Dirichlet series
| L(s) = 1 | − 3·2-s − 8·3-s + 3·4-s + 5-s + 24·6-s + 8-s + 36·9-s − 3·10-s + 5·11-s − 24·12-s − 8·15-s − 6·16-s − 7·17-s − 108·18-s + 24·19-s + 3·20-s − 15·22-s + 23-s − 8·24-s − 18·25-s − 120·27-s − 11·29-s + 24·30-s + 30·31-s + 3·32-s − 40·33-s + 21·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 4.61·3-s + 3/2·4-s + 0.447·5-s + 9.79·6-s + 0.353·8-s + 12·9-s − 0.948·10-s + 1.50·11-s − 6.92·12-s − 2.06·15-s − 3/2·16-s − 1.69·17-s − 25.4·18-s + 5.50·19-s + 0.670·20-s − 3.19·22-s + 0.208·23-s − 1.63·24-s − 3.59·25-s − 23.0·27-s − 2.04·29-s + 4.38·30-s + 5.38·31-s + 0.530·32-s − 6.96·33-s + 3.60·34-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
| \(L(1)\) | \(\approx\) | \(0.3059697127\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3059697127\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 3 T + 3 p T^{2} + p^{3} T^{3} + 9 T^{4} + 3 p^{2} T^{5} + 27 T^{6} + 59 T^{7} + 93 T^{8} + 59 p T^{9} + 27 p^{2} T^{10} + 3 p^{5} T^{11} + 9 p^{4} T^{12} + p^{8} T^{13} + 3 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 - T + 19 T^{2} - 17 T^{3} + 161 T^{4} - 192 T^{5} + 927 T^{6} - 1582 T^{7} + 4718 T^{8} - 1582 p T^{9} + 927 p^{2} T^{10} - 192 p^{3} T^{11} + 161 p^{4} T^{12} - 17 p^{5} T^{13} + 19 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 25 T^{2} + p T^{3} + 5 p^{2} T^{4} + 304 T^{5} + 157 p T^{6} + 4509 T^{7} + 3708 T^{8} + 4509 p T^{9} + 157 p^{3} T^{10} + 304 p^{3} T^{11} + 5 p^{6} T^{12} + p^{6} T^{13} + 25 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 - 5 T + 65 T^{2} - 281 T^{3} + 2021 T^{4} - 7532 T^{5} + 38975 T^{6} - 124436 T^{7} + 512616 T^{8} - 124436 p T^{9} + 38975 p^{2} T^{10} - 7532 p^{3} T^{11} + 2021 p^{4} T^{12} - 281 p^{5} T^{13} + 65 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 43 T^{2} - 10 T^{3} + 1062 T^{4} - 887 T^{5} + 18051 T^{6} - 22315 T^{7} + 251414 T^{8} - 22315 p T^{9} + 18051 p^{2} T^{10} - 887 p^{3} T^{11} + 1062 p^{4} T^{12} - 10 p^{5} T^{13} + 43 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 7 T + 94 T^{2} + 515 T^{3} + 3983 T^{4} + 19188 T^{5} + 110490 T^{6} + 27950 p T^{7} + 2210030 T^{8} + 27950 p^{2} T^{9} + 110490 p^{2} T^{10} + 19188 p^{3} T^{11} + 3983 p^{4} T^{12} + 515 p^{5} T^{13} + 94 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 24 T + 353 T^{2} - 3710 T^{3} + 31022 T^{4} - 214835 T^{5} + 1278735 T^{6} - 6653837 T^{7} + 30755986 T^{8} - 6653837 p T^{9} + 1278735 p^{2} T^{10} - 214835 p^{3} T^{11} + 31022 p^{4} T^{12} - 3710 p^{5} T^{13} + 353 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - T + 3 p T^{2} + 65 T^{3} + 3165 T^{4} + 2232 T^{5} + 113855 T^{6} + 101490 T^{7} + 2806012 T^{8} + 101490 p T^{9} + 113855 p^{2} T^{10} + 2232 p^{3} T^{11} + 3165 p^{4} T^{12} + 65 p^{5} T^{13} + 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 11 T + 179 T^{2} + 1685 T^{3} + 15871 T^{4} + 120326 T^{5} + 863835 T^{6} + 5257502 T^{7} + 30730100 T^{8} + 5257502 p T^{9} + 863835 p^{2} T^{10} + 120326 p^{3} T^{11} + 15871 