Dirichlet series
| L(s) = 1 | − 3-s − 2·7-s − 7·9-s + 5·13-s − 3·17-s − 8·19-s + 2·21-s + 23-s − 20·25-s + 8·27-s − 29-s + 3·31-s − 23·37-s − 5·39-s − 10·41-s − 2·43-s − 8·47-s − 15·49-s + 3·51-s + 6·53-s + 8·57-s − 11·59-s − 9·61-s + 14·63-s + 9·67-s − 69-s − 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 7/3·9-s + 1.38·13-s − 0.727·17-s − 1.83·19-s + 0.436·21-s + 0.208·23-s − 4·25-s + 1.53·27-s − 0.185·29-s + 0.538·31-s − 3.78·37-s − 0.800·39-s − 1.56·41-s − 0.304·43-s − 1.16·47-s − 2.14·49-s + 0.420·51-s + 0.824·53-s + 1.05·57-s − 1.43·59-s − 1.15·61-s + 1.76·63-s + 1.09·67-s − 0.120·69-s − 0.356·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 11^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.45295\times 10^{14}\) |
| Root analytic conductor: | \(8.56915\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{16} \cdot 11^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( 1 \) |
| 11 | \( 1 \) | |
| 19 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + T + 8 T^{2} + 7 T^{3} + 44 T^{4} + 43 T^{5} + 172 T^{6} + 20 p^{2} T^{7} + 194 p T^{8} + 20 p^{3} T^{9} + 172 p^{2} T^{10} + 43 p^{3} T^{11} + 44 p^{4} T^{12} + 7 p^{5} T^{13} + 8 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 4 p T^{2} + 227 T^{4} + 9 T^{5} + 348 p T^{6} + 117 T^{7} + 10037 T^{8} + 117 p T^{9} + 348 p^{3} T^{10} + 9 p^{3} T^{11} + 227 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 + 2 T + 19 T^{2} + 5 p T^{3} + 23 p T^{4} + 58 p T^{5} + 1040 T^{6} + 4216 T^{7} + 7230 T^{8} + 4216 p T^{9} + 1040 p^{2} T^{10} + 58 p^{4} T^{11} + 23 p^{5} T^{12} + 5 p^{6} T^{13} + 19 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 5 T + 81 T^{2} - 380 T^{3} + 3163 T^{4} - 13089 T^{5} + 75796 T^{6} - 266090 T^{7} + 1200903 T^{8} - 266090 p T^{9} + 75796 p^{2} T^{10} - 13089 p^{3} T^{11} + 3163 p^{4} T^{12} - 380 p^{5} T^{13} + 81 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 3 T + 90 T^{2} + 18 p T^{3} + 4039 T^{4} + 13482 T^{5} + 117610 T^{6} + 349674 T^{7} + 2382387 T^{8} + 349674 p T^{9} + 117610 p^{2} T^{10} + 13482 p^{3} T^{11} + 4039 p^{4} T^{12} + 18 p^{6} T^{13} + 90 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - T + 66 T^{2} + 38 T^{3} + 2788 T^{4} + 3433 T^{5} + 85078 T^{6} + 149265 T^{7} + 2106312 T^{8} + 149265 p T^{9} + 85078 p^{2} T^{10} + 3433 p^{3} T^{11} + 2788 p^{4} T^{12} + 38 p^{5} T^{13} + 66 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + T + 102 T^{2} - 89 T^{3} + 5062 T^{4} - 9415 T^{5} + 206425 T^{6} - 294075 T^{7} + 7022841 T^{8} - 294075 p T^{9} + 206425 p^{2} T^{10} - 9415 p^{3} T^{11} + 5062 p^{4} T^{12} - 89 p^{5} T^{13} + 102 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 3 T + 123 T^{2} - 110 T^{3} + 7214 T^{4} + 1431 T^{5} + 331842 T^{6} + 49211 T^{7} + 12185094 T^{8} + 49211 p T^{9} + 331842 p^{2} T^{10} + 1431 p^{3} T^{11} + 7214 p^{4} T^{12} - 110 p^{5} T^{13} + 123 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 23 T + 354 T^{2} + 3525 T^{3} + 25302 T^{4} + 106911 T^{5} + 16297 T^{6} - 4177975 T^{7} - 35886669 T^{8} - 4177975 p T^{9} + 16297 p^{2} T^{10} + 106911 p^{3} T^{11} + 25302 p^{4} T^{12} + 3525 p^{5} T^{13} + 354 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 10 T + 310 T^{2} + 2492 T^{3} + 42490 T^{4} + 280320 T^{5} + 3385219 T^{6} + 18338293 T^{7} + 171798013 T^{8} + 18338293 p T^{9} + 3385219 p^{2} T^{10} + 280320 p^{3} T^{11} + 42490 p^{4} T^{12} + 2492 p^{5} T^{13} + 310 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 2 T + 198 T^{2} + 198 T^{3} + 19305 T^{4} + 8940 T^{5} + 1292900 T^{6} + 371785 T^{7} + 64394934 T^{8} + 371785 p T^{9} + 1292900 p^{2} T^{10} + 8940 p^{3} T^{11} + 19305 p^{4} T^{12} + 198 p^{5} T^{13} + 198 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 8 T + 310 T^{2} + 2194 T^{3} + 45010 T^{4} + 274308 T^{5} + 3939265 T^{6} + 20257895 T^{7} + 226060750 T^{8} + 20257895 p T^{9} + 3939265 p^{2} T^{10} + 274308 p^{3} T^{11} + 45010 p^{4} T^{12} + 2194 p^{5} T^{13} + 310 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 6 T + 288 T^{2} - 1851 T^{3} + 40945 T^{4} - 255597 T^{5} + 3736324 T^{6} - 20900610 T^{7} + 236079357 T^{8} - 20900610 p T^{9} + 3736324 p^{2} T^{10} - 255597 p^{3} T^{11} + 40945 p^{4} T^{12} - 1851 p^{5} T^{13} + 288 