Dirichlet series
| L(s) = 1 | − 8·2-s + 4·3-s + 36·4-s − 2·5-s − 32·6-s + 8·7-s − 120·8-s + 16·10-s − 12·11-s + 144·12-s + 10·13-s − 64·14-s − 8·15-s + 330·16-s − 6·17-s − 72·20-s + 32·21-s + 96·22-s − 20·23-s − 480·24-s − 14·25-s − 80·26-s − 18·27-s + 288·28-s + 8·29-s + 64·30-s + 18·31-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 2.30·3-s + 18·4-s − 0.894·5-s − 13.0·6-s + 3.02·7-s − 42.4·8-s + 5.05·10-s − 3.61·11-s + 41.5·12-s + 2.77·13-s − 17.1·14-s − 2.06·15-s + 82.5·16-s − 1.45·17-s − 16.0·20-s + 6.98·21-s + 20.4·22-s − 4.17·23-s − 97.9·24-s − 2.79·25-s − 15.6·26-s − 3.46·27-s + 54.4·28-s + 1.48·29-s + 11.6·30-s + 3.23·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 7^{8} \cdot 19^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.03556\times 10^{12}\) |
| Root analytic conductor: | \(6.35266\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 7^{8} \cdot 19^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.883382915\) |
| \(L(\frac12)\) | \(\approx\) | \(1.883382915\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{8} \) |
| 7 | \( ( 1 - T )^{8} \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 - 4 T + 16 T^{2} - 46 T^{3} + 41 p T^{4} - 284 T^{5} + 614 T^{6} - 1168 T^{7} + 2153 T^{8} - 1168 p T^{9} + 614 p^{2} T^{10} - 284 p^{3} T^{11} + 41 p^{5} T^{12} - 46 p^{5} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 2 T + 18 T^{2} + 24 T^{3} + 26 p T^{4} + 154 T^{5} + 688 T^{6} + 1042 T^{7} + 3611 T^{8} + 1042 p T^{9} + 688 p^{2} T^{10} + 154 p^{3} T^{11} + 26 p^{5} T^{12} + 24 p^{5} T^{13} + 18 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 12 T + 116 T^{2} + 768 T^{3} + 4526 T^{4} + 21916 T^{5} + 97384 T^{6} + 33964 p T^{7} + 1325887 T^{8} + 33964 p^{2} T^{9} + 97384 p^{2} T^{10} + 21916 p^{3} T^{11} + 4526 p^{4} T^{12} + 768 p^{5} T^{13} + 116 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 114 T^{2} - 740 T^{3} + 5007 T^{4} - 24730 T^{5} + 738 p^{2} T^{6} - 495920 T^{7} + 1995105 T^{8} - 495920 p T^{9} + 738 p^{4} T^{10} - 24730 p^{3} T^{11} + 5007 p^{4} T^{12} - 740 p^{5} T^{13} + 114 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 6 T + 78 T^{2} + 252 T^{3} + 2471 T^{4} + 5966 T^{5} + 62810 T^{6} + 144916 T^{7} + 1281969 T^{8} + 144916 p T^{9} + 62810 p^{2} T^{10} + 5966 p^{3} T^{11} + 2471 p^{4} T^{12} + 252 p^{5} T^{13} + 78 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 20 T + 304 T^{2} + 3240 T^{3} + 29462 T^{4} + 220540 T^{5} + 1462152 T^{6} + 8359220 T^{7} + 42920255 T^{8} + 8359220 p T^{9} + 1462152 p^{2} T^{10} + 220540 p^{3} T^{11} + 29462 p^{4} T^{12} + 3240 p^{5} T^{13} + 304 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 8 T + 180 T^{2} - 1200 T^{3} + 15190 T^{4} - 85544 T^{5} + 790792 T^{6} - 3751320 T^{7} + 27679655 T^{8} - 3751320 p T^{9} + 790792 p^{2} T^{10} - 85544 p^{3} T^{11} + 15190 p^{4} T^{12} - 1200 p^{5} T^{13} + 180 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 18 T + 326 