p^{4} T^{12} + 1685 p^{5} T^{13} + 179 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 30 T + 494 T^{2} - 5767 T^{3} + 53036 T^{4} - 409236 T^{5} + 2761738 T^{6} - 16906143 T^{7} + 96763478 T^{8} - 16906143 p T^{9} + 2761738 p^{2} T^{10} - 409236 p^{3} T^{11} + 53036 p^{4} T^{12} - 5767 p^{5} T^{13} + 494 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 11 T + 189 T^{2} - 1649 T^{3} + 17551 T^{4} - 135404 T^{5} + 1078209 T^{6} - 7208484 T^{7} + 46667124 T^{8} - 7208484 p T^{9} + 1078209 p^{2} T^{10} - 135404 p^{3} T^{11} + 17551 p^{4} T^{12} - 1649 p^{5} T^{13} + 189 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 10 T + 232 T^{2} - 1869 T^{3} + 25858 T^{4} - 173932 T^{5} + 1795868 T^{6} - 10291247 T^{7} + 87043884 T^{8} - 10291247 p T^{9} + 1795868 p^{2} T^{10} - 173932 p^{3} T^{11} + 25858 p^{4} T^{12} - 1869 p^{5} T^{13} + 232 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 24 T + 480 T^{2} - 6573 T^{3} + 79642 T^{4} - 786764 T^{5} + 7047644 T^{6} - 54159389 T^{7} + 381070708 T^{8} - 54159389 p T^{9} + 7047644 p^{2} T^{10} - 786764 p^{3} T^{11} + 79642 p^{4} T^{12} - 6573 p^{5} T^{13} + 480 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 3 T + 73 T^{2} + 628 T^{3} + 6578 T^{4} + 33029 T^{5} + 475607 T^{6} + 2491768 T^{7} + 21399806 T^{8} + 2491768 p T^{9} + 475607 p^{2} T^{10} + 33029 p^{3} T^{11} + 6578 p^{4} T^{12} + 628 p^{5} T^{13} + 73 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 25 T + 579 T^{2} + 8774 T^{3} + 120610 T^{4} + 1329957 T^{5} + 13375065 T^{6} + 114505786 T^{7} + 896980008 T^{8} + 114505786 p T^{9} + 13375065 p^{2} T^{10} + 1329957 p^{3} T^{11} + 120610 p^{4} T^{12} + 8774 p^{5} T^{13} + 579 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 45 T + 1223 T^{2} - 23998 T^{3} + 374940 T^{4} - 4846851 T^{5} + 53304093 T^{6} - 504932174 T^{7} + 4157961070 T^{8} - 504932174 p T^{9} + 53304093 p^{2} T^{10} - 4846851 p^{3} T^{11} + 374940 p^{4} T^{12} - 23998 p^{5} T^{13} + 1223 p^{6} T^{14} - 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 16 T + 449 T^{2} - 5808 T^{3} + 91382 T^{4} - 952529 T^{5} + 10887925 T^{6} - 91603311 T^{7} + 824334746 T^{8} - 91603311 p T^{9} + 10887925 p^{2} T^{10} - 952529 p^{3} T^{11} + 91382 p^{4} T^{12} - 5808 p^{5} T^{13} + 449 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 18 T + 475 T^{2} - 6870 T^{3} + 105772 T^{4} - 1204725 T^{5} + 13774321 T^{6} - 126193127 T^{7} + 1141351048 T^{8} - 126193127 p T^{9} + 13774321 p^{2} T^{10} - 1204725 p^{3} T^{11} + 105772 p^{4} T^{12} - 6870 p^{5} T^{13} + 475 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 21 T + 408 T^{2} - 4842 T^{3} + 53937 T^{4} - 469505 T^{5} + 3864512 T^{6} - 29321392 T^{7} + 233979356 T^{8} - 29321392 p T^{9} + 3864512 p^{2} T^{10} - 469505 p^{3} T^{11} + 53937 p^{4} T^{12} - 4842 p^{5} T^{13} + 408 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 8 T + 361 T^{2} + 2398 T^{3} + 62862 T^{4} + 365975 T^{5} + 7401469 T^{6} + 38062427 T^{7} + 634690566 T^{8} + 38062427 