Dirichlet series
| L(s) = 1 | − 3-s − 4·7-s − 11·9-s − 9·13-s − 4·17-s + 4·19-s + 4·21-s − 2·23-s + 10·27-s − 5·29-s + 11·31-s + 8·37-s + 9·39-s − 4·43-s − 14·47-s − 25·49-s + 4·51-s − 11·53-s − 4·57-s − 15·59-s − 13·61-s + 44·63-s − 19·67-s + 2·69-s + 40·71-s − 31·73-s + 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s − 3.66·9-s − 2.49·13-s − 0.970·17-s + 0.917·19-s + 0.872·21-s − 0.417·23-s + 1.92·27-s − 0.928·29-s + 1.97·31-s + 1.31·37-s + 1.44·39-s − 0.609·43-s − 2.04·47-s − 3.57·49-s + 0.560·51-s − 1.51·53-s − 0.529·57-s − 1.95·59-s − 1.66·61-s + 5.54·63-s − 2.32·67-s + 0.240·69-s + 4.74·71-s − 3.62·73-s + 0.225·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{16} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 5^{16} \cdot 37^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.48617\times 10^{14}\) |
| Root analytic conductor: | \(7.68695\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{24} \cdot 5^{16} \cdot 37^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 5 | \( 1 \) | |
| 37 | \( ( 1 - T )^{8} \) | |
| good | 3 | \( 1 + T + 4 p T^{2} + 13 T^{3} + 25 p T^{4} + 83 T^{5} + 4 p^{4} T^{6} + 340 T^{7} + 362 p T^{8} + 340 p T^{9} + 4 p^{6} T^{10} + 83 p^{3} T^{11} + 25 p^{5} T^{12} + 13 p^{5} T^{13} + 4 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 4 T + 41 T^{2} + 134 T^{3} + 760 T^{4} + 2141 T^{5} + 8811 T^{6} + 3103 p T^{7} + 72190 T^{8} + 3103 p^{2} T^{9} + 8811 p^{2} T^{10} + 2141 p^{3} T^{11} + 760 p^{4} T^{12} + 134 p^{5} T^{13} + 41 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 40 T^{2} + 21 T^{3} + 756 T^{4} + 1183 T^{5} + 9179 T^{6} + 24891 T^{7} + 95896 T^{8} + 24891 p T^{9} + 9179 p^{2} T^{10} + 1183 p^{3} T^{11} + 756 p^{4} T^{12} + 21 p^{5} T^{13} + 40 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 9 T + 99 T^{2} + 634 T^{3} + 4216 T^{4} + 20993 T^{5} + 104419 T^{6} + 419854 T^{7} + 1666702 T^{8} + 419854 p T^{9} + 104419 p^{2} T^{10} + 20993 p^{3} T^{11} + 4216 p^{4} T^{12} + 634 p^{5} T^{13} + 99 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 4 T + 71 T^{2} + 145 T^{3} + 2168 T^{4} + 1557 T^{5} + 2687 p T^{6} - 217 T^{7} + 823058 T^{8} - 217 p T^{9} + 2687 p^{3} T^{10} + 1557 p^{3} T^{11} + 2168 p^{4} T^{12} + 145 p^{5} T^{13} + 71 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 4 T + 89 T^{2} - 356 T^{3} + 3589 T^{4} - 15109 T^{5} + 92127 T^{6} - 407004 T^{7} + 1868187 T^{8} - 407004 p T^{9} + 92127 p^{2} T^{10} - 15109 p^{3} T^{11} + 3589 p^{4} T^{12} - 356 p^{5} T^{13} + 89 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + 2 T + 94 T^{2} + 301 T^{3} + 4971 T^{4} + 16391 T^{5} + 182755 T^{6} + 565688 T^{7} + 4833338 T^{8} + 565688 p T^{9} + 182755 p^{2} T^{10} + 16391 p^{3} T^{11} + 4971 p^{4} T^{12} + 301 p^{5} T^{13} + 94 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 5 T + 187 T^{2} + 800 T^{3} + 16086 T^{4} + 58905 T^{5} + 839763 