Dirichlet series
| L(s) = 1 | − 8·3-s − 7-s + 36·9-s − 3·11-s − 8·13-s − 7·17-s + 8·21-s + 9·23-s − 17·25-s − 120·27-s + 17·29-s − 23·31-s + 24·33-s + 8·37-s + 64·39-s − 8·41-s − 2·43-s − 34·47-s − 25·49-s + 56·51-s + 12·53-s − 16·59-s − 2·61-s − 36·63-s + 21·67-s − 72·69-s − 29·71-s + ⋯ |
| L(s) = 1 | − 4.61·3-s − 0.377·7-s + 12·9-s − 0.904·11-s − 2.21·13-s − 1.69·17-s + 1.74·21-s + 1.87·23-s − 3.39·25-s − 23.0·27-s + 3.15·29-s − 4.13·31-s + 4.17·33-s + 1.31·37-s + 10.2·39-s − 1.24·41-s − 0.304·43-s − 4.95·47-s − 3.57·49-s + 7.84·51-s + 1.64·53-s − 2.08·59-s − 0.256·61-s − 4.53·63-s + 2.56·67-s − 8.66·69-s − 3.44·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 3^{8} \cdot 167^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.10062\times 10^{12}\) |
| Root analytic conductor: | \(5.65721\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{24} \cdot 3^{8} \cdot 167^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 167 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + 17 T^{2} - 3 T^{3} + 173 T^{4} - 62 T^{5} + 1217 T^{6} - 551 T^{7} + 6789 T^{8} - 551 p T^{9} + 1217 p^{2} T^{10} - 62 p^{3} T^{11} + 173 p^{4} T^{12} - 3 p^{5} T^{13} + 17 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + T + 26 T^{2} + 10 T^{3} + 293 T^{4} - 170 T^{5} + 1973 T^{6} - 3807 T^{7} + 11987 T^{8} - 3807 p T^{9} + 1973 p^{2} T^{10} - 170 p^{3} T^{11} + 293 p^{4} T^{12} + 10 p^{5} T^{13} + 26 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 3 T + 41 T^{2} + 53 T^{3} + 839 T^{4} + 654 T^{5} + 13251 T^{6} + 4788 T^{7} + 14132 p T^{8} + 4788 p T^{9} + 13251 p^{2} T^{10} + 654 p^{3} T^{11} + 839 p^{4} T^{12} + 53 p^{5} T^{13} + 41 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 8 T + 57 T^{2} + 370 T^{3} + 2162 T^{4} + 10395 T^{5} + 46959 T^{6} + 194063 T^{7} + 739966 T^{8} + 194063 p T^{9} + 46959 p^{2} T^{10} + 10395 p^{3} T^{11} + 2162 p^{4} T^{12} + 370 p^{5} T^{13} + 57 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 7 T + 98 T^{2} + 559 T^{3} + 4645 T^{4} + 22152 T^{5} + 138662 T^{6} + 556330 T^{7} + 2823464 T^{8} + 556330 p T^{9} + 138662 p^{2} T^{10} + 22152 p^{3} T^{11} + 4645 p^{4} T^{12} + 559 p^{5} T^{13} + 98 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 59 T^{2} + 162 T^{3} + 1930 T^{4} + 21 p^{2} T^{5} + 59167 T^{6} + 175147 T^{7} + 1382418 T^{8} + 175147 p T^{9} + 59167 p^{2} T^{10} + 21 p^{5} T^{11} + 1930 p^{4} T^{12} + 162 p^{5} T^{13} + 59 p^{6} T^{14} + p^{8} T^{16} \) | |
| 23 | \( 1 - 9 T + 189 T^{2} - 1307 T^{3} + 15143 T^{4} - 83642 T^{5} + 688539 T^{6} - 3080972 T^{7} + 19690828 T^{8} - 3080972 p T^{9} + 688539 p^{2} T^{10} - 83642 p^{3} T^{11} + 15143 p^{4} T^{12} - 1307 p^{5} T^{13} + 189 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 17 T + 223 T^{2} - 1799 T^{3} + 11399 T^{4} - 45172 T^{5} + 82549 T^{6} + 525004 T^{7} - 4747660 T^{8} + 525004 p T^{9} + 82549 p^{2} T^{10} - 45172 p^{3} T^{11} + 11399 p^{4} T^{12} - 1799 p^{5} T^{13} + 223 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 