Dirichlet series
| L(s) = 1 | + 5·3-s + 10·7-s + 5·9-s − 11-s − 10·13-s − 15·17-s + 10·19-s + 50·21-s + 30·23-s − 15·27-s + 10·29-s + 9·31-s − 5·33-s + 10·37-s − 50·39-s − 4·41-s + 30·47-s + 20·49-s − 75·51-s − 10·53-s + 50·57-s + 5·59-s + 6·61-s + 50·63-s + 10·67-s + 150·69-s + 9·71-s + ⋯ |
| L(s) = 1 | + 2.88·3-s + 3.77·7-s + 5/3·9-s − 0.301·11-s − 2.77·13-s − 3.63·17-s + 2.29·19-s + 10.9·21-s + 6.25·23-s − 2.88·27-s + 1.85·29-s + 1.61·31-s − 0.870·33-s + 1.64·37-s − 8.00·39-s − 0.624·41-s + 4.37·47-s + 20/7·49-s − 10.5·51-s − 1.37·53-s + 6.62·57-s + 0.650·59-s + 0.768·61-s + 6.29·63-s + 1.22·67-s + 18.0·69-s + 1.06·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.65278\times 10^{15}\) |
| Root analytic conductor: | \(8.93590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 5^{32} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(218.4581129\) |
| \(L(\frac12)\) | \(\approx\) | \(218.4581129\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| good | 3 | \( 1 - 5 T + 20 T^{2} - 20 p T^{3} + 161 T^{4} - 125 p T^{5} + 815 T^{6} - 1580 T^{7} + 2881 T^{8} - 1580 p T^{9} + 815 p^{2} T^{10} - 125 p^{4} T^{11} + 161 p^{4} T^{12} - 20 p^{6} T^{13} + 20 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 10 T + 80 T^{2} - 65 p T^{3} + 318 p T^{4} - 9040 T^{5} + 32660 T^{6} - 102420 T^{7} + 289171 T^{8} - 102420 p T^{9} + 32660 p^{2} T^{10} - 9040 p^{3} T^{11} + 318 p^{5} T^{12} - 65 p^{6} T^{13} + 80 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + T + 30 T^{2} - 5 T^{3} + 530 T^{4} - 32 p T^{5} + 8358 T^{6} - 2865 T^{7} + 108615 T^{8} - 2865 p T^{9} + 8358 p^{2} T^{10} - 32 p^{4} T^{11} + 530 p^{4} T^{12} - 5 p^{5} T^{13} + 30 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 10 T + 115 T^{2} + 770 T^{3} + 5201 T^{4} + 26485 T^{5} + 131900 T^{6} + 41330 p T^{7} + 2120011 T^{8} + 41330 p^{2} T^{9} + 131900 p^{2} T^{10} + 26485 p^{3} T^{11} + 5201 p^{4} T^{12} + 770 p^{5} T^{13} + 115 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 15 T + 195 T^{2} + 1665 T^{3} + 12881 T^{4} + 79230 T^{5} + 450900 T^{6} + 2158500 T^{7} + 9636021 T^{8} + 2158500 p T^{9} + 450900 p^{2} T^{10} + 79230 p^{3} T^{11} + 12881 p^{4} T^{12} + 1665 p^{5} T^{13} + 195 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 10 T + 137 T^{2} - 875 T^{3} + 7113 T^{4} - 34660 T^{5} + 219869 T^{6} - 898650 T^{7} + 4848685 T^{8} - 898650 p T^{9} + 219869 p^{2} T^{10} - 34660 p^{3} T^{11} + 7113 p^{4} T^{12} - 875 p^{5} T^{13} + 137 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 30 T + 535 T^{2} - 6765 T^{3} + 67381 T^{4} - 550125 T^{5} + 3801715 T^{6} - 22541700 T^{7} + 116002711 T^{8} - 22541700 p T^{9} + 3801715 p^{2} T^{10} - 550125 p^{3} T^{11} + 67381 p^{4} T^{12} - 6765 p^{5} T^{13} + 535 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 10 T + 207 T^{2} - 1495 T^{3} + 18338 T^{4} - 106235 T^{5} + 973524 T^{6} - 4656000 T^{7} + 34241385 T^{8} - 4656000 p T^{9} + 973524 p^{2} T^{10} - 106235 p^{3} T^{11} + 18338 