Dirichlet series
| L(s) = 1 | − 8·2-s − 8·3-s + 36·4-s + 64·6-s − 4·7-s − 120·8-s + 36·9-s + 6·11-s − 288·12-s − 2·13-s + 32·14-s + 330·16-s − 14·17-s − 288·18-s + 10·19-s + 32·21-s − 48·22-s − 12·23-s + 960·24-s + 16·26-s − 120·27-s − 144·28-s + 10·29-s + 16·31-s − 792·32-s − 48·33-s + 112·34-s + ⋯ |
| L(s) = 1 | − 5.65·2-s − 4.61·3-s + 18·4-s + 26.1·6-s − 1.51·7-s − 42.4·8-s + 12·9-s + 1.80·11-s − 83.1·12-s − 0.554·13-s + 8.55·14-s + 82.5·16-s − 3.39·17-s − 67.8·18-s + 2.29·19-s + 6.98·21-s − 10.2·22-s − 2.50·23-s + 195.·24-s + 3.13·26-s − 23.0·27-s − 27.2·28-s + 1.85·29-s + 2.87·31-s − 140.·32-s − 8.35·33-s + 19.2·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.46348\times 10^{11}\) |
| Root analytic conductor: | \(5.47210\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.08656904530\) |
| \(L(\frac12)\) | \(\approx\) | \(0.08656904530\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( ( 1 + T )^{8} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + 23 T^{2} + 48 T^{3} + 191 T^{4} + 264 T^{5} + 1285 T^{6} + 2084 T^{7} + 10564 T^{8} + 2084 p T^{9} + 1285 p^{2} T^{10} + 264 p^{3} T^{11} + 191 p^{4} T^{12} + 48 p^{5} T^{13} + 23 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 6 T + 45 T^{2} - 240 T^{3} + 1205 T^{4} - 4658 T^{5} + 20073 T^{6} - 68300 T^{7} + 239780 T^{8} - 68300 p T^{9} + 20073 p^{2} T^{10} - 4658 p^{3} T^{11} + 1205 p^{4} T^{12} - 240 p^{5} T^{13} + 45 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 T + 47 T^{2} + 176 T^{3} + 1191 T^{4} + 5212 T^{5} + 25325 T^{6} + 85818 T^{7} + 405224 T^{8} + 85818 p T^{9} + 25325 p^{2} T^{10} + 5212 p^{3} T^{11} + 1191 p^{4} T^{12} + 176 p^{5} T^{13} + 47 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 14 T + 9 p T^{2} + 1158 T^{3} + 7331 T^{4} + 38214 T^{5} + 179335 T^{6} + 760774 T^{7} + 3198344 T^{8} + 760774 p T^{9} + 179335 p^{2} T^{10} + 38214 p^{3} T^{11} + 7331 p^{4} T^{12} + 1158 p^{5} T^{13} + 9 p^{7} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 10 T + 102 T^{2} - 670 T^{3} + 4268 T^{4} - 22710 T^{5} + 117514 T^{6} - 560450 T^{7} + 2521030 T^{8} - 560450 p T^{9} + 117514 p^{2} T^{10} - 22710 p^{3} T^{11} + 4268 p^{4} T^{12} - 670 p^{5} T^{13} + 102 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 12 T + 132 T^{2} + 716 T^{3} + 4016 T^{4} + 10172 T^{5} + 50860 T^{6} + 53308 T^{7} + 886334 T^{8} + 53308 p T^{9} + 50860 p^{2} T^{10} + 10172 p^{3} T^{11} + 4016 p^{4} T^{12} + 716 p^{5} T^{13} + 132 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 10 T + 197 T^{2} - 1510 T^{3} + 17343 T^{4} - 107610 T^{5} + 913319 T^{6} - 4694150 T^{7} + 32002680 T^{8} - 4694150 p T^{9} + 913319 p^{2} T^{10} - 107610 p^{3} T^{11} + 17343 p^{4} T^{12} - 1510 p^{5} T^{13} + 197 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 16 T + 245 T^{2} - 2700 T^{3} + 25605 T^{4} - 210758 T^{5} + 1537653 T^{6} - 9946750 T^{7} + 58857900 T^{8} - 9946750 p T^{9} + 1537653 p^{2} T^{10} - 210758 p^{3} T^{11} + 25605 p^{4} T^{12} - 2700 p^{5} T^{13} + 245 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 6 T + 243 T^{2} - 1412 T^{3} + 27071 T^{4} - 147396 T^{5} + 1822865 T^{6} - 8822506 T^{7} + 81640584 T^{8} - 8822506 p T^{9} + 1822865 p^{2} T^{10} - 147396 p^{3} T^{11} + 27071 p^{4} T^{12} - 1412 p^{5} T^{13} + 243 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 6 T + 145 T^{2} - 850 T^{3} + 10255 T^{4} - 69718 T^{5} + 526063 T^{6} - 4134450 T^{7} + 23274800 T^{8} - 4134450 p T^{9} + 526063 p^{2} T^{10} - 69718 p^{3} T^{11} + 10255 p^{4} T^{12} - 850 p^{5} T^{13} + 145 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 2 T + 182 T^{2} + 166 T^{3} + 15756 T^{4} - 5778 T^{5} + 902810 T^{6} - 1211542 T^{7} + 41601734 T^{8} - 1211542 p T^{9} + 902810 p^{2} T^{10} - 5778 p^{3} T^{11} + 15756 p^{4} T^{12} + 166 p^{5} T^{13} + 182 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 14 T + 158 T^{2} + 1418 T^{3} + 16156 T^{4} + 125194 T^{5} + 1006370 T^{6} + 7336014 T^{7} + 59392774 T^{8} + 7336014 p T^{9} + 1006370 p^{2} T^{10} + 125194 p^{3} T^{11} + 16156 p^{4} T^{12} + 1418 p^{5} T^{13} + 158 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 12 T + 352 T^{2} + 3946 T^{3} + 57861 T^{4} + 577532 T^{5} + 5731520 T^{6} + 48919988 T^{7} + 371517029 T^{8} + 48919988 p T^{9} + 5731520 p^{2} T^{10} + 577532 p^{3} T^{11} + 57861 