Dirichlet series
| L(s) = 1 | + 4·2-s + 2·4-s − 8·7-s − 12·8-s − 2·11-s − 16·13-s − 32·14-s − 16·16-s + 16·17-s − 14·19-s − 8·22-s + 14·23-s − 64·26-s − 16·28-s − 2·29-s − 22·31-s + 6·32-s + 64·34-s − 28·37-s − 56·38-s − 8·41-s − 20·43-s − 4·44-s + 56·46-s + 10·47-s + 4·49-s − 32·52-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s − 3.02·7-s − 4.24·8-s − 0.603·11-s − 4.43·13-s − 8.55·14-s − 4·16-s + 3.88·17-s − 3.21·19-s − 1.70·22-s + 2.91·23-s − 12.5·26-s − 3.02·28-s − 0.371·29-s − 3.95·31-s + 1.06·32-s + 10.9·34-s − 4.60·37-s − 9.08·38-s − 1.24·41-s − 3.04·43-s − 0.603·44-s + 8.25·46-s + 1.45·47-s + 4/7·49-s − 4.43·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.65652\times 10^{13}\) |
| Root analytic conductor: | \(6.70192\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - p^{2} T + 7 p T^{2} - 9 p^{2} T^{3} + 21 p^{2} T^{4} - 83 p T^{5} + 303 T^{6} - 61 p^{3} T^{7} + 733 T^{8} - 61 p^{4} T^{9} + 303 p^{2} T^{10} - 83 p^{4} T^{11} + 21 p^{6} T^{12} - 9 p^{7} T^{13} + 7 p^{7} T^{14} - p^{9} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 8 T + 60 T^{2} + 284 T^{3} + 1286 T^{4} + 4612 T^{5} + 16181 T^{6} + 47820 T^{7} + 137369 T^{8} + 47820 p T^{9} + 16181 p^{2} T^{10} + 4612 p^{3} T^{11} + 1286 p^{4} T^{12} + 284 p^{5} T^{13} + 60 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T + 45 T^{2} + 64 T^{3} + 1106 T^{4} + 1142 T^{5} + 18113 T^{6} + 14326 T^{7} + 226623 T^{8} + 14326 p T^{9} + 18113 p^{2} T^{10} + 1142 p^{3} T^{11} + 1106 p^{4} T^{12} + 64 p^{5} T^{13} + 45 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 16 T + 177 T^{2} + 112 p T^{3} + 9832 T^{4} + 56184 T^{5} + 278231 T^{6} + 1208392 T^{7} + 4631687 T^{8} + 1208392 p T^{9} + 278231 p^{2} T^{10} + 56184 p^{3} T^{11} + 9832 p^{4} T^{12} + 112 p^{6} T^{13} + 177 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 16 T + 178 T^{2} - 86 p T^{3} + 10156 T^{4} - 60156 T^{5} + 318645 T^{6} - 1508276 T^{7} + 6538409 T^{8} - 1508276 p T^{9} + 318645 p^{2} T^{10} - 60156 p^{3} T^{11} + 10156 p^{4} T^{12} - 86 p^{6} T^{13} + 178 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 14 T + 163 T^{2} + 1346 T^{3} + 9763 T^{4} + 59294 T^{5} + 327860 T^{6} + 1614360 T^{7} + 7387711 T^{8} + 1614360 p T^{9} + 327860 p^{2} T^{10} + 59294 p^{3} T^{11} + 9763 p^{4} T^{12} + 1346 p^{5} T^{13} + 163 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 14 T + 195 T^{2} - 1768 T^{3} + 14846 T^{4} - 101386 T^{5} + 640059 T^{6} - 3514140 T^{7} + 17918999 T^{8} - 3514140 p T^{9} + 640059 p^{2} T^{10} - 101386 p^{3} T^{11} + 14846 p^{4} T^{12} - 1768 p^{5} T^{13} + 195 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 2 T + 126 T^{2} + 244 T^{3} + 8060 T^{4} + 14162 T^{5} + 12255 p T^{6} + 535870 T^{7} + 11787061 T^{8} + 535870 p T^{9} + 12255 p^{3} T^{10} + 14162 p^{3} T^{11} + 8060 p^{4} T^{12} + 244 p^{5} T^{13} + 126 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 22 T + 417 T^{2} + 5322 T^{3} + 58928 