p^{4} T^{12} - 1495 p^{5} T^{13} + 207 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 9 T + 165 T^{2} - 1120 T^{3} + 12915 T^{4} - 2362 p T^{5} + 659698 T^{6} - 3221745 T^{7} + 24086635 T^{8} - 3221745 p T^{9} + 659698 p^{2} T^{10} - 2362 p^{4} T^{11} + 12915 p^{4} T^{12} - 1120 p^{5} T^{13} + 165 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 10 T + 205 T^{2} - 1605 T^{3} + 19241 T^{4} - 124765 T^{5} + 1155900 T^{6} - 6414745 T^{7} + 49874231 T^{8} - 6414745 p T^{9} + 1155900 p^{2} T^{10} - 124765 p^{3} T^{11} + 19241 p^{4} T^{12} - 1605 p^{5} T^{13} + 205 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 4 T + 135 T^{2} + 5 p T^{3} + 8810 T^{4} + 142 p T^{5} + 512313 T^{6} + 539955 T^{7} + 24859965 T^{8} + 539955 p T^{9} + 512313 p^{2} T^{10} + 142 p^{4} T^{11} + 8810 p^{4} T^{12} + 5 p^{6} T^{13} + 135 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 245 T^{2} + 180 T^{3} + 28176 T^{4} + 33015 T^{5} + 2039890 T^{6} + 2606415 T^{7} + 103354571 T^{8} + 2606415 p T^{9} + 2039890 p^{2} T^{10} + 33015 p^{3} T^{11} + 28176 p^{4} T^{12} + 180 p^{5} T^{13} + 245 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 - 30 T + 690 T^{2} - 10845 T^{3} + 145496 T^{4} - 1580760 T^{5} + 15227925 T^{6} - 2658405 p T^{7} + 922196961 T^{8} - 2658405 p^{2} T^{9} + 15227925 p^{2} T^{10} - 1580760 p^{3} T^{11} + 145496 p^{4} T^{12} - 10845 p^{5} T^{13} + 690 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 10 T + 285 T^{2} + 2845 T^{3} + 39866 T^{4} + 360485 T^{5} + 3615555 T^{6} + 27790515 T^{7} + 228358671 T^{8} + 27790515 p T^{9} + 3615555 p^{2} T^{10} + 360485 p^{3} T^{11} + 39866 p^{4} T^{12} + 2845 p^{5} T^{13} + 285 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 5 T + 372 T^{2} - 1910 T^{3} + 1087 p T^{4} - 321430 T^{5} + 6757764 T^{6} - 30827475 T^{7} + 479489235 T^{8} - 30827475 p T^{9} + 6757764 p^{2} T^{10} - 321430 p^{3} T^{11} + 1087 p^{5} T^{12} - 1910 p^{5} T^{13} + 372 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 6 T + 315 T^{2} - 1430 T^{3} + 46515 T^{4} - 161153 T^{5} + 4429318 T^{6} - 12243180 T^{7} + 309414985 T^{8} - 12243180 p T^{9} + 4429318 p^{2} T^{10} - 161153 p^{3} T^{11} + 46515 p^{4} T^{12} - 1430 p^{5} T^{13} + 315 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 10 T + 355 T^{2} - 2615 T^{3} + 56936 T^{4} - 328690 T^{5} + 5800145 T^{6} - 27554285 T^{7} + 437541211 T^{8} - 27554285 p T^{9} + 5800145 p^{2} T^{10} - 328690 p^{3} T^{11} + 56936 p^{4} T^{12} - 2615 p^{5} T^{13} + 355 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 9 T + 435 T^{2} - 3015 T^{3} + 84560 T^{4} - 460242 T^{5} + 10018518 T^{6} - 44477955 T^{7} + 830638095 T^{8} - 44477955 p T^{9} + 10018518 p^{2} T^{10} - 460242 p^{3} T^{11} + 84560 p^{4} T^{12} - 3015 p^{5} T^{13} + 435 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 270 T^{2} - 185 T^{3} + 35526 T^{4} - 55370 T^{5} + 3355690 T^{6} - 7429800 T^{7} + 264804421 T^{8} - 7429800 p T^{9} + 3355690 p^{2} T^{10} - 55370 p^{3} T^{11} + 35526 p^{4} T^{12} - 185 p^{5} T^{13} + 270 p^{6} T^{14} + p^{8} T^{16} \) | |
