Dirichlet series
| L(s) = 1 | + 2·2-s + 3·3-s − 2·4-s + 3·5-s + 6·6-s − 2·7-s − 11·8-s − 5·9-s + 6·10-s + 18·11-s − 6·12-s + 8·13-s − 4·14-s + 9·15-s − 11·16-s + 14·17-s − 10·18-s − 6·19-s − 6·20-s − 6·21-s + 36·22-s + 22·23-s − 33·24-s − 9·25-s + 16·26-s − 21·27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s − 4-s + 1.34·5-s + 2.44·6-s − 0.755·7-s − 3.88·8-s − 5/3·9-s + 1.89·10-s + 5.42·11-s − 1.73·12-s + 2.21·13-s − 1.06·14-s + 2.32·15-s − 2.75·16-s + 3.39·17-s − 2.35·18-s − 1.37·19-s − 1.34·20-s − 1.30·21-s + 7.67·22-s + 4.58·23-s − 6.73·24-s − 9/5·25-s + 3.13·26-s − 4.04·27-s + 0.755·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(31^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.20227\times 10^{7}\) |
| Root analytic conductor: | \(2.77013\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 31^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(42.80155881\) |
| \(L(\frac12)\) | \(\approx\) | \(42.80155881\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - p T + 3 p T^{2} - 5 T^{3} + 11 T^{4} - T^{5} + 15 T^{6} + 9 T^{7} + 27 T^{8} + 9 p T^{9} + 15 p^{2} T^{10} - p^{3} T^{11} + 11 p^{4} T^{12} - 5 p^{5} T^{13} + 3 p^{7} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 - p T + 14 T^{2} - 4 p^{2} T^{3} + 37 p T^{4} - 26 p^{2} T^{5} + 560 T^{6} - 335 p T^{7} + 1987 T^{8} - 335 p^{2} T^{9} + 560 p^{2} T^{10} - 26 p^{5} T^{11} + 37 p^{5} T^{12} - 4 p^{7} T^{13} + 14 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 - 3 T + 18 T^{2} - 36 T^{3} + 31 p T^{4} - 261 T^{5} + 993 T^{6} - 1428 T^{7} + 5151 T^{8} - 1428 p T^{9} + 993 p^{2} T^{10} - 261 p^{3} T^{11} + 31 p^{5} T^{12} - 36 p^{5} T^{13} + 18 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 + 2 T + 31 T^{2} + 60 T^{3} + 506 T^{4} + 881 T^{5} + 5550 T^{6} + 8441 T^{7} + 44837 T^{8} + 8441 p T^{9} + 5550 p^{2} T^{10} + 881 p^{3} T^{11} + 506 p^{4} T^{12} + 60 p^{5} T^{13} + 31 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 18 T + 219 T^{2} - 1875 T^{3} + 12998 T^{4} - 73485 T^{5} + 354666 T^{6} - 132522 p T^{7} + 5210553 T^{8} - 132522 p^{2} T^{9} + 354666 p^{2} T^{10} - 73485 p^{3} T^{11} + 12998 p^{4} T^{12} - 1875 p^{5} T^{13} + 219 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 8 T + 100 T^{2} - 567 T^{3} + 4139 T^{4} - 18554 T^{5} + 100344 T^{6} - 368819 T^{7} + 1591961 T^{8} - 368819 p T^{9} + 100344 p^{2} T^{10} - 18554 p^{3} T^{11} + 4139 p^{4} T^{12} - 567 p^{5} T^{13} + 100 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 14 T + 171 T^{2} - 1412 T^{3} + 10571 T^{4} - 64147 T^{5} + 358770 T^{6} - 1712910 T^{7} + 7592697 T^{8} - 1712910 p T^{9} + 358770 p^{2} T^{10} - 64147 p^{3} T^{11} + 10571 p^{4} T^{12} - 1412 p^{5} T^{13} + 171 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 6 T + 126 T^{2} + 550 T^{3} + 6723 T^{4} + 22465 T^{5} + 213499 T^{6} + 577881 T^{7} + 4727383 T^{8} + 577881 p T^{9} + 213499 p^{2} T^{10} + 22465 p^{3} T^{11} + 6723 p^{4} T^{12} + 550 p^{5} T^{13} + 126 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 22 T + 372 T^{2} - 4321 T^{3} + 42281 T^{4} - 