Dirichlet series
| L(s) = 1 | − 8·4-s − 8·9-s + 12·11-s + 36·16-s − 20·19-s − 20·29-s + 32·31-s + 64·36-s + 12·41-s − 96·44-s + 30·49-s + 32·61-s − 120·64-s + 12·71-s + 160·76-s − 20·79-s + 36·81-s − 40·89-s − 96·99-s + 12·101-s − 40·109-s + 160·116-s + 18·121-s − 256·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4·4-s − 8/3·9-s + 3.61·11-s + 9·16-s − 4.58·19-s − 3.71·29-s + 5.74·31-s + 32/3·36-s + 1.87·41-s − 14.4·44-s + 30/7·49-s + 4.09·61-s − 15·64-s + 1.42·71-s + 18.3·76-s − 2.25·79-s + 4·81-s − 4.23·89-s − 9.64·99-s + 1.19·101-s − 3.83·109-s + 14.8·116-s + 1.63·121-s − 22.9·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{64}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.17766\times 10^{23}\) |
| Root analytic conductor: | \(5.47210\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{64} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07347560114\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07347560114\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( ( 1 + T^{2} )^{8} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 50516750 p^{2} T^{18} + 6275029 p^{4} T^{20} - 716280 p^{6} T^{22} + 74703 p^{8} T^{24} - 6940 p^{10} T^{26} + 527 p^{12} T^{28} - 30 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 6 T + 45 T^{2} - 240 T^{3} + 1205 T^{4} - 4658 T^{5} + 20073 T^{6} - 68300 T^{7} + 239780 T^{8} - 68300 p T^{9} + 20073 p^{2} T^{10} - 4658 p^{3} T^{11} + 1205 p^{4} T^{12} - 240 p^{5} T^{13} + 45 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 - 90 T^{2} + 23 p^{2} T^{4} - 110780 T^{6} + 2431583 T^{8} - 3472280 p T^{10} + 743871389 T^{12} - 11084458050 T^{14} + 150617413480 T^{16} - 11084458050 p^{2} T^{18} + 743871389 p^{4} T^{20} - 3472280 p^{7} T^{22} + 2431583 p^{8} T^{24} - 110780 p^{10} T^{26} + 23 p^{14} T^{28} - 90 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 110 T^{2} + 5647 T^{4} - 191000 T^{6} + 5211463 T^{8} - 127367860 T^{10} + 2784558349 T^{12} - 53708825350 T^{14} + 943711137080 T^{16} - 53708825350 p^{2} T^{18} + 2784558349 p^{4} T^{20} - 127367860 p^{6} T^{22} + 5211463 p^{8} T^{24} - 191000 p^{10} T^{26} + 5647 p^{12} T^{28} - 110 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 10 T + 102 T^{2} + 670 T^{3} + 4268 T^{4} + 22710 T^{5} + 117514 T^{6} + 560450 T^{7} + 2521030 T^{8} + 560450 p T^{9} + 117514 p^{2} T^{10} + 22710 p^{3} T^{11} + 4268 p^{4} T^{12} + 670 p^{5} T^{13} + 102 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 120 T^{2} + 8272 T^{4} - 405160 T^{6} + 15482268 T^{8} - 487076920 T^{10} + 13245805744 T^{12} - 326418457000 T^{14} + 7636031189830 T^{16} - 326418457000 p^{2} T^{18} + 13245805744 p^{4} T^{20} - 487076920 p^{6} T^{22} + 15482268 p^{8} T^{24} - 405160 p^{10} T^{26} + 8272 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 10 T + 197 T^{2} + 1510 T^{3} + 17343 T^{4} + 107610 T^{5} + 913319 T^{6} + 4694150 T^{7} + 32002680 T^{8} + 4694150 p T^{9} + 913319 p^{2} T^{10} + 107610 p^{3} T^{11} + 17343 p^{4} T^{12} + 1510 p^{5} T^{13} + 197 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 16 T + 245 T^{2} - 2700 T^{3} + 25605 T^{4} - 210758 T^{5} + 1537653 T^{6} - 9946750 T^{7} + 58857900 T^{8} - 9946750 p T^{9} + 1537653 