T^{2} - 3752 T^{3} + 39786 T^{4} - 336994 T^{5} + 2593784 T^{6} - 17075006 T^{7} + 101927147 T^{8} - 17075006 p T^{9} + 2593784 p^{2} T^{10} - 336994 p^{3} T^{11} + 39786 p^{4} T^{12} - 3752 p^{5} T^{13} + 326 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 36 T + 728 T^{2} - 10272 T^{3} + 112766 T^{4} - 1016596 T^{5} + 7881880 T^{6} - 54398996 T^{7} + 344111879 T^{8} - 54398996 p T^{9} + 7881880 p^{2} T^{10} - 1016596 p^{3} T^{11} + 112766 p^{4} T^{12} - 10272 p^{5} T^{13} + 728 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 4 T + 180 T^{2} - 1070 T^{3} + 16515 T^{4} - 107552 T^{5} + 1095378 T^{6} - 6183220 T^{7} + 53448045 T^{8} - 6183220 p T^{9} + 1095378 p^{2} T^{10} - 107552 p^{3} T^{11} + 16515 p^{4} T^{12} - 1070 p^{5} T^{13} + 180 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 16 T + 186 T^{2} + 1944 T^{3} + 19463 T^{4} + 163136 T^{5} + 1289884 T^{6} + 9383952 T^{7} + 64639653 T^{8} + 9383952 p T^{9} + 1289884 p^{2} T^{10} + 163136 p^{3} T^{11} + 19463 p^{4} T^{12} + 1944 p^{5} T^{13} + 186 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 10 T + 106 T^{2} + 900 T^{3} + 8922 T^{4} + 54670 T^{5} + 410288 T^{6} + 2540490 T^{7} + 19998195 T^{8} + 2540490 p T^{9} + 410288 p^{2} T^{10} + 54670 p^{3} T^{11} + 8922 p^{4} T^{12} + 900 p^{5} T^{13} + 106 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 32 T + 672 T^{2} - 10616 T^{3} + 140046 T^{4} - 1567432 T^{5} + 15375400 T^{6} - 132999768 T^{7} + 1027275399 T^{8} - 132999768 p T^{9} + 15375400 p^{2} T^{10} - 1567432 p^{3} T^{11} + 140046 p^{4} T^{12} - 10616 p^{5} T^{13} + 672 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 2 T + 350 T^{2} + 620 T^{3} + 57630 T^{4} + 89486 T^{5} + 5893712 T^{6} + 7900890 T^{7} + 413798455 T^{8} + 7900890 p T^{9} + 5893712 p^{2} T^{10} + 89486 p^{3} T^{11} + 57630 p^{4} T^{12} + 620 p^{5} T^{13} + 350 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 2 T + 406 T^{2} - 508 T^{3} + 1226 p T^{4} - 54166 T^{5} + 8280744 T^{6} - 59494 p T^{7} + 610317667 T^{8} - 59494 p^{2} T^{9} + 8280744 p^{2} T^{10} - 54166 p^{3} T^{11} + 1226 p^{5} T^{12} - 508 p^{5} T^{13} + 406 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 44 T + 1198 T^{2} - 23708 T^{3} + 377831 T^{4} - 5021804 T^{5} + 57346020 T^{6} - 569653264 T^{7} + 4972474249 T^{8} - 569653264 p T^{9} + 57346020 p^{2} T^{10} - 5021804 p^{3} T^{11} + 377831 p^{4} T^{12} - 23708 p^{5} T^{13} + 1198 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 8 T + 456 T^{2} - 3432 T^{3} + 97396 T^{4} - 662184 T^{5} + 12657944 T^{6} - 74559816 T^{7} + 1091090982 T^{8} - 74559816 p T^{9} + 12657944 p^{2} T^{10} - 662184 p^{3} T^{11} + 97396 p^{4} T^{12} - 3432 p^{5} T^{13} + 456 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 30 T + 574 T^{2} - 9060 T^{3} + 127167 T^{4} - 1530370 T^{5} + 16391722 T^{6} - 161356800 T^{7} + 1454975085 T^{8} - 161356800 p T^{9} + 16391722 p^{2} T^{10} - 1530370 p^{3} T^{11} + 127167 p^{4} T^{12} - 