334832 T^{5} + 2312310 T^{6} - 13546893 T^{7} + 70125249 T^{8} - 13546893 p T^{9} + 2312310 p^{2} T^{10} - 334832 p^{3} T^{11} + 42281 p^{4} T^{12} - 4321 p^{5} T^{13} + 372 p^{6} T^{14} - 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 12 T + 228 T^{2} - 2067 T^{3} + 22391 T^{4} - 160170 T^{5} + 1262877 T^{6} - 7289670 T^{7} + 45291747 T^{8} - 7289670 p T^{9} + 1262877 p^{2} T^{10} - 160170 p^{3} T^{11} + 22391 p^{4} T^{12} - 2067 p^{5} T^{13} + 228 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 8 T + 184 T^{2} + 1003 T^{3} + 14093 T^{4} + 50978 T^{5} + 642206 T^{6} + 1564129 T^{7} + 23810023 T^{8} + 1564129 p T^{9} + 642206 p^{2} T^{10} + 50978 p^{3} T^{11} + 14093 p^{4} T^{12} + 1003 p^{5} T^{13} + 184 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 22 T + 498 T^{2} + 6811 T^{3} + 89027 T^{4} + 877490 T^{5} + 8169387 T^{6} + 61298622 T^{7} + 433574745 T^{8} + 61298622 p T^{9} + 8169387 p^{2} T^{10} + 877490 p^{3} T^{11} + 89027 p^{4} T^{12} + 6811 p^{5} T^{13} + 498 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 2 T + 205 T^{2} + 403 T^{3} + 21134 T^{4} + 40961 T^{5} + 1453424 T^{6} + 2649346 T^{7} + 72598441 T^{8} + 2649346 p T^{9} + 1453424 p^{2} T^{10} + 40961 p^{3} T^{11} + 21134 p^{4} T^{12} + 403 p^{5} T^{13} + 205 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 18 T + 411 T^{2} + 4965 T^{3} + 66911 T^{4} + 618324 T^{5} + 6152625 T^{6} + 45556584 T^{7} + 358374597 T^{8} + 45556584 p T^{9} + 6152625 p^{2} T^{10} + 618324 p^{3} T^{11} + 66911 p^{4} T^{12} + 4965 p^{5} T^{13} + 411 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 244 T^{2} - 1605 T^{3} + 30376 T^{4} - 203598 T^{5} + 2505355 T^{6} - 16037007 T^{7} + 152391427 T^{8} - 16037007 p T^{9} + 2505355 p^{2} T^{10} - 203598 p^{3} T^{11} + 30376 p^{4} T^{12} - 1605 p^{5} T^{13} + 244 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 4 T + 309 T^{2} + 1189 T^{3} + 45401 T^{4} + 174557 T^{5} + 4316301 T^{6} + 15922002 T^{7} + 295944207 T^{8} + 15922002 p T^{9} + 4316301 p^{2} T^{10} + 174557 p^{3} T^{11} + 45401 p^{4} T^{12} + 1189 p^{5} T^{13} + 309 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 30 T + 776 T^{2} - 13665 T^{3} + 208005 T^{4} - 2593710 T^{5} + 28390399 T^{6} - 268109040 T^{7} + 2240276459 T^{8} - 268109040 p T^{9} + 28390399 p^{2} T^{10} - 2593710 p^{3} T^{11} + 208005 p^{4} T^{12} - 13665 p^{5} T^{13} + 776 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 13 T + 427 T^{2} + 4781 T^{3} + 85148 T^{4} + 816694 T^{5} + 10335824 T^{6} + 84073193 T^{7} + 837177553 T^{8} + 84073193 p T^{9} + 10335824 p^{2} T^{10} + 816694 p^{3} T^{11} + 85148 p^{4} T^{12} + 4781 p^{5} T^{13} + 427 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + T + 465 T^{2} + 325 T^{3} + 99635 T^{4} + 48353 T^{5} + 12913848 T^{6} + 4606650 T^{7} + 1111294875 T^{8} + 4606650 p T^{9} + 12913848 p^{2} T^{10} + 48353 p^{3} T^{11} + 99635 p^{4} T^{12} + 325 p^{5} T^{13} + 465 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 2 T + 280 T^{2} - 423 T^{3} + 39254 T^{4} - 42716 T^{5} + 3669099 T^{6} - 2910611 