p^{2} T^{10} - 210758 p^{3} T^{11} + 25605 p^{4} T^{12} - 2700 p^{5} T^{13} + 245 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 - 450 T^{2} + 96247 T^{4} - 13039740 T^{6} + 1259916223 T^{8} - 2508225800 p T^{10} + 5445063403189 T^{12} - 5178960850 p^{3} T^{14} + 10575876674386280 T^{16} - 5178960850 p^{5} T^{18} + 5445063403189 p^{4} T^{20} - 2508225800 p^{7} T^{22} + 1259916223 p^{8} T^{24} - 13039740 p^{10} T^{26} + 96247 p^{12} T^{28} - 450 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 6 T + 145 T^{2} - 850 T^{3} + 10255 T^{4} - 69718 T^{5} + 526063 T^{6} - 4134450 T^{7} + 23274800 T^{8} - 4134450 p T^{9} + 526063 p^{2} T^{10} - 69718 p^{3} T^{11} + 10255 p^{4} T^{12} - 850 p^{5} T^{13} + 145 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 147613243096600 p^{2} T^{18} + 2846509586204 p^{4} T^{20} - 47508203160 p^{6} T^{22} + 666842308 p^{8} T^{24} - 7536360 p^{10} T^{26} + 63972 p^{12} T^{28} - 360 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( 1 - 120 T^{2} + 17572 T^{4} - 1601880 T^{6} + 137356228 T^{8} - 9599418920 T^{10} + 613416365084 T^{12} - 33575205819400 T^{14} + 1700876128099830 T^{16} - 33575205819400 p^{2} T^{18} + 613416365084 p^{4} T^{20} - 9599418920 p^{6} T^{22} + 137356228 p^{8} T^{24} - 1601880 p^{10} T^{26} + 17572 p^{12} T^{28} - 120 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 560 T^{2} + 144922 T^{4} - 22765500 T^{6} + 2393957203 T^{8} - 175164644160 T^{10} + 9059081039884 T^{12} - 352699017663600 T^{14} + 14677013509359205 T^{16} - 352699017663600 p^{2} T^{18} + 9059081039884 p^{4} T^{20} - 175164644160 p^{6} T^{22} + 2393957203 p^{8} T^{24} - 22765500 p^{10} T^{26} + 144922 p^{12} T^{28} - 560 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 247 T^{2} - 500 T^{3} + 28643 T^{4} - 123000 T^{5} + 2182549 T^{6} - 13248500 T^{7} + 135310720 T^{8} - 13248500 p T^{9} + 2182549 p^{2} T^{10} - 123000 p^{3} T^{11} + 28643 p^{4} T^{12} - 500 p^{5} T^{13} + 247 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 16 T + 335 T^{2} - 3580 T^{3} + 40995 T^{4} - 303668 T^{5} + 2560793 T^{6} - 14411280 T^{7} + 131714360 T^{8} - 14411280 p T^{9} + 2560793 p^{2} T^{10} - 303668 p^{3} T^{11} + 40995 p^{4} T^{12} - 3580 p^{5} T^{13} + 335 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 - 440 T^{2} + 112372 T^{4} - 20390680 T^{6} + 2900260388 T^{8} - 337708700840 T^{10} + 33102219933324 T^{12} - 2771008225944200 T^{14} + 199830173598131830 T^{16} - 2771008225944200 p^{2} T^{18} + 33102219933324 p^{4} T^{20} - 337708700840 p^{6} T^{22} + 2900260388 p^{8} T^{24} - 20390680 p^{10} T^{26} + 112372 p^{12} T^{28} - 440 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 6 T + 390 T^{2} - 2050 T^{3} + 74900 T^{4} - 339378 T^{5} + 9085578 T^{6} - 35263990 T^{7} + 765200950 T^{8} - 35263990 p T^{9} + 9085578 p^{2} T^{10} - 339378 p^{3} T^{11} + 74900 p^{4} T^{12} - 2050 p^{5} T^{13} + 390 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 - 590 T^{2} + 178647 T^{4} - 36798660 T^{6} + 5771348543 T^{8} - 731373593640 T^{10} + 77567862900069 T^{12} - 7031544086002150 T^{14} + 551057873643279880 T^{16} - 7031544086002150 p^{2} T^{18} + 77567862900069 p^{4} T^{20} - 731373593640 p^{6} T^{22} + 