Dirichlet series
| L(s) = 1 | + 4·3-s + 8·5-s + 8·7-s + 8·9-s + 12·11-s + 32·15-s − 4·19-s + 32·21-s + 8·23-s + 36·25-s + 12·27-s − 8·29-s + 32·31-s + 48·33-s + 64·35-s − 16·37-s − 8·41-s − 20·43-s + 64·45-s + 32·49-s + 16·53-s + 96·55-s − 16·57-s − 28·59-s + 64·63-s − 12·67-s + 32·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.57·5-s + 3.02·7-s + 8/3·9-s + 3.61·11-s + 8.26·15-s − 0.917·19-s + 6.98·21-s + 1.66·23-s + 36/5·25-s + 2.30·27-s − 1.48·29-s + 5.74·31-s + 8.35·33-s + 10.8·35-s − 2.63·37-s − 1.24·41-s − 3.04·43-s + 9.54·45-s + 32/7·49-s + 2.19·53-s + 12.9·55-s − 2.11·57-s − 3.64·59-s + 8.06·63-s − 1.46·67-s + 3.85·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{72}\) |
| Sign: | $1$ |
| Analytic conductor: | \(78050.6\) |
| Root analytic conductor: | \(2.02196\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{72} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(90.20983504\) |
| \(L(\frac12)\) | \(\approx\) | \(90.20983504\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + 16 T^{4} - 44 T^{5} + 88 T^{6} - 100 T^{7} + 142 T^{8} - 100 p T^{9} + 88 p^{2} T^{10} - 44 p^{3} T^{11} + 16 p^{4} T^{12} - 4 p^{6} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 7 | \( 1 - 8 T + 32 T^{2} - 104 T^{3} + 36 p T^{4} - 8 p^{2} T^{5} + 480 T^{6} - 24 p T^{7} - 26 p^{2} T^{8} - 24 p^{2} T^{9} + 480 p^{2} T^{10} - 8 p^{5} T^{11} + 36 p^{5} T^{12} - 104 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - 12 T + 72 T^{2} - 28 p T^{3} + 1104 T^{4} - 3444 T^{5} + 9496 T^{6} - 23308 T^{7} + 63694 T^{8} - 23308 p T^{9} + 9496 p^{2} T^{10} - 3444 p^{3} T^{11} + 1104 p^{4} T^{12} - 28 p^{6} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T^{2} + 16 T^{3} + 8 T^{4} - 848 T^{5} + 1876 T^{6} + 4480 T^{7} - 5522 T^{8} + 4480 p T^{9} + 1876 p^{2} T^{10} - 848 p^{3} T^{11} + 8 p^{4} T^{12} + 16 p^{5} T^{13} - 4 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 72 T^{2} + 2620 T^{4} - 64376 T^{6} + 1215110 T^{8} - 64376 p^{2} T^{10} + 2620 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 + 4 T + 24 T^{2} + 28 T^{3} + 80 T^{4} - 580 T^{5} - 9592 T^{6} - 13468 T^{7} - 163698 T^{8} - 13468 p T^{9} - 9592 p^{2} T^{10} - 580 p^{3} T^{11} + 80 p^{4} T^{12} + 28 p^{5} T^{13} + 24 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T + 32 T^{2} - 200 T^{3} + 2108 T^{4} - 11560 T^{5} + 45024 T^{6} - 277224 T^{7} + 1704454 T^{8} - 277224 p T^{9} + 45024 p^{2} T^{10} - 11560 p^{3} T^{11} + 2108 p^{4} T^{12} - 200 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 8 T + 44 T^{2} + 152 T^{3} + 392 T^{4} - 24 p T^{5} - 22492 T^{6} - 315304 T^{7} - 2293522 T^{8} - 315304 p T^{9} - 22492 p^{2} T^{10} - 24 p^{4} T^{11} + 392 p^{4} T^{12} + 152 p^{5} T^{13} + 44 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) | |
| 37 | \( 1 + 16 T + 204 T^{2} + 1504 T^{3} + 9032 T^{4} + 11552 T^{5} - 6028 p T^{6} - 3586384 T^{7} - 24284370 T^{8} - 3586384 p T^{9} - 6028 p^{3} T^{10} + 11552 p^{3} T^{11} + 9032 p^{4} T^{12} + 1504 p^{5} T^{13} + 204 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 1788 T^{4} + 23048 T^{5} + 131040 T^{6} + 18264 p T^{7} + 2278 p^{2} T^{8} + 18264 p^{2} T^{9} + 131040 p^{2} T^{10} + 23048 p^{3} T^{11} + 1788 p^{4} T^{12} + 88 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 20 T + 216 T^{2} + 2012 T^{3} + 17168 T^{4} + 134172 T^{5} + 978248 T^{6} + 6990356 T^{7} + 48253518 T^{8} + 6990356 p T^{9} + 978248 p^{2} T^{10} + 134172 p^{3} T^{11} + 17168 p^{4} T^{12} + 2012 p^{5} T^{13} + 216 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 312 T^{2} + 44348 T^{4} - 3794696 T^{6} + 215798406 T^{8} - 3794696 p^{2} T^{10} + 44348 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} \) | |