T^{6} + 2610422 T^{7} + 29357790 T^{8} + 2610422 p T^{9} + 839763 p^{2} T^{10} + 58905 p^{3} T^{11} + 16086 p^{4} T^{12} + 800 p^{5} T^{13} + 187 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 11 T + 231 T^{2} - 1905 T^{3} + 23233 T^{4} - 153870 T^{5} + 1371031 T^{6} - 7421606 T^{7} + 52211608 T^{8} - 7421606 p T^{9} + 1371031 p^{2} T^{10} - 153870 p^{3} T^{11} + 23233 p^{4} T^{12} - 1905 p^{5} T^{13} + 231 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 224 T^{2} - 141 T^{3} + 24629 T^{4} - 22712 T^{5} + 1723583 T^{6} - 1673307 T^{7} + 83770693 T^{8} - 1673307 p T^{9} + 1723583 p^{2} T^{10} - 22712 p^{3} T^{11} + 24629 p^{4} T^{12} - 141 p^{5} T^{13} + 224 p^{6} T^{14} + p^{8} T^{16} \) | |
| 43 | \( 1 + 4 T + 76 T^{2} + 193 T^{3} + 5489 T^{4} + 24195 T^{5} + 355439 T^{6} + 947630 T^{7} + 14585842 T^{8} + 947630 p T^{9} + 355439 p^{2} T^{10} + 24195 p^{3} T^{11} + 5489 p^{4} T^{12} + 193 p^{5} T^{13} + 76 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 14 T + 203 T^{2} + 2202 T^{3} + 22790 T^{4} + 190523 T^{5} + 1638719 T^{6} + 11878747 T^{7} + 86001310 T^{8} + 11878747 p T^{9} + 1638719 p^{2} T^{10} + 190523 p^{3} T^{11} + 22790 p^{4} T^{12} + 2202 p^{5} T^{13} + 203 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T + 229 T^{2} + 1735 T^{3} + 22744 T^{4} + 123313 T^{5} + 1409954 T^{6} + 6077569 T^{7} + 74018624 T^{8} + 6077569 p T^{9} + 1409954 p^{2} T^{10} + 123313 p^{3} T^{11} + 22744 p^{4} T^{12} + 1735 p^{5} T^{13} + 229 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 15 T + 407 T^{2} + 4664 T^{3} + 74618 T^{4} + 688341 T^{5} + 8132690 T^{6} + 61792736 T^{7} + 583695956 T^{8} + 61792736 p T^{9} + 8132690 p^{2} T^{10} + 688341 p^{3} T^{11} + 74618 p^{4} T^{12} + 4664 p^{5} T^{13} + 407 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 13 T + 163 T^{2} + 1165 T^{3} + 16487 T^{4} + 142792 T^{5} + 1563853 T^{6} + 9653588 T^{7} + 89810140 T^{8} + 9653588 p T^{9} + 1563853 p^{2} T^{10} + 142792 p^{3} T^{11} + 16487 p^{4} T^{12} + 1165 p^{5} T^{13} + 163 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 19 T + 559 T^{2} + 7561 T^{3} + 127663 T^{4} + 1338243 T^{5} + 16489908 T^{6} + 139349862 T^{7} + 1359453328 T^{8} + 139349862 p T^{9} + 16489908 p^{2} T^{10} + 1338243 p^{3} T^{11} + 127663 p^{4} T^{12} + 7561 p^{5} T^{13} + 559 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 40 T + 899 T^{2} - 14968 T^{3} + 209296 T^{4} - 2532989 T^{5} + 27167623 T^{6} - 262647011 T^{7} + 2318120662 T^{8} - 262647011 p T^{9} + 27167623 p^{2} T^{10} - 2532989 p^{3} T^{11} + 209296 p^{4} T^{12} - 14968 p^{5} T^{13} + 899 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 31 T + 859 T^{2} + 15266 T^{3} + 247690 T^{4} + 3153920 T^{5} + 37367805 T^{6} + 369631911 T^{7} + 3429783898 T^{8} + 369631911 p T^{9} + 37367805 p^{2} T^{10} + 3153920 p^{3} T^{11} + 247690 p^{4} T^{12} + 15266 p^{5} T^{13} + 859 