p T^{9} + 7401469 p^{2} T^{10} + 365975 p^{3} T^{11} + 62862 p^{4} T^{12} + 2398 p^{5} T^{13} + 361 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T + 266 T^{2} - 1825 T^{3} + 37702 T^{4} - 229058 T^{5} + 4155060 T^{6} - 23409469 T^{7} + 372961876 T^{8} - 23409469 p T^{9} + 4155060 p^{2} T^{10} - 229058 p^{3} T^{11} + 37702 p^{4} T^{12} - 1825 p^{5} T^{13} + 266 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 7 T + 397 T^{2} - 3500 T^{3} + 78910 T^{4} - 711159 T^{5} + 10765451 T^{6} - 84640014 T^{7} + 1058869618 T^{8} - 84640014 p T^{9} + 10765451 p^{2} T^{10} - 711159 p^{3} T^{11} + 78910 p^{4} T^{12} - 3500 p^{5} T^{13} + 397 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 26 T + 761 T^{2} - 12182 T^{3} + 203638 T^{4} - 2354455 T^{5} + 29224825 T^{6} - 274376689 T^{7} + 2923563118 T^{8} - 274376689 p T^{9} + 29224825 p^{2} T^{10} - 2354455 p^{3} T^{11} + 203638 p^{4} T^{12} - 12182 p^{5} T^{13} + 761 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 3 T + 402 T^{2} + 2024 T^{3} + 73933 T^{4} + 595959 T^{5} + 8542284 T^{6} + 96913774 T^{7} + 827247384 T^{8} + 96913774 p T^{9} + 8542284 p^{2} T^{10} + 595959 p^{3} T^{11} + 73933 p^{4} T^{12} + 2024 p^{5} T^{13} + 402 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−5.22061089917088030692738564224, −4.55537245627840723830107567493, −4.51110909665787298087745877900, −4.39889890004506302519836338935, −4.32670282031134736508934335469, −4.27989840796402986794224599293, −4.13363276892208760848199104633, −3.91999242961956315001015563311, −3.86851488137129750590093800513, −3.49788557831194291943802895429, −3.36893062202818075015025262614, −3.30087998387398286227482758946, −3.06311918691360932992139361023, −2.61679274720593880949555858948, −2.55464713013913032691131064228, −2.28013743575295345331782002545, −2.12037246582956806235305618331, −1.79828460544709132150943368733, −1.78416591347316346911234336077, −1.15672382508157023825111599354, −1.14352365349629479797529080130, −1.11372361667743692415632884796, −0.822701727877174890280783313858, −0.60537520634470805490272800781, −0.43088335864384602893007309816, 0.43088335864384602893007309816, 0.60537520634470805490272800781, 0.822701727877174890280783313858, 1.11372361667743692415632884796, 1.14352365349629479797529080130, 1.15672382508157023825111599354, 1.78416591347316346911234336077, 1.79828460544709132150943368733, 2.12037246582956806235305618331, 2.28013743575295345331782002545, 2.55464713013913032691131064228, 2.61679274720593880949555858948, 3.06311918691360932992139361023, 3.30087998387398286227482758946, 3.36893062202818075015025262614, 3.49788557831194291943802895429, 3.86851488137129750590093800513, 3.91999242961956315001015563311, 4.13363276892208760848199104633, 4.27989840796402986794224599293, 4.32670282031134736508934335469, 4.39889890004506302519836338935, 4.51110909665787298087745877900, 4.55537245627840723830107567493, 5.22061089917088030692738564224
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.