9060 p^{5} T^{13} + 574 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 60 T + 2102 T^{2} - 51840 T^{3} + 997523 T^{4} - 15610220 T^{5} + 204832004 T^{6} - 2285132800 T^{7} + 21900771645 T^{8} - 2285132800 p T^{9} + 204832004 p^{2} T^{10} - 15610220 p^{3} T^{11} + 997523 p^{4} T^{12} - 51840 p^{5} T^{13} + 2102 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 28 T + 652 T^{2} + 11004 T^{3} + 164896 T^{4} + 2150268 T^{5} + 25012900 T^{6} + 264499292 T^{7} + 2507137614 T^{8} + 264499292 p T^{9} + 25012900 p^{2} T^{10} + 2150268 p^{3} T^{11} + 164896 p^{4} T^{12} + 11004 p^{5} T^{13} + 652 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 22 T + 490 T^{2} + 6460 T^{3} + 87855 T^{4} + 831766 T^{5} + 8879342 T^{6} + 71319460 T^{7} + 754689725 T^{8} + 71319460 p T^{9} + 8879342 p^{2} T^{10} + 831766 p^{3} T^{11} + 87855 p^{4} T^{12} + 6460 p^{5} T^{13} + 490 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 6 T + 398 T^{2} + 112 T^{3} + 62206 T^{4} - 357874 T^{5} + 6478280 T^{6} - 69951234 T^{7} + 634218239 T^{8} - 69951234 p T^{9} + 6478280 p^{2} T^{10} - 357874 p^{3} T^{11} + 62206 p^{4} T^{12} + 112 p^{5} T^{13} + 398 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.13727205280318044048315872234, −3.02387283139055381053749117111, −2.74398112880199557346757952366, −2.71456318448955604776613106825, −2.70335552819145512581823012860, −2.62624127376001482030594308418, −2.60526047965859677184350803286, −2.52600230417370256470172181398, −2.33926870642585444066321420424, −2.16064853242910039788305766533, −2.12707277081151705032251195409, −2.11395523546879854857847764992, −1.98651378832727534590570166946, −1.87661191131047925370073000302, −1.85045236123485947571519063533, −1.61684260431507643574646817898, −1.26117692178616481589387862483, −1.24031535032165674362879338410, −0.974587470149560669034672658663, −0.919999169307741399171680945965, −0.75061134409588181678720423345, −0.58013871594329329865771334900, −0.54964171623714267952748879559, −0.43834136206667458660224708644, −0.23499228923635906824931134792, 0.23499228923635906824931134792, 0.43834136206667458660224708644, 0.54964171623714267952748879559, 0.58013871594329329865771334900, 0.75061134409588181678720423345, 0.919999169307741399171680945965, 0.974587470149560669034672658663, 1.24031535032165674362879338410, 1.26117692178616481589387862483, 1.61684260431507643574646817898, 1.85045236123485947571519063533, 1.87661191131047925370073000302, 1.98651378832727534590570166946, 2.11395523546879854857847764992, 2.12707277081151705032251195409, 2.16064853242910039788305766533, 2.33926870642585444066321420424, 2.52600230417370256470172181398, 2.60526047965859677184350803286, 2.62624127376001482030594308418, 2.70335552819145512581823012860, 2.71456318448955604776613106825, 2.74398112880199557346757952366, 3.02387283139055381053749117111, 3.13727205280318044048315872234
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.