T^{4} + 528532 T^{5} + 4176395 T^{6} + 908680 p T^{7} + 168675651 T^{8} + 908680 p^{2} T^{9} + 4176395 p^{2} T^{10} + 528532 p^{3} T^{11} + 58928 p^{4} T^{12} + 5322 p^{5} T^{13} + 417 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 28 T + 500 T^{2} + 6824 T^{3} + 76826 T^{4} + 732792 T^{5} + 6085461 T^{6} + 44405120 T^{7} + 286972229 T^{8} + 44405120 p T^{9} + 6085461 p^{2} T^{10} + 732792 p^{3} T^{11} + 76826 p^{4} T^{12} + 6824 p^{5} T^{13} + 500 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 8 T + 272 T^{2} + 1868 T^{3} + 33458 T^{4} + 200348 T^{5} + 2488680 T^{6} + 12815300 T^{7} + 123432171 T^{8} + 12815300 p T^{9} + 2488680 p^{2} T^{10} + 200348 p^{3} T^{11} + 33458 p^{4} T^{12} + 1868 p^{5} T^{13} + 272 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 20 T + 350 T^{2} + 4260 T^{3} + 46371 T^{4} + 424180 T^{5} + 3588460 T^{6} + 26681640 T^{7} + 185967861 T^{8} + 26681640 p T^{9} + 3588460 p^{2} T^{10} + 424180 p^{3} T^{11} + 46371 p^{4} T^{12} + 4260 p^{5} T^{13} + 350 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T + 190 T^{2} - 1310 T^{3} + 19256 T^{4} - 129400 T^{5} + 1467855 T^{6} - 8170760 T^{7} + 76541721 T^{8} - 8170760 p T^{9} + 1467855 p^{2} T^{10} - 129400 p^{3} T^{11} + 19256 p^{4} T^{12} - 1310 p^{5} T^{13} + 190 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 44 T + 1170 T^{2} - 22238 T^{3} + 334716 T^{4} - 4144336 T^{5} + 43500629 T^{6} - 391956900 T^{7} + 3063022409 T^{8} - 391956900 p T^{9} + 43500629 p^{2} T^{10} - 4144336 p^{3} T^{11} + 334716 p^{4} T^{12} - 22238 p^{5} T^{13} + 1170 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 14 T + 443 T^{2} + 4896 T^{3} + 85513 T^{4} + 771014 T^{5} + 9617080 T^{6} + 71396820 T^{7} + 697201411 T^{8} + 71396820 p T^{9} + 9617080 p^{2} T^{10} + 771014 p^{3} T^{11} + 85513 p^{4} T^{12} + 4896 p^{5} T^{13} + 443 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 20 T + 352 T^{2} + 3040 T^{3} + 27198 T^{4} + 141700 T^{5} + 1692024 T^{6} + 13498660 T^{7} + 2503715 p T^{8} + 13498660 p T^{9} + 1692024 p^{2} T^{10} + 141700 p^{3} T^{11} + 27198 p^{4} T^{12} + 3040 p^{5} T^{13} + 352 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 16 T + 398 T^{2} + 3852 T^{3} + 49991 T^{4} + 250216 T^{5} + 2139560 T^{6} - 2434984 T^{7} + 36983209 T^{8} - 2434984 p T^{9} + 2139560 p^{2} T^{10} + 250216 p^{3} T^{11} + 49991 p^{4} T^{12} + 3852 p^{5} T^{13} + 398 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 16 T + 490 T^{2} + 5790 T^{3} + 102500 T^{4} + 968088 T^{5} + 12744273 T^{6} + 100209740 T^{7} + 1076051405 T^{8} + 100209740 p T^{9} + 12744273 p^{2} T^{10} + 968088 p^{3} T^{11} + 102500 p^{4} T^{12} + 5790 p^{5} T^{13} + 490 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 24 T + 550 T^{2} + 7048 T^{3} + 88491 T^{4} + 715096 T^{5} + 6315244 T^{6} + 35963680 T^{7} + 349140469 T^{8} + 35963680 p T^{9} + 6315244 p^{2} T^{10} + 715096 p^{3} T^{11} + 88491 p^{4} T^{12} + 7048 p^{5} T^{13} + 550 