| 79 | \( 1 - 20 T + 517 T^{2} - 7710 T^{3} + 118553 T^{4} - 1415495 T^{5} + 16387224 T^{6} - 163269350 T^{7} + 1541554535 T^{8} - 163269350 p T^{9} + 16387224 p^{2} T^{10} - 1415495 p^{3} T^{11} + 118553 p^{4} T^{12} - 7710 p^{5} T^{13} + 517 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 40 T + 1095 T^{2} - 21430 T^{3} + 350651 T^{4} - 4819475 T^{5} + 58790820 T^{6} - 630350640 T^{7} + 6097566381 T^{8} - 630350640 p T^{9} + 58790820 p^{2} T^{10} - 4819475 p^{3} T^{11} + 350651 p^{4} T^{12} - 21430 p^{5} T^{13} + 1095 p^{6} T^{14} - 40 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 5 T + 417 T^{2} + 2660 T^{3} + 85673 T^{4} + 651880 T^{5} + 11547054 T^{6} + 92814975 T^{7} + 1162452885 T^{8} + 92814975 p T^{9} + 11547054 p^{2} T^{10} + 651880 p^{3} T^{11} + 85673 p^{4} T^{12} + 2660 p^{5} T^{13} + 417 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 455 T^{2} - 1875 T^{3} + 99696 T^{4} - 615180 T^{5} + 15954925 T^{6} - 88841085 T^{7} + 1873622411 T^{8} - 88841085 p T^{9} + 15954925 p^{2} T^{10} - 615180 p^{3} T^{11} + 99696 p^{4} T^{12} - 1875 p^{5} T^{13} + 455 p^{6} T^{14} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.13333928077853965413166095844, −2.77398874097503373106923543338, −2.69965750075695921407884484034, −2.69475242097452430260583585026, −2.67483088668300646057002736339, −2.61527180156462722484376292351, −2.60839895518116391784141701561, −2.50772450429575725113993959931, −2.35729340908865837597768515424, −2.05147821787418019361099686621, −1.96227003610402243767824356632, −1.90199193737984577870221863037, −1.87467895337901913100457728307, −1.82205604370001989082608496086, −1.78582458748539910249917493807, −1.59402329731831750047781991999, −1.30515460554300633249344669054, −0.997344529629633134800370388459, −0.927959329550112401093090746079, −0.894786437594806613657316792975, −0.893869139237711460074373060683, −0.78470304251297588239721788651, −0.54942713787176081375595469113, −0.46536893337921514522341194860, −0.26151085613594820245784517116, 0.26151085613594820245784517116, 0.46536893337921514522341194860, 0.54942713787176081375595469113, 0.78470304251297588239721788651, 0.893869139237711460074373060683, 0.894786437594806613657316792975, 0.927959329550112401093090746079, 0.997344529629633134800370388459, 1.30515460554300633249344669054, 1.59402329731831750047781991999, 1.78582458748539910249917493807, 1.82205604370001989082608496086, 1.87467895337901913100457728307, 1.90199193737984577870221863037, 1.96227003610402243767824356632, 2.05147821787418019361099686621, 2.35729340908865837597768515424, 2.50772450429575725113993959931, 2.60839895518116391784141701561, 2.61527180156462722484376292351, 2.67483088668300646057002736339, 2.69475242097452430260583585026, 2.69965750075695921407884484034, 2.77398874097503373106923543338, 3.13333928077853965413166095844
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.