T^{7} + 282548711 T^{8} - 2910611 p T^{9} + 3669099 p^{2} T^{10} - 42716 p^{3} T^{11} + 39254 p^{4} T^{12} - 423 p^{5} T^{13} + 280 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 8 T + 343 T^{2} - 2748 T^{3} + 65666 T^{4} - 478310 T^{5} + 8349177 T^{6} - 54496370 T^{7} + 769049897 T^{8} - 54496370 p T^{9} + 8349177 p^{2} T^{10} - 478310 p^{3} T^{11} + 65666 p^{4} T^{12} - 2748 p^{5} T^{13} + 343 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 39 T + 12 p T^{2} - 18231 T^{3} + 273488 T^{4} - 3449334 T^{5} + 38635404 T^{6} - 390596853 T^{7} + 3688239693 T^{8} - 390596853 p T^{9} + 38635404 p^{2} T^{10} - 3449334 p^{3} T^{11} + 273488 p^{4} T^{12} - 18231 p^{5} T^{13} + 12 p^{7} T^{14} - 39 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 27 T + 807 T^{2} - 15033 T^{3} + 266447 T^{4} - 3727035 T^{5} + 48700278 T^{6} - 532857324 T^{7} + 5459588145 T^{8} - 532857324 p T^{9} + 48700278 p^{2} T^{10} - 3727035 p^{3} T^{11} + 266447 p^{4} T^{12} - 15033 p^{5} T^{13} + 807 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 34 T + 1021 T^{2} - 20487 T^{3} + 372881 T^{4} - 5460517 T^{5} + 73647420 T^{6} - 843517135 T^{7} + 8954673947 T^{8} - 843517135 p T^{9} + 73647420 p^{2} T^{10} - 5460517 p^{3} T^{11} + 372881 p^{4} T^{12} - 20487 p^{5} T^{13} + 1021 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.48922036255695188032153995746, −3.90440327798996540820770290020, −3.85644654283247669571367407202, −3.67785749657087078758425042079, −3.61225763734995469468947103785, −3.55246081584157521437001666567, −3.52161221072777681190294212969, −3.51512656782749023188541822393, −3.21565094478984252232131943120, −3.18216519625795522332704430782, −3.12208589020671323357587230479, −3.10796975895595825101672747725, −2.99888128042516508548917096363, −2.73718964775739896649560550894, −2.28106842267315120489838433733, −2.07670455890525745557489366450, −1.91189006466009234453826698555, −1.90785407613630421997222776427, −1.89096130756384857595321329742, −1.53075198023429447349881932914, −1.24713037714290168964586397242, −1.06112232391071405361692778038, −0.74250389482280370000216607371, −0.64716136886203316775996400597, −0.59097615621094013380322393746, 0.59097615621094013380322393746, 0.64716136886203316775996400597, 0.74250389482280370000216607371, 1.06112232391071405361692778038, 1.24713037714290168964586397242, 1.53075198023429447349881932914, 1.89096130756384857595321329742, 1.90785407613630421997222776427, 1.91189006466009234453826698555, 2.07670455890525745557489366450, 2.28106842267315120489838433733, 2.73718964775739896649560550894, 2.99888128042516508548917096363, 3.10796975895595825101672747725, 3.12208589020671323357587230479, 3.18216519625795522332704430782, 3.21565094478984252232131943120, 3.51512656782749023188541822393, 3.52161221072777681190294212969, 3.55246081584157521437001666567, 3.61225763734995469468947103785, 3.67785749657087078758425042079, 3.85644654283247669571367407202, 3.90440327798996540820770290020, 4.48922036255695188032153995746
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.