Dirichlet series
| L(s) = 1 | + 8·2-s − 3·3-s + 36·4-s − 8·5-s − 24·6-s − 4·7-s + 120·8-s − 7·9-s − 64·10-s + 8·11-s − 108·12-s − 7·13-s − 32·14-s + 24·15-s + 330·16-s + 2·17-s − 56·18-s − 19-s − 288·20-s + 12·21-s + 64·22-s − 16·23-s − 360·24-s + 36·25-s − 56·26-s + 26·27-s − 144·28-s + ⋯ |
| L(s) = 1 | + 5.65·2-s − 1.73·3-s + 18·4-s − 3.57·5-s − 9.79·6-s − 1.51·7-s + 42.4·8-s − 7/3·9-s − 20.2·10-s + 2.41·11-s − 31.1·12-s − 1.94·13-s − 8.55·14-s + 6.19·15-s + 82.5·16-s + 0.485·17-s − 13.1·18-s − 0.229·19-s − 64.3·20-s + 2.61·21-s + 13.6·22-s − 3.33·23-s − 73.4·24-s + 36/5·25-s − 10.9·26-s + 5.00·27-s − 27.2·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 73^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 73^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 73^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.85720\times 10^{14}\) |
| Root analytic conductor: | \(8.00748\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{8} \cdot 11^{8} \cdot 73^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 - T )^{8} \) |
| 5 | \( ( 1 + T )^{8} \) | |
| 11 | \( ( 1 - T )^{8} \) | |
| 73 | \( ( 1 + T )^{8} \) | |
| good | 3 | \( 1 + p T + 16 T^{2} + 43 T^{3} + 131 T^{4} + 299 T^{5} + 692 T^{6} + 1306 T^{7} + 2495 T^{8} + 1306 p T^{9} + 692 p^{2} T^{10} + 299 p^{3} T^{11} + 131 p^{4} T^{12} + 43 p^{5} T^{13} + 16 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 4 T + 39 T^{2} + 130 T^{3} + 730 T^{4} + 2028 T^{5} + 8594 T^{6} + 20310 T^{7} + 70727 T^{8} + 20310 p T^{9} + 8594 p^{2} T^{10} + 2028 p^{3} T^{11} + 730 p^{4} T^{12} + 130 p^{5} T^{13} + 39 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 7 T + 67 T^{2} + 300 T^{3} + 1982 T^{4} + 7601 T^{5} + 40912 T^{6} + 10194 p T^{7} + 604403 T^{8} + 10194 p^{2} T^{9} + 40912 p^{2} T^{10} + 7601 p^{3} T^{11} + 1982 p^{4} T^{12} + 300 p^{5} T^{13} + 67 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T + 87 T^{2} - 275 T^{3} + 3600 T^{4} - 14460 T^{5} + 95900 T^{6} - 412102 T^{7} + 1865923 T^{8} - 412102 p T^{9} + 95900 p^{2} T^{10} - 14460 p^{3} T^{11} + 3600 p^{4} T^{12} - 275 p^{5} T^{13} + 87 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + T + 51 T^{2} - 14 T^{3} + 1773 T^{4} - 942 T^{5} + 48269 T^{6} - 22923 T^{7} + 1061824 T^{8} - 22923 p T^{9} + 48269 p^{2} T^{10} - 942 p^{3} T^{11} + 1773 p^{4} T^{12} - 14 p^{5} T^{13} + 51 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 16 T + 215 T^{2} + 2098 T^{3} + 17938 T^{4} + 128384 T^{5} + 826714 T^{6} + 4650122 T^{7} + 23729431 T^{8} + 4650122 p T^{9} + 826714 p^{2} T^{10} + 128384 p^{3} T^{11} + 17938 p^{4} T^{12} + 2098 p^{5} T^{13} + 215 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T + 161 T^{2} + 690 T^{3} + 12634 T^{4} + 47243 T^{5} + 629688 T^{6} + 2039088 T^{7} + 21727555 T^{8} + 2039088 p T^{9} + 629688 p^{2} T^{10} + 47243 p^{3} T^{11} + 12634 p^{4} T^{12} + 690 p^{5} T^{13} + 161 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 22 T + 334 T^{2} + 3358 T^{3} + 28660 T^{4} + 202126 T^{5} + 1372098 T^{6} + 8366694 T^{7} + 49526326 T^{8} + 8366694 p T^{9} + 1372098 p^{2} T^{10} + 202126 p^{3} T^{11} + 28660 p^{4} T^{12} + 3358 p^{5} T^{13} + 334 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 9 T + 162 T^{2} + 1209 T^{3} + 319 p T^{4} + 70331 T^{5} + 543704 T^{6} + 2700436 T^{7} + 20588233 T^{8} + 2700436 p T^{9} + 543704 p^{2} T^{10} + 70331 p^{3} T^{11} + 319 p^{5} T^{12} + 1209 p^{5} T^{13} + 162 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 6 T + 144 T^{2} - 1090 T^{3} + 13037 T^{4} - 90504 T^{5} + 860550 T^{6} - 5145808 T^{7} + 40878108 T^{8} - 5145808 p T^{9} + 860550 p^{2} T^{10} - 90504 p^{3} T^{11} + 13037 p^{4} T^{12} - 1090 p^{5} T^{13} + 144 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 15 T + 262 T^{2} - 2469 T^{3} + 23975 T^{4} - 162649 T^{5} + 1157290 T^{6} - 6664323 T^{7} + 45736272 T^{8} - 6664323 p T^{9} + 1157290 p^{2} T^{10} - 162649 p^{3} T^{11} + 23975 p^{4} T^{12} - 2469 p^{5} T^{13} + 262 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 7 T + 262 T^{2} + 1995 T^{3} + 30421 T^{4} + 253907 T^{5} + 2170494 T^{6} + 18851311 T^{7} + 114269916 T^{8} + 18851311 p T^{9} + 2170494 p^{2} T^{10} + 253907 p^{3} T^{11} + 30421 p^{4} T^{12} + 1995 p^{5} T^{13} + 262 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 