23 T + 291 T^{2} + 2559 T^{3} + 19167 T^{4} + 133797 T^{5} + 902061 T^{6} + 5635784 T^{7} + 32738735 T^{8} + 5635784 p T^{9} + 902061 p^{2} T^{10} + 133797 p^{3} T^{11} + 19167 p^{4} T^{12} + 2559 p^{5} T^{13} + 291 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 8 T + 223 T^{2} - 1097 T^{3} + 18733 T^{4} - 47772 T^{5} + 848091 T^{6} - 655265 T^{7} + 30323167 T^{8} - 655265 p T^{9} + 848091 p^{2} T^{10} - 47772 p^{3} T^{11} + 18733 p^{4} T^{12} - 1097 p^{5} T^{13} + 223 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 8 T + 176 T^{2} + 1141 T^{3} + 16364 T^{4} + 93186 T^{5} + 1035222 T^{6} + 5288225 T^{7} + 49290458 T^{8} + 5288225 p T^{9} + 1035222 p^{2} T^{10} + 93186 p^{3} T^{11} + 16364 p^{4} T^{12} + 1141 p^{5} T^{13} + 176 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 2 T + 162 T^{2} + 289 T^{3} + 15170 T^{4} + 24474 T^{5} + 990912 T^{6} + 1421947 T^{7} + 48358106 T^{8} + 1421947 p T^{9} + 990912 p^{2} T^{10} + 24474 p^{3} T^{11} + 15170 p^{4} T^{12} + 289 p^{5} T^{13} + 162 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 34 T + 671 T^{2} + 9008 T^{3} + 92085 T^{4} + 741241 T^{5} + 5008223 T^{6} + 30523154 T^{7} + 196537247 T^{8} + 30523154 p T^{9} + 5008223 p^{2} T^{10} + 741241 p^{3} T^{11} + 92085 p^{4} T^{12} + 9008 p^{5} T^{13} + 671 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 12 T + 287 T^{2} - 2704 T^{3} + 40155 T^{4} - 320963 T^{5} + 3624181 T^{6} - 24616764 T^{7} + 227551679 T^{8} - 24616764 p T^{9} + 3624181 p^{2} T^{10} - 320963 p^{3} T^{11} + 40155 p^{4} T^{12} - 2704 p^{5} T^{13} + 287 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 16 T + 35 T^{2} - 1428 T^{3} - 7709 T^{4} + 96093 T^{5} + 1088279 T^{6} - 2962600 T^{7} - 81956161 T^{8} - 2962600 p T^{9} + 1088279 p^{2} T^{10} + 96093 p^{3} T^{11} - 7709 p^{4} T^{12} - 1428 p^{5} T^{13} + 35 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 2 T + 217 T^{2} - 120 T^{3} + 20030 T^{4} - 78881 T^{5} + 1125723 T^{6} - 9676201 T^{7} + 58648922 T^{8} - 9676201 p T^{9} + 1125723 p^{2} T^{10} - 78881 p^{3} T^{11} + 20030 p^{4} T^{12} - 120 p^{5} T^{13} + 217 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 21 T + 574 T^{2} - 8515 T^{3} + 136593 T^{4} - 1558001 T^{5} + 18283819 T^{6} - 166136202 T^{7} + 1528683795 T^{8} - 166136202 p T^{9} + 18283819 p^{2} T^{10} - 1558001 p^{3} T^{11} + 136593 p^{4} T^{12} - 8515 p^{5} T^{13} + 574 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 29 T + 698 T^{2} + 162 p T^{3} + 167613 T^{4} + 2002193 T^{5} + 21936902 T^{6} + 209526928 T^{7} + 1874048096 T^{8} + 209526928 p T^{9} + 21936902 p^{2} T^{10} + 2002193 p^{3} T^{11} + 167613 p^{4} T^{12} + 162 p^{6} T^{13} + 698 p^{6} T^{14} + 29 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 38 T + 1109 T^{2} + 22304 T^{3} + 378276 T^{4} + 5224831 T^{5} + 63151333 T^{6} + 652952139 T^{7} + 5994430214 T^{8} + 652952139 p T^{9} + 63151333 p^{2} T^{10} + 5224831 p^{3} T^{11} + 378276 p^{4} T^{12} + 