5771348543 p^{8} T^{24} - 36798660 p^{10} T^{26} + 178647 p^{12} T^{28} - 590 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 + 10 T + 227 T^{2} + 1720 T^{3} + 32163 T^{4} + 256360 T^{5} + 3728769 T^{6} + 26100650 T^{7} + 323096880 T^{8} + 26100650 p T^{9} + 3728769 p^{2} T^{10} + 256360 p^{3} T^{11} + 32163 p^{4} T^{12} + 1720 p^{5} T^{13} + 227 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 - 490 T^{2} + 107247 T^{4} - 14373340 T^{6} + 1475841863 T^{8} - 146194063040 T^{10} + 14859792976549 T^{12} - 1411441844814250 T^{14} + 121859257800782680 T^{16} - 1411441844814250 p^{2} T^{18} + 14859792976549 p^{4} T^{20} - 146194063040 p^{6} T^{22} + 1475841863 p^{8} T^{24} - 14373340 p^{10} T^{26} + 107247 p^{12} T^{28} - 490 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 20 T + 537 T^{2} + 8210 T^{3} + 124663 T^{4} + 1568070 T^{5} + 17732339 T^{6} + 192143300 T^{7} + 1812907320 T^{8} + 192143300 p T^{9} + 17732339 p^{2} T^{10} + 1568070 p^{3} T^{11} + 124663 p^{4} T^{12} + 8210 p^{5} T^{13} + 537 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 660 T^{2} + 206842 T^{4} - 40899980 T^{6} + 5901330683 T^{8} - 716216832660 T^{10} + 83710235703084 T^{12} - 9634323674140400 T^{14} + 1005189451203013405 T^{16} - 9634323674140400 p^{2} T^{18} + 83710235703084 p^{4} T^{20} - 716216832660 p^{6} T^{22} + 5901330683 p^{8} T^{24} - 40899980 p^{10} T^{26} + 206842 p^{12} T^{28} - 660 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.02839903985366106422149008552, −1.97970676420663850405863268477, −1.89919407455727009563796822500, −1.82035880800641605007804230492, −1.71552123569408679336538730336, −1.67417065179895896293403150135, −1.60502955011433113408841793821, −1.56932192834651735687046237353, −1.44926520594701462106325654162, −1.38349716839177237346537013178, −1.26412028825572977392149177959, −1.19074058289614382677974888959, −1.18131703825804560583152672820, −1.01841461152955186108256530988, −0.868630038845163019664082211158, −0.866803226892878690149120954925, −0.859138679468406913755472085560, −0.71365725160788574132299184096, −0.70115145067728828216308499376, −0.58442273206283195867640901518, −0.54944710975251981141843118293, −0.32008783593779358946772505901, −0.26683598499237643748916925623, −0.097204747282683375743304357255, −0.04027438930330953810016800009, 0.04027438930330953810016800009, 0.097204747282683375743304357255, 0.26683598499237643748916925623, 0.32008783593779358946772505901, 0.54944710975251981141843118293, 0.58442273206283195867640901518, 0.70115145067728828216308499376, 0.71365725160788574132299184096, 0.859138679468406913755472085560, 0.866803226892878690149120954925, 0.868630038845163019664082211158, 1.01841461152955186108256530988, 1.18131703825804560583152672820, 1.19074058289614382677974888959, 1.26412028825572977392149177959, 1.38349716839177237346537013178, 1.44926520594701462106325654162, 1.56932192834651735687046237353, 1.60502955011433113408841793821, 1.67417065179895896293403150135, 1.71552123569408679336538730336, 1.82035880800641605007804230492, 1.89919407455727009563796822500, 1.97970676420663850405863268477, 2.02839903985366106422149008552
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.