^{4} - 877603 T^{5} + 15080094 T^{6} - 90574920 T^{7} + 1280201724 T^{8} - 90574920 p T^{9} + 15080094 p^{2} T^{10} - 877603 p^{3} T^{11} + 118293 p^{4} T^{12} - 5116 p^{5} T^{13} + 534 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 20 T + 415 T^{2} + 5020 T^{3} + 914 p T^{4} + 623383 T^{5} + 6619959 T^{6} + 52721729 T^{7} + 512092382 T^{8} + 52721729 p T^{9} + 6619959 p^{2} T^{10} + 623383 p^{3} T^{11} + 914 p^{5} T^{12} + 5020 p^{5} T^{13} + 415 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 8 T + 168 T^{2} - 573 T^{3} + 22756 T^{4} - 69216 T^{5} + 2102300 T^{6} - 2692689 T^{7} + 175872908 T^{8} - 2692689 p T^{9} + 2102300 p^{2} T^{10} - 69216 p^{3} T^{11} + 22756 p^{4} T^{12} - 573 p^{5} T^{13} + 168 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 33 T + 921 T^{2} - 17544 T^{3} + 295768 T^{4} - 4076215 T^{5} + 50726539 T^{6} - 543414180 T^{7} + 5301266758 T^{8} - 543414180 p T^{9} + 50726539 p^{2} T^{10} - 4076215 p^{3} T^{11} + 295768 p^{4} T^{12} - 17544 p^{5} T^{13} + 921 p^{6} T^{14} - 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 4 T + 513 T^{2} - 2094 T^{3} + 127678 T^{4} - 487693 T^{5} + 19972765 T^{6} - 67413913 T^{7} + 2130074694 T^{8} - 67413913 p T^{9} + 19972765 p^{2} T^{10} - 487693 p^{3} T^{11} + 127678 p^{4} T^{12} - 2094 p^{5} T^{13} + 513 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 31 T + 826 T^{2} + 15984 T^{3} + 265829 T^{4} + 3793587 T^{5} + 48345108 T^{6} + 548722798 T^{7} + 5707233480 T^{8} + 548722798 p T^{9} + 48345108 p^{2} T^{10} + 3793587 p^{3} T^{11} + 265829 p^{4} T^{12} + 15984 p^{5} T^{13} + 826 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.96228589868605946244496327237, −4.37569208187352812838792154738, −4.19556847275688664166823455813, −4.16812277718471080733588299125, −4.12495320168778160808051108239, −4.01868059447442464479935752588, −3.97229682754120291568822780851, −3.74744282677466531578371290004, −3.59861200800168779269325296310, −3.53916232795658892693599336086, −3.25921305314036931415082652878, −3.24365416381162138141229986540, −3.23956110216432821994786094980, −3.01720658284449821481665002957, −2.80882726370989470699938180848, −2.58747673715040758605280383165, −2.30793838144421451543205804476, −2.30660584616972270330118159367, −2.17428675272539384050335391025, −1.78029379854353458805021807651, −1.71871479474509125842573720152, −1.39142383348563492958854457305, −1.33571232167049086179262942673, −1.05926625841929963317103349374, −1.01997264317665096264760027267, 1.01997264317665096264760027267, 1.05926625841929963317103349374, 1.33571232167049086179262942673, 1.39142383348563492958854457305, 1.71871479474509125842573720152, 1.78029379854353458805021807651, 2.17428675272539384050335391025, 2.30660584616972270330118159367, 2.30793838144421451543205804476, 2.58747673715040758605280383165, 2.80882726370989470699938180848, 3.01720658284449821481665002957, 3.23956110216432821994786094980, 3.24365416381162138141229986540, 3.25921305314036931415082652878, 3.53916232795658892693599336086, 3.59861200800168779269325296310, 3.74744282677466531578371290004, 3.97229682754120291568822780851, 4.01868059447442464479935752588, 4.12495320168778160808051108239, 4.16812277718471080733588299125, 4.19556847275688664166823455813, 4.37569208187352812838792154738, 4.96228589868605946244496327237
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.