Dirichlet series
| L(s) = 1 | − 5-s − 9·9-s + 9·11-s + 4·17-s + 8·19-s + 25·23-s − 12·25-s + 27-s + 6·29-s + 13·37-s − 16·41-s + 17·43-s + 9·45-s − 24·47-s + 2·53-s − 9·55-s + 2·59-s − 13·61-s + 2·67-s + 10·71-s + 5·73-s + 16·79-s + 33·81-s − 43·83-s − 4·85-s + 8·89-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 3·9-s + 2.71·11-s + 0.970·17-s + 1.83·19-s + 5.21·23-s − 2.39·25-s + 0.192·27-s + 1.11·29-s + 2.13·37-s − 2.49·41-s + 2.59·43-s + 1.34·45-s − 3.50·47-s + 0.274·53-s − 1.21·55-s + 0.260·59-s − 1.66·61-s + 0.244·67-s + 1.18·71-s + 0.585·73-s + 1.80·79-s + 11/3·81-s − 4.71·83-s − 0.433·85-s + 0.847·89-s − 0.820·95-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 7^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56507\times 10^{14}\) |
| Root analytic conductor: | \(7.71184\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 7^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(54.76683685\) |
| \(L(\frac12)\) | \(\approx\) | \(54.76683685\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 19 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 + p^{2} T^{2} - T^{3} + 16 p T^{4} - 11 T^{5} + 203 T^{6} - 46 T^{7} + 694 T^{8} - 46 p T^{9} + 203 p^{2} T^{10} - 11 p^{3} T^{11} + 16 p^{5} T^{12} - p^{5} T^{13} + p^{8} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + T + 13 T^{2} + 26 T^{3} + 112 T^{4} + 248 T^{5} + 851 T^{6} + 1537 T^{7} + 5006 T^{8} + 1537 p T^{9} + 851 p^{2} T^{10} + 248 p^{3} T^{11} + 112 p^{4} T^{12} + 26 p^{5} T^{13} + 13 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 9 T + 82 T^{2} - 459 T^{3} + 2417 T^{4} - 9780 T^{5} + 38622 T^{6} - 128886 T^{7} + 453764 T^{8} - 128886 p T^{9} + 38622 p^{2} T^{10} - 9780 p^{3} T^{11} + 2417 p^{4} T^{12} - 459 p^{5} T^{13} + 82 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 63 T^{2} + T^{3} + 1844 T^{4} + 265 T^{5} + 34701 T^{6} + 9002 T^{7} + 498638 T^{8} + 9002 p T^{9} + 34701 p^{2} T^{10} + 265 p^{3} T^{11} + 1844 p^{4} T^{12} + p^{5} T^{13} + 63 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 4 T + 58 T^{2} - 185 T^{3} + 121 p T^{4} - 6234 T^{5} + 53760 T^{6} - 139113 T^{7} + 1017128 T^{8} - 139113 p T^{9} + 53760 p^{2} T^{10} - 6234 p^{3} T^{11} + 121 p^{5} T^{12} - 185 p^{5} T^{13} + 58 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 25 T + 413 T^{2} - 4867 T^{3} + 46668 T^{4} - 369316 T^{5} + 2515264 T^{6} - 14747354 T^{7} + 75826491 T^{8} - 14747354 p T^{9} + 2515264 p^{2} T^{10} - 369316 p^{3} T^{11} + 46668 p^{4} T^{12} - 4867 p^{5} T^{13} + 413 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 6 T + 127 T^{2} - 889 T^{3} + 8908 T^{4} - 58681 T^{5} + 443085 T^{6} - 2388316 T^{7} + 15502590 T^{8} - 2388316 p T^{9} + 443085 p^{2} T^{10} - 58681 p^{3} T^{11} + 8908 p^{4} T^{12} - 889 p^{5} T^{13} + 127 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 148 T^{2} - 116 T^{3} + 10252 T^{4} - 17596 T^{5} + 457996 T^{6} - 1076104 T^{7} + 15671846 T^{8} - 1076104 p T^{9} + 457996 p^{2} T^{10} - 17596 p^{3} T^{11} + 10252 p^{4} T^{12} - 116 p^{5} T^{13} + 148 p^{6} T^{14} + p^{8} T^{16} \) | |
| 37 | \( 1 - 13 T + 207 T^{2} - 1750 T^{3} + 16402 T^{4} - 112836 T^{5} + 816117 T^{6} - 5071053 T^{7} + 32170002 T^{8} - 5071053 p T^{9} + 816117 p^{2} T^{10} - 112836 p^{3} T^{11} + 16402 p^{4} T^{12} - 1750 p^{5} T^{13} + 207 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( ( 1 + 2 T + p T^{2} )^{8} \) | |
| 43 | \( 1 - 17 T + 355 T^{2} - 4036 T^{3} + 49646 T^{4} - 429858 T^{5} + 3936957 T^{6} - 642057 p T^{7} + 204548050 T^{8} - 642057 p^{2} T^{9} + 3936957 p^{2} T^{10} - 429858 p^{3} T^{11} + 49646 p^{4} T^{12} - 4036 p^{5} T^{13} + 355 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 24 T + 392 T^{2} + 4364 T^{3} + 37602 T^{4} + 233364 T^{5} + 1036208 T^{6} + 2240436 T^{7} + 2928963 T^{8} + 2240436 p T^{9} + 1036208 p^{2} T^{10} + 233364 p^{3} T^{11} + 37602 p^{4} T^{12} + 4364 p^{5} T^{13} + 392 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 2 T + 129 T^{2} + 37 T^{3} + 10574 T^{4} - 3559 T^{5} + 772851 T^{6} - 539736 T^{7} + 43318986 T^{8} - 539736 p T^{9} + 772851 p^{2} T^{10} - 3559 p^{3} T^{11} + 10574 p^{4} T^{12} + 37 p^{5} T^{13} + 129 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 2 T + 