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 2 T + 241 T^{2} + 317 T^{3} + 25518 T^{4} + 160045 T^{5} + 1915546 T^{6} + 21066986 T^{7} + 144183384 T^{8} + 21066986 p T^{9} + 1915546 p^{2} T^{10} + 160045 p^{3} T^{11} + 25518 p^{4} T^{12} + 317 p^{5} T^{13} + 241 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 4 T + 301 T^{2} + 857 T^{3} + 54134 T^{4} + 95849 T^{5} + 6417081 T^{6} + 7294227 T^{7} + 604402462 T^{8} + 7294227 p T^{9} + 6417081 p^{2} T^{10} + 95849 p^{3} T^{11} + 54134 p^{4} T^{12} + 857 p^{5} T^{13} + 301 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 12 T + 459 T^{2} + 4425 T^{3} + 102144 T^{4} + 851303 T^{5} + 14968945 T^{6} + 109132431 T^{7} + 17613520 p T^{8} + 109132431 p T^{9} + 14968945 p^{2} T^{10} + 851303 p^{3} T^{11} + 102144 p^{4} T^{12} + 4425 p^{5} T^{13} + 459 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 11 T + 341 T^{2} + 3139 T^{3} + 37601 T^{4} + 191096 T^{5} - 201829 T^{6} - 22517198 T^{7} - 298694996 T^{8} - 22517198 p T^{9} - 201829 p^{2} T^{10} + 191096 p^{3} T^{11} + 37601 p^{4} T^{12} + 3139 p^{5} T^{13} + 341 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.63781537093458223077848955815, −3.35358602094162403631116762944, −3.16866834861953392050049366688, −3.11060510655001701741078058305, −3.09263506739564643488899512771, −3.06323099618941609541453997654, −3.06085589887160388742016133900, −2.99852503550551270884118000171, −2.99734256247562265776340215907, −2.61434727497127814684422254156, −2.49599059664715106792597623423, −2.49253999766850301263740328006, −2.37758830003560983719967997036, −2.20968707240768147442259439429, −2.17833575734121189749515951834, −2.13389746703319454506044827658, −2.11093992332776658930164184034, −1.57990519623249631007725704567, −1.57425735451752144925734281963, −1.39545628431842754335440431609, −1.30669521302015643659372614173, −1.30388085362307466487649111931, −1.03752905638199006409131040907, −0.964386981791585366381639066511, −0.879741498050797199283884864291, 0, 0, 0, 0, 0, 0, 0, 0, 0.879741498050797199283884864291, 0.964386981791585366381639066511, 1.03752905638199006409131040907, 1.30388085362307466487649111931, 1.30669521302015643659372614173, 1.39545628431842754335440431609, 1.57425735451752144925734281963, 1.57990519623249631007725704567, 2.11093992332776658930164184034, 2.13389746703319454506044827658, 2.17833575734121189749515951834, 2.20968707240768147442259439429, 2.37758830003560983719967997036, 2.49253999766850301263740328006, 2.49599059664715106792597623423, 2.61434727497127814684422254156, 2.99734256247562265776340215907, 2.99852503550551270884118000171, 3.06085589887160388742016133900, 3.06323099618941609541453997654, 3.09263506739564643488899512771, 3.11060510655001701741078058305, 3.16866834861953392050049366688, 3.35358602094162403631116762944, 3.63781537093458223077848955815
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.