Dirichlet series
| L(s) = 1 | − 3·2-s + 4-s − 7·5-s − 4·7-s + 7·8-s + 21·10-s − 13·11-s + 12·14-s − 12·16-s − 11·17-s + 12·19-s − 7·20-s + 39·22-s − 7·23-s + 2·25-s − 4·28-s − 29-s − 2·31-s + 5·32-s + 33·34-s + 28·35-s − 9·37-s − 36·38-s − 49·40-s − 4·41-s + 2·43-s − 13·44-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1/2·4-s − 3.13·5-s − 1.51·7-s + 2.47·8-s + 6.64·10-s − 3.91·11-s + 3.20·14-s − 3·16-s − 2.66·17-s + 2.75·19-s − 1.56·20-s + 8.31·22-s − 1.45·23-s + 2/5·25-s − 0.755·28-s − 0.185·29-s − 0.359·31-s + 0.883·32-s + 5.65·34-s + 4.73·35-s − 1.47·37-s − 5.83·38-s − 7.74·40-s − 0.624·41-s + 0.304·43-s − 1.95·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 167^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.30416\times 10^{8}\) |
| Root analytic conductor: | \(3.46432\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{16} \cdot 167^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 3 | \( 1 \) |
| 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
−4.47359803057553380593562120737, −4.21214785433358538680212174091, −4.16837443573461332776476995166, −4.09457942562365233702474863380, −3.99568266980407447474982506034, −3.97422943838303345423765729202, −3.87258016820167414157685474251, −3.62075113296081681394132974309, −3.47995770873603120605776420204, −3.27011146559701584941552877409, −3.22672459281132677601446632195, −3.18827240366616004563314202157, −2.94860772884330872805199608782, −2.93696984479184100922822521497, −2.66322195289194495484643904959, −2.63649176875822374910558471541, −2.47124475662266408276292988105, −2.40718851079089703494911177236, −2.07614741485300965740819436455, −1.85491585856113394675826333237, −1.57527024851417019426865779214, −1.51553927771919470926631901187, −1.44750589976310097162020082419, −1.25618656847262821760993426240, −1.17540177295079368397446694749, 0, 0, 0, 0, 0, 0, 0, 0, 1.17540177295079368397446694749, 1.25618656847262821760993426240, 1.44750589976310097162020082419, 1.51553927771919470926631901187, 1.57527024851417019426865779214, 1.85491585856113394675826333237, 2.07614741485300965740819436455, 2.40718851079089703494911177236, 2.47124475662266408276292988105, 2.63649176875822374910558471541, 2.66322195289194495484643904959, 2.93696984479184100922822521497, 2.94860772884330872805199608782, 3.18827240366616004563314202157, 3.22672459281132677601446632195, 3.27011146559701584941552877409, 3.47995770873603120605776420204, 3.62075113296081681394132974309, 3.87258016820167414157685474251, 3.97422943838303345423765729202, 3.99568266980407447474982506034, 4.09457942562365233702474863380, 4.16837443573461332776476995166, 4.21214785433358538680212174091, 4.47359803057553380593562120737
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.