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 11 T + 241 T^{2} + 2503 T^{3} + 32722 T^{4} + 306036 T^{5} + 3124393 T^{6} + 24716759 T^{7} + 215138794 T^{8} + 24716759 p T^{9} + 3124393 p^{2} T^{10} + 306036 p^{3} T^{11} + 32722 p^{4} T^{12} + 2503 p^{5} T^{13} + 241 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 9 T + 154 T^{2} + 593 T^{3} + 4180 T^{4} - 12475 T^{5} + 126800 T^{6} + 828998 T^{7} + 25877264 T^{8} + 828998 p T^{9} + 126800 p^{2} T^{10} - 12475 p^{3} T^{11} + 4180 p^{4} T^{12} + 593 p^{5} T^{13} + 154 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 9 T + 439 T^{2} - 2957 T^{3} + 83641 T^{4} - 433124 T^{5} + 9534884 T^{6} - 39694529 T^{7} + 750538988 T^{8} - 39694529 p T^{9} + 9534884 p^{2} T^{10} - 433124 p^{3} T^{11} + 83641 p^{4} T^{12} - 2957 p^{5} T^{13} + 439 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 3 T + 276 T^{2} + 1494 T^{3} + 39868 T^{4} + 274827 T^{5} + 4174408 T^{6} + 29266251 T^{7} + 337639002 T^{8} + 29266251 p T^{9} + 4174408 p^{2} T^{10} + 274827 p^{3} T^{11} + 39868 p^{4} T^{12} + 1494 p^{5} T^{13} + 276 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 6 T + 124 T^{2} - 280 T^{3} + 12379 T^{4} + 8411 T^{5} + 436523 T^{6} + 4562438 T^{7} + 33905516 T^{8} + 4562438 p T^{9} + 436523 p^{2} T^{10} + 8411 p^{3} T^{11} + 12379 p^{4} T^{12} - 280 p^{5} T^{13} + 124 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 10 T + 377 T^{2} + 3440 T^{3} + 71090 T^{4} + 586157 T^{5} + 8878108 T^{6} + 64695196 T^{7} + 808083210 T^{8} + 64695196 p T^{9} + 8878108 p^{2} T^{10} + 586157 p^{3} T^{11} + 71090 p^{4} T^{12} + 3440 p^{5} T^{13} + 377 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 34 T + 726 T^{2} + 12379 T^{3} + 179869 T^{4} + 2284748 T^{5} + 26250952 T^{6} + 272176566 T^{7} + 2582839200 T^{8} + 272176566 p T^{9} + 26250952 p^{2} T^{10} + 2284748 p^{3} T^{11} + 179869 p^{4} T^{12} + 12379 p^{5} T^{13} + 726 p^{6} T^{14} + 34 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 11 T + 295 T^{2} + 2441 T^{3} + 44808 T^{4} + 289771 T^{5} + 5309585 T^{6} + 33112371 T^{7} + 529051579 T^{8} + 33112371 p T^{9} + 5309585 p^{2} T^{10} + 289771 p^{3} T^{11} + 44808 p^{4} T^{12} + 2441 p^{5} T^{13} + 295 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 6 T + 292 T^{2} + 800 T^{3} + 31447 T^{4} + 336836 T^{5} + 5171862 T^{6} + 22336085 T^{7} + 756982203 T^{8} + 22336085 p T^{9} + 5171862 p^{2} T^{10} + 336836 p^{3} T^{11} + 31447 p^{4} T^{12} + 800 p^{5} T^{13} + 292 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.56333714999103952349493704732, −3.26243083740737820125144961038, −3.22453368792389223588126393605, −3.21767215509619826869562647340, −3.15494561902823857447490834039, −3.06396046310134759518014181074, −2.91683009372135575609406667836, −2.83837837201287712678355707413, −2.78327026281947281820824517105, −2.53578324232743137566296023863, −2.33616458210893507144648833198, −2.30719668433841760238144389191, −2.28877052449814410829467427202, −2.20990752024427745071733437185, −2.07145872674546938160484674255, −2.02075905000205630434553115490, −1.70794319722794039560200242922, −1.70102509160247505788665236834, −1.47378257502837363730887421162, −1.35799861380384137452084794463, −1.29055174636256270175702237489, −1.27335954035492267493293154461, −1.13005227136266392084434818798, −1.11166742571010440030493122417, −0.72005204725810784774770555916, 0, 0, 0, 0, 0, 0, 0, 0, 0.72005204725810784774770555916, 1.11166742571010440030493122417, 1.13005227136266392084434818798, 1.27335954035492267493293154461, 1.29055174636256270175702237489, 1.35799861380384137452084794463, 1.47378257502837363730887421162, 1.70102509160247505788665236834, 1.70794319722794039560200242922, 2.02075905000205630434553115490, 2.07145872674546938160484674255, 2.20990752024427745071733437185, 2.28877052449814410829467427202, 2.30719668433841760238144389191, 2.33616458210893507144648833198, 2.53578324232743137566296023863, 2.78327026281947281820824517105, 2.83837837201287712678355707413, 2.91683009372135575609406667836, 3.06396046310134759518014181074, 3.15494561902823857447490834039, 3.21767215509619826869562647340, 3.22453368792389223588126393605, 3.26243083740737820125144961038, 3.56333714999103952349493704732
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.