22304 p^{5} T^{13} + 1109 p^{6} T^{14} + 38 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 12 T + 320 T^{2} + 1893 T^{3} + 34010 T^{4} + 101016 T^{5} + 2948272 T^{6} + 11033963 T^{7} + 281803786 T^{8} + 11033963 p T^{9} + 2948272 p^{2} T^{10} + 101016 p^{3} T^{11} + 34010 p^{4} T^{12} + 1893 p^{5} T^{13} + 320 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 32 T + 889 T^{2} + 16658 T^{3} + 281459 T^{4} + 3833129 T^{5} + 47889213 T^{6} + 508519056 T^{7} + 4992494871 T^{8} + 508519056 p T^{9} + 47889213 p^{2} T^{10} + 3833129 p^{3} T^{11} + 281459 p^{4} T^{12} + 16658 p^{5} T^{13} + 889 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 11 T + 522 T^{2} - 5609 T^{3} + 136333 T^{4} - 1294131 T^{5} + 22088115 T^{6} - 179114938 T^{7} + 2381362433 T^{8} - 179114938 p T^{9} + 22088115 p^{2} T^{10} - 1294131 p^{3} T^{11} + 136333 p^{4} T^{12} - 5609 p^{5} T^{13} + 522 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 8 T + 554 T^{2} - 4269 T^{3} + 144129 T^{4} - 1076092 T^{5} + 23543035 T^{6} - 162571999 T^{7} + 2688085377 T^{8} - 162571999 p T^{9} + 23543035 p^{2} T^{10} - 1076092 p^{3} T^{11} + 144129 p^{4} T^{12} - 4269 p^{5} T^{13} + 554 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.25498212732539563949499261786, −3.72794673410962390377519013607, −3.58916438459899756328508685010, −3.56237467931551269863850639720, −3.51203127470065696007442229053, −3.50085153701939746562981102608, −3.42677472917950704605935964772, −3.19263689206877121010231434894, −3.09802222346028727644355786001, −2.78604943467038034466762795978, −2.65883176599612271944190032975, −2.51337817973767537309853150524, −2.46625463191709365157757796545, −2.46072328029049094524317000605, −2.39585103641443528590676532108, −2.12542214792273632987141212186, −2.11380999689056007242769978928, −1.62000984213570729064124599714, −1.53693840862355384030799935764, −1.50132622261890115402843219037, −1.37766075895618324646946513733, −1.29072487708857856261308499777, −1.24245531371421576255507177656, −1.19215116960988816790743645844, −1.01087658527285487659632088287, 0, 0, 0, 0, 0, 0, 0, 0, 1.01087658527285487659632088287, 1.19215116960988816790743645844, 1.24245531371421576255507177656, 1.29072487708857856261308499777, 1.37766075895618324646946513733, 1.50132622261890115402843219037, 1.53693840862355384030799935764, 1.62000984213570729064124599714, 2.11380999689056007242769978928, 2.12542214792273632987141212186, 2.39585103641443528590676532108, 2.46072328029049094524317000605, 2.46625463191709365157757796545, 2.51337817973767537309853150524, 2.65883176599612271944190032975, 2.78604943467038034466762795978, 3.09802222346028727644355786001, 3.19263689206877121010231434894, 3.42677472917950704605935964772, 3.50085153701939746562981102608, 3.51203127470065696007442229053, 3.56237467931551269863850639720, 3.58916438459899756328508685010, 3.72794673410962390377519013607, 4.25498212732539563949499261786
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.