| 53 | \( 1 - 16 T + 60 T^{2} - 192 T^{3} + 5896 T^{4} - 69312 T^{5} + 601556 T^{6} - 2947504 T^{7} + 8421422 T^{8} - 2947504 p T^{9} + 601556 p^{2} T^{10} - 69312 p^{3} T^{11} + 5896 p^{4} T^{12} - 192 p^{5} T^{13} + 60 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 28 T + 456 T^{2} + 5844 T^{3} + 63760 T^{4} + 602388 T^{5} + 5104280 T^{6} + 41213596 T^{7} + 323336270 T^{8} + 41213596 p T^{9} + 5104280 p^{2} T^{10} + 602388 p^{3} T^{11} + 63760 p^{4} T^{12} + 5844 p^{5} T^{13} + 456 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 156 T^{2} + 528 T^{3} + 12168 T^{4} + 91824 T^{5} + 700212 T^{6} + 9406080 T^{7} + 36137966 T^{8} + 9406080 p T^{9} + 700212 p^{2} T^{10} + 91824 p^{3} T^{11} + 12168 p^{4} T^{12} + 528 p^{5} T^{13} + 156 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 + 12 T + 216 T^{2} + 2196 T^{3} + 23760 T^{4} + 224244 T^{5} + 2110536 T^{6} + 20206380 T^{7} + 158903054 T^{8} + 20206380 p T^{9} + 2110536 p^{2} T^{10} + 224244 p^{3} T^{11} + 23760 p^{4} T^{12} + 2196 p^{5} T^{13} + 216 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 21308 T^{4} - 62712 T^{5} - 415008 T^{6} + 11818440 T^{7} - 144298362 T^{8} + 11818440 p T^{9} - 415008 p^{2} T^{10} - 62712 p^{3} T^{11} + 21308 p^{4} T^{12} - 2904 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 32 T + 512 T^{2} - 96 p T^{3} + 95708 T^{4} - 1109024 T^{5} + 11042304 T^{6} - 107800160 T^{7} + 989395590 T^{8} - 107800160 p T^{9} + 11042304 p^{2} T^{10} - 1109024 p^{3} T^{11} + 95708 p^{4} T^{12} - 96 p^{6} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 120 T^{2} + 10940 T^{4} + 170552 T^{6} - 23710074 T^{8} + 170552 p^{2} T^{10} + 10940 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) | |
| 83 | \( 1 + 36 T + 8 p T^{2} + 7852 T^{3} + 62480 T^{4} + 334156 T^{5} + 1295752 T^{6} + 6295044 T^{7} + 54709902 T^{8} + 6295044 p T^{9} + 1295752 p^{2} T^{10} + 334156 p^{3} T^{11} + 62480 p^{4} T^{12} + 7852 p^{5} T^{13} + 8 p^{7} T^{14} + 36 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 16 T + 128 T^{2} - 2096 T^{3} + 18652 T^{4} + 22640 T^{5} - 553088 T^{6} + 12396240 T^{7} - 224072442 T^{8} + 12396240 p T^{9} - 553088 p^{2} T^{10} + 22640 p^{3} T^{11} + 18652 p^{4} T^{12} - 2096 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( ( 1 - 16 T + 428 T^{2} - 4368 T^{3} + 63222 T^{4} - 4368 p T^{5} + 428 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
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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
−4.86225649528618927880186561605, −4.57743073755929591897358371402, −4.57023994977045490777977992854, −4.49528751850476131014802070235, −4.12034024001927669804832613771, −4.08712587760529242563133046164, −3.92625749656851438211193264133, −3.72116423077193514677147882553, −3.63518705261291005567348330064, −3.57488289095398309448282903540, −3.16991187470117014546410056357, −3.00234853888873776465978660604, −2.91519779611334118898603250632, −2.78246256488193253827444013473, −2.73478246322379693595472037234, −2.23158340691775057509727742936, −2.21359562302623711392501068250, −2.16234318048058964640696623561, −1.94962413249165835278545105062, −1.66621819722535568361411526299, −1.58986431222424686484865740674, −1.34397974764690524523124868783, −1.32198757493050711272928574702, −1.05561598336297316492855715690, −1.03675258619460326302507771177, 1.03675258619460326302507771177, 1.05561598336297316492855715690, 1.32198757493050711272928574702, 1.34397974764690524523124868783, 1.58986431222424686484865740674, 1.66621819722535568361411526299, 1.94962413249165835278545105062, 2.16234318048058964640696623561, 2.21359562302623711392501068250, 2.23158340691775057509727742936, 2.73478246322379693595472037234, 2.78246256488193253827444013473, 2.91519779611334118898603250632, 3.00234853888873776465978660604, 3.16991187470117014546410056357, 3.57488289095398309448282903540, 3.63518705261291005567348330064, 3.72116423077193514677147882553, 3.92625749656851438211193264133, 4.08712587760529242563133046164, 4.12034024001927669804832613771, 4.49528751850476131014802070235, 4.57023994977045490777977992854, 4.57743073755929591897358371402, 4.86225649528618927880186561605
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.