161 T^{2} - 671 T^{3} + 14184 T^{4} - 82025 T^{5} + 987807 T^{6} - 6008672 T^{7} + 60662110 T^{8} - 6008672 p T^{9} + 987807 p^{2} T^{10} - 82025 p^{3} T^{11} + 14184 p^{4} T^{12} - 671 p^{5} T^{13} + 161 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 13 T + 330 T^{2} + 3221 T^{3} + 51309 T^{4} + 421256 T^{5} + 5202698 T^{6} + 36815298 T^{7} + 375128276 T^{8} + 36815298 p T^{9} + 5202698 p^{2} T^{10} + 421256 p^{3} T^{11} + 51309 p^{4} T^{12} + 3221 p^{5} T^{13} + 330 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 2 T + 453 T^{2} - 869 T^{3} + 94066 T^{4} - 164739 T^{5} + 11710995 T^{6} - 17822484 T^{7} + 956952666 T^{8} - 17822484 p T^{9} + 11710995 p^{2} T^{10} - 164739 p^{3} T^{11} + 94066 p^{4} T^{12} - 869 p^{5} T^{13} + 453 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 10 T + 328 T^{2} - 2494 T^{3} + 51564 T^{4} - 327342 T^{5} + 78728 p T^{6} - 31264890 T^{7} + 457121126 T^{8} - 31264890 p T^{9} + 78728 p^{3} T^{10} - 327342 p^{3} T^{11} + 51564 p^{4} T^{12} - 2494 p^{5} T^{13} + 328 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 5 T + 227 T^{2} - 759 T^{3} + 19426 T^{4} + 26030 T^{5} + 597516 T^{6} + 13022320 T^{7} - 270615 T^{8} + 13022320 p T^{9} + 597516 p^{2} T^{10} + 26030 p^{3} T^{11} + 19426 p^{4} T^{12} - 759 p^{5} T^{13} + 227 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 412 T^{2} - 5444 T^{3} + 81524 T^{4} - 930572 T^{5} + 10484580 T^{6} - 103560600 T^{7} + 965277206 T^{8} - 103560600 p T^{9} + 10484580 p^{2} T^{10} - 930572 p^{3} T^{11} + 81524 p^{4} T^{12} - 5444 p^{5} T^{13} + 412 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 43 T + 1290 T^{2} + 27837 T^{3} + 495581 T^{4} + 7318764 T^{5} + 93657586 T^{6} + 1035827238 T^{7} + 10099433844 T^{8} + 1035827238 p T^{9} + 93657586 p^{2} T^{10} + 7318764 p^{3} T^{11} + 495581 p^{4} T^{12} + 27837 p^{5} T^{13} + 1290 p^{6} T^{14} + 43 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 8 T + 448 T^{2} - 3580 T^{3} + 98092 T^{4} - 710908 T^{5} + 13956896 T^{6} - 88249184 T^{7} + 16134934 p T^{8} - 88249184 p T^{9} + 13956896 p^{2} T^{10} - 710908 p^{3} T^{11} + 98092 p^{4} T^{12} - 3580 p^{5} T^{13} + 448 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 12 T + 608 T^{2} + 5812 T^{3} + 170620 T^{4} + 1355452 T^{5} + 29452384 T^{6} + 196855172 T^{7} + 3436368006 T^{8} + 196855172 p T^{9} + 29452384 p^{2} T^{10} + 1355452 p^{3} T^{11} + 170620 p^{4} T^{12} + 5812 p^{5} T^{13} + 608 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.29207584995394540115068442489, −2.97025566599721680851085903937, −2.95743450417319163042219427380, −2.93966823292409946157197193714, −2.80398561467016136394838585001, −2.75882413259309349605950079167, −2.72069388028561751606906667875, −2.60359025475657129036670078771, −2.50910114169373286090819234576, −2.05781482670116854294927745620, −1.92100403643489553231850075471, −1.91874588274641452236736494683, −1.88039250670180818372252089599, −1.77318043839249126537234618105, −1.65922171927053373976948418559, −1.45500311099123761259675854012, −1.29095645664281311727707889651, −1.12376992463193954143160693599, −1.09802029650983048767998412188, −0.979391882788534858588754955441, −0.56757633580984542772623305889, −0.53934515358848748387159892336, −0.53325613418273381647312438425, −0.50755109545045332146904247087, −0.42456785024167406924971147747, 0.42456785024167406924971147747, 0.50755109545045332146904247087, 0.53325613418273381647312438425, 0.53934515358848748387159892336, 0.56757633580984542772623305889, 0.979391882788534858588754955441, 1.09802029650983048767998412188, 1.12376992463193954143160693599, 1.29095645664281311727707889651, 1.45500311099123761259675854012, 1.65922171927053373976948418559, 1.77318043839249126537234618105, 1.88039250670180818372252089599, 1.91874588274641452236736494683, 1.92100403643489553231850075471, 2.05781482670116854294927745620, 2.50910114169373286090819234576, 2.60359025475657129036670078771, 2.72069388028561751606906667875, 2.75882413259309349605950079167, 2.80398561467016136394838585001, 2.93966823292409946157197193714, 2.95743450417319163042219427380, 2.97025566599721680851085903937, 3.29207584995394540115068442489
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.