Dirichlet series
| L(s) = 1 | + 4·2-s − 8·3-s + 2·4-s − 32·6-s + 8·7-s − 12·8-s + 36·9-s + 2·11-s − 16·12-s + 16·13-s + 32·14-s − 16·16-s + 16·17-s + 144·18-s − 14·19-s − 64·21-s + 8·22-s + 14·23-s + 96·24-s + 64·26-s − 120·27-s + 16·28-s + 2·29-s − 22·31-s + 6·32-s − 16·33-s + 64·34-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 4.61·3-s + 4-s − 13.0·6-s + 3.02·7-s − 4.24·8-s + 12·9-s + 0.603·11-s − 4.61·12-s + 4.43·13-s + 8.55·14-s − 4·16-s + 3.88·17-s + 33.9·18-s − 3.21·19-s − 13.9·21-s + 1.70·22-s + 2.91·23-s + 19.5·24-s + 12.5·26-s − 23.0·27-s + 3.02·28-s + 0.371·29-s − 3.95·31-s + 1.06·32-s − 2.78·33-s + 10.9·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.52480\times 10^{9}\) |
| Root analytic conductor: | \(3.86936\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(26.44887833\) |
| \(L(\frac12)\) | \(\approx\) | \(26.44887833\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{8} \) |
| 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
−4.11047631084457508527443908011, −3.82685654000343620602718516805, −3.72760112637302660053887752762, −3.71123637667986172197398487790, −3.69090689988160441000742583989, −3.47464187506563361008315325574, −3.46697208012111020525918845717, −3.44125351403919307805852314845, −3.20251081279167544972989090713, −2.57875479947324953413614594800, −2.54211683051286553015614944683, −2.50003513441518027904301912754, −2.44561084554159418761349840366, −2.13051149110699871948062925657, −1.88810531120861454427528544026, −1.76264489528467325352558284408, −1.53941946295797567691624508938, −1.38869621811313374227001036047, −1.35743388700795010778657274901, −0.987353738408260084385826513754, −0.931835801920470098868480984877, −0.910298035463927915950174985674, −0.74361059488048017420941187906, −0.68504878245081997097536531131, −0.41277562080690542901938272003, 0.41277562080690542901938272003, 0.68504878245081997097536531131, 0.74361059488048017420941187906, 0.910298035463927915950174985674, 0.931835801920470098868480984877, 0.987353738408260084385826513754, 1.35743388700795010778657274901, 1.38869621811313374227001036047, 1.53941946295797567691624508938, 1.76264489528467325352558284408, 1.88810531120861454427528544026, 2.13051149110699871948062925657, 2.44561084554159418761349840366, 2.50003513441518027904301912754, 2.54211683051286553015614944683, 2.57875479947324953413614594800, 3.20251081279167544972989090713, 3.44125351403919307805852314845, 3.46697208012111020525918845717, 3.47464187506563361008315325574, 3.69090689988160441000742583989, 3.71123637667986172197398487790, 3.72760112637302660053887752762, 3.82685654000343620602718516805, 4.11047631084457508527443908011
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.