p^{4} T^{12} + 3946 p^{5} T^{13} + 352 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 247 T^{2} + 500 T^{3} + 28643 T^{4} + 123000 T^{5} + 2182549 T^{6} + 13248500 T^{7} + 135310720 T^{8} + 13248500 p T^{9} + 2182549 p^{2} T^{10} + 123000 p^{3} T^{11} + 28643 p^{4} T^{12} + 500 p^{5} T^{13} + 247 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 - 16 T + 335 T^{2} - 3580 T^{3} + 40995 T^{4} - 303668 T^{5} + 2560793 T^{6} - 14411280 T^{7} + 131714360 T^{8} - 14411280 p T^{9} + 2560793 p^{2} T^{10} - 303668 p^{3} T^{11} + 40995 p^{4} T^{12} - 3580 p^{5} T^{13} + 335 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 6 T + 238 T^{2} - 1202 T^{3} + 35076 T^{4} - 154266 T^{5} + 3495250 T^{6} - 13666446 T^{7} + 270524214 T^{8} - 13666446 p T^{9} + 3495250 p^{2} T^{10} - 154266 p^{3} T^{11} + 35076 p^{4} T^{12} - 1202 p^{5} T^{13} + 238 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 6 T + 390 T^{2} - 2050 T^{3} + 74900 T^{4} - 339378 T^{5} + 9085578 T^{6} - 35263990 T^{7} + 765200950 T^{8} - 35263990 p T^{9} + 9085578 p^{2} T^{10} - 339378 p^{3} T^{11} + 74900 p^{4} T^{12} - 2050 p^{5} T^{13} + 390 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 8 T + 327 T^{2} - 3024 T^{3} + 60051 T^{4} - 506008 T^{5} + 7383005 T^{6} - 54348912 T^{7} + 633329824 T^{8} - 54348912 p T^{9} + 7383005 p^{2} T^{10} - 506008 p^{3} T^{11} + 60051 p^{4} T^{12} - 3024 p^{5} T^{13} + 327 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 10 T + 227 T^{2} - 1720 T^{3} + 32163 T^{4} - 256360 T^{5} + 3728769 T^{6} - 26100650 T^{7} + 323096880 T^{8} - 26100650 p T^{9} + 3728769 p^{2} T^{10} - 256360 p^{3} T^{11} + 32163 p^{4} T^{12} - 1720 p^{5} T^{13} + 227 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 22 T + 487 T^{2} + 7796 T^{3} + 106551 T^{4} + 1298832 T^{5} + 14259445 T^{6} + 141748378 T^{7} + 1361172004 T^{8} + 141748378 p T^{9} + 14259445 p^{2} T^{10} + 1298832 p^{3} T^{11} + 106551 p^{4} T^{12} + 7796 p^{5} T^{13} + 487 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 20 T + 537 T^{2} - 8210 T^{3} + 124663 T^{4} - 1568070 T^{5} + 17732339 T^{6} - 192143300 T^{7} + 1812907320 T^{8} - 192143300 p T^{9} + 17732339 p^{2} T^{10} - 1568070 p^{3} T^{11} + 124663 p^{4} T^{12} - 8210 p^{5} T^{13} + 537 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 16 T + 458 T^{2} - 4612 T^{3} + 72331 T^{4} - 417776 T^{5} + 4642080 T^{6} - 3332736 T^{7} + 188812609 T^{8} - 3332736 p T^{9} + 4642080 p^{2} T^{10} - 417776 p^{3} T^{11} + 72331 p^{4} T^{12} - 4612 p^{5} T^{13} + 458 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.60363212329171392478321715641, −3.17939728068629323199994087327, −3.03997507207469277726267083097, −2.83171450164717666645008857327, −2.80217147621100278949773246331, −2.77160724666157675633142735290, −2.70979558305371269384873308549, −2.70090730864254303154845837031, −2.69302164781324524846291613049, −1.97665373014372503092642370775, −1.92842968314910691022016797499, −1.91173725695410114690232985009, −1.84761573707582582185933118066, −1.70751230159946075707467253181, −1.65856023189404819839939921472, −1.58064045692853118509980946767, −1.52611911187121715581523877238, −0.943408881415150737903748193077, −0.895736477056963700658806738673, −0.827278709667328748558529711562, −0.62707482916999882208419498011, −0.54071937685458495413681151446, −0.51497873240470903446743274959, −0.49434924355337649794413937413, −0.21511894490375709999383468632, 0.21511894490375709999383468632, 0.49434924355337649794413937413, 0.51497873240470903446743274959, 0.54071937685458495413681151446, 0.62707482916999882208419498011, 0.827278709667328748558529711562, 0.895736477056963700658806738673, 0.943408881415150737903748193077, 1.52611911187121715581523877238, 1.58064045692853118509980946767, 1.65856023189404819839939921472, 1.70751230159946075707467253181, 1.84761573707582582185933118066, 1.91173725695410114690232985009, 1.92842968314910691022016797499, 1.97665373014372503092642370775, 2.69302164781324524846291613049, 2.70090730864254303154845837031, 2.70979558305371269384873308549, 2.77160724666157675633142735290, 2.80217147621100278949773246331, 2.83171450164717666645008857327, 3.03997507207469277726267083097, 3.17939728068629323199994087327, 3.60363212329171392478321715641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.