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 30 T + 777 T^{2} + 13910 T^{3} + 223888 T^{4} + 2947580 T^{5} + 35493519 T^{6} + 367201700 T^{7} + 3499703455 T^{8} + 367201700 p T^{9} + 35493519 p^{2} T^{10} + 2947580 p^{3} T^{11} + 223888 p^{4} T^{12} + 13910 p^{5} T^{13} + 777 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 12 T + 476 T^{2} - 4412 T^{3} + 103454 T^{4} - 9656 p T^{5} + 14395152 T^{6} - 95922016 T^{7} + 1413679143 T^{8} - 95922016 p T^{9} + 14395152 p^{2} T^{10} - 9656 p^{4} T^{11} + 103454 p^{4} T^{12} - 4412 p^{5} T^{13} + 476 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 16 T + 673 T^{2} + 8514 T^{3} + 195458 T^{4} + 2007806 T^{5} + 32882225 T^{6} + 277843390 T^{7} + 3579994211 T^{8} + 277843390 p T^{9} + 32882225 p^{2} T^{10} + 2007806 p^{3} T^{11} + 195458 p^{4} T^{12} + 8514 p^{5} T^{13} + 673 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 16 T + 668 T^{2} + 8432 T^{3} + 199226 T^{4} + 2062176 T^{5} + 35473680 T^{6} + 305197056 T^{7} + 4175116899 T^{8} + 305197056 p T^{9} + 35473680 p^{2} T^{10} + 2062176 p^{3} T^{11} + 199226 p^{4} T^{12} + 8432 p^{5} T^{13} + 668 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.74837470802506201045884310084, −3.46196269590455854044740922474, −3.44803047216322019523907820180, −3.44157841241721532591980545688, −3.43601633928350384411022539120, −3.34017554221074255744730213298, −3.27796150745624589964063510393, −3.20504966079342169783581106854, −2.98386264132959046967222499832, −2.80893155974429368709069062370, −2.79266617093213924950787111456, −2.67298803981766120882725320108, −2.57612565900687711245122363190, −2.45582049848557124046809373933, −2.34769278117501035282396288534, −2.03431065185049761826468809371, −2.00427923379074501316633970766, −1.96212421465342290514383918831, −1.77638125921845938596899113866, −1.48154711135775154691756126099, −1.46124073744230312248724763113, −1.34741450172329226834674701088, −1.17444287872324185785904481832, −0.988175731362714418909239816920, −0.983317184296303353983659568152, 0, 0, 0, 0, 0, 0, 0, 0, 0.983317184296303353983659568152, 0.988175731362714418909239816920, 1.17444287872324185785904481832, 1.34741450172329226834674701088, 1.46124073744230312248724763113, 1.48154711135775154691756126099, 1.77638125921845938596899113866, 1.96212421465342290514383918831, 2.00427923379074501316633970766, 2.03431065185049761826468809371, 2.34769278117501035282396288534, 2.45582049848557124046809373933, 2.57612565900687711245122363190, 2.67298803981766120882725320108, 2.79266617093213924950787111456, 2.80893155974429368709069062370, 2.98386264132959046967222499832, 3.20504966079342169783581106854, 3.27796150745624589964063510393, 3.34017554221074255744730213298, 3.43601633928350384411022539120, 3.44157841241721532591980545688, 3.44803047216322019523907820180, 3.46196269590455854044740922474, 3.74837470802506201045884310084
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.