320 T^{2} + 3487 T^{3} + 50391 T^{4} + 482211 T^{5} + 93984 p T^{6} + 39159279 T^{7} + 324405684 T^{8} + 39159279 p T^{9} + 93984 p^{3} T^{10} + 482211 p^{3} T^{11} + 50391 p^{4} T^{12} + 3487 p^{5} T^{13} + 320 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 11 T + 343 T^{2} + 3408 T^{3} + 965 p T^{4} + 505572 T^{5} + 100557 p T^{6} + 45585531 T^{7} + 419913232 T^{8} + 45585531 p T^{9} + 100557 p^{3} T^{10} + 505572 p^{3} T^{11} + 965 p^{5} T^{12} + 3408 p^{5} T^{13} + 343 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 22 T + 506 T^{2} + 7169 T^{3} + 100581 T^{4} + 1078379 T^{5} + 11376318 T^{6} + 98501842 T^{7} + 841524336 T^{8} + 98501842 p T^{9} + 11376318 p^{2} T^{10} + 1078379 p^{3} T^{11} + 100581 p^{4} T^{12} + 7169 p^{5} T^{13} + 506 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 21 T + 471 T^{2} - 5426 T^{3} + 64296 T^{4} - 413881 T^{5} + 3052424 T^{6} - 5885828 T^{7} + 74969915 T^{8} - 5885828 p T^{9} + 3052424 p^{2} T^{10} - 413881 p^{3} T^{11} + 64296 p^{4} T^{12} - 5426 p^{5} T^{13} + 471 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 28 T + 695 T^{2} + 11735 T^{3} + 181080 T^{4} + 2257974 T^{5} + 25955826 T^{6} + 254189068 T^{7} + 2306774727 T^{8} + 254189068 p T^{9} + 25955826 p^{2} T^{10} + 2257974 p^{3} T^{11} + 181080 p^{4} T^{12} + 11735 p^{5} T^{13} + 695 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 28 T + 591 T^{2} + 9330 T^{3} + 133929 T^{4} + 1676654 T^{5} + 19084085 T^{6} + 192955768 T^{7} + 1797064564 T^{8} + 192955768 p T^{9} + 19084085 p^{2} T^{10} + 1676654 p^{3} T^{11} + 133929 p^{4} T^{12} + 9330 p^{5} T^{13} + 591 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 5 T + 312 T^{2} + 2479 T^{3} + 53993 T^{4} + 474905 T^{5} + 7047782 T^{6} + 53925918 T^{7} + 691982713 T^{8} + 53925918 p T^{9} + 7047782 p^{2} T^{10} + 474905 p^{3} T^{11} + 53993 p^{4} T^{12} + 2479 p^{5} T^{13} + 312 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 17 T + 449 T^{2} + 4110 T^{3} + 59774 T^{4} + 191457 T^{5} + 2466910 T^{6} - 23509738 T^{7} + 2756913 T^{8} - 23509738 p T^{9} + 2466910 p^{2} T^{10} + 191457 p^{3} T^{11} + 59774 p^{4} T^{12} + 4110 p^{5} T^{13} + 449 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - T + 210 T^{2} - 2597 T^{3} + 29359 T^{4} - 462359 T^{5} + 5477340 T^{6} - 50089406 T^{7} + 653169355 T^{8} - 50089406 p T^{9} + 5477340 p^{2} T^{10} - 462359 p^{3} T^{11} + 29359 p^{4} T^{12} - 2597 p^{5} T^{13} + 210 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.74043082939076416475916590331, −3.48920965141794189713905025614, −3.43854613945333040118732473546, −3.33425053923658427148164723562, −3.22279222341481889798905340461, −3.17604303090631890574967582915, −3.17160550736089319849221125684, −3.05584564613456140200299715297, −3.03856724601349542777585672178, −2.75755761354628477612681182216, −2.59975860241770046505031756019, −2.51217840698437289754918072160, −2.50304882710865674502772806909, −2.30705552486425455278738945687, −2.29886113632289948062847249971, −2.20996922594764356470486315936, −2.08992506675116784979214253514, −1.69368374657155312228848295381, −1.68089226563104619562818281528, −1.56712506708635705000840552632, −1.35806664456403025001626320796, −1.28581116565976922865393402638, −1.24789810930615784572117824367, −1.10275852888043396366776975481, −0.992726136455475809906788034697, 0, 0, 0, 0, 0, 0, 0, 0, 0.992726136455475809906788034697, 1.10275852888043396366776975481, 1.24789810930615784572117824367, 1.28581116565976922865393402638, 1.35806664456403025001626320796, 1.56712506708635705000840552632, 1.68089226563104619562818281528, 1.69368374657155312228848295381, 2.08992506675116784979214253514, 2.20996922594764356470486315936, 2.29886113632289948062847249971, 2.30705552486425455278738945687, 2.50304882710865674502772806909, 2.51217840698437289754918072160, 2.59975860241770046505031756019, 2.75755761354628477612681182216, 3.03856724601349542777585672178, 3.05584564613456140200299715297, 3.17160550736089319849221125684, 3.17604303090631890574967582915, 3.22279222341481889798905340461, 3.33425053923658427148164723562, 3.43854613945333040118732473546, 3.48920965141794189713905025614, 3.74043082939076416475916590331
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.