Dirichlet series
| L(s) = 1 | − 2·2-s − 9·4-s + 20·8-s + 14·9-s − 16·13-s + 37·16-s + 4·17-s − 28·18-s + 20·19-s + 42·25-s + 32·26-s − 98·32-s − 8·34-s − 126·36-s − 40·38-s + 8·43-s + 14·47-s + 35·49-s − 84·50-s + 144·52-s + 26·53-s − 30·59-s − 85·64-s + 10·67-s − 36·68-s + 280·72-s − 180·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 9/2·4-s + 7.07·8-s + 14/3·9-s − 4.43·13-s + 37/4·16-s + 0.970·17-s − 6.59·18-s + 4.58·19-s + 42/5·25-s + 6.27·26-s − 17.3·32-s − 1.37·34-s − 21·36-s − 6.48·38-s + 1.21·43-s + 2.04·47-s + 5·49-s − 11.8·50-s + 19.9·52-s + 3.57·53-s − 3.90·59-s − 10.6·64-s + 1.22·67-s − 4.36·68-s + 32.9·72-s − 20.6·76-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(11^{16} \cdot 17^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(610.800\) |
| Root analytic conductor: | \(1.22196\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 11^{16} \cdot 17^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6021242257\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6021242257\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + T^{2} )^{8} \) |
| 17 | \( 1 - 4 T + 44 T^{2} - 148 T^{3} + 724 T^{4} - 2124 T^{5} + 4980 T^{6} - 19548 T^{7} + 25878 T^{8} - 19548 p T^{9} + 4980 p^{2} T^{10} - 2124 p^{3} T^{11} + 724 p^{4} T^{12} - 148 p^{5} T^{13} + 44 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 2 | \( ( 1 + T + 3 p T^{2} + 3 p T^{3} + 5 p^{2} T^{4} + 11 p T^{5} + 13 p^{2} T^{6} + 31 p T^{7} + 29 p^{2} T^{8} + 31 p^{2} T^{9} + 13 p^{4} T^{10} + 11 p^{4} T^{11} + 5 p^{6} T^{12} + 3 p^{6} T^{13} + 3 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 3 | \( 1 - 14 T^{2} + 115 T^{4} - 718 T^{6} + 3721 T^{8} - 16628 T^{10} + 65582 T^{12} - 77096 p T^{14} + 731818 T^{16} - 77096 p^{3} T^{18} + 65582 p^{4} T^{20} - 16628 p^{6} T^{22} + 3721 p^{8} T^{24} - 718 p^{10} T^{26} + 115 p^{12} T^{28} - 14 p^{14} T^{30} + p^{16} T^{32} \) | |
| 5 | \( 1 - 42 T^{2} + 891 T^{4} - 2542 p T^{6} + 137233 T^{8} - 1195544 T^{10} + 8719246 T^{12} - 10865884 p T^{14} + 292052674 T^{16} - 10865884 p^{3} T^{18} + 8719246 p^{4} T^{20} - 1195544 p^{6} T^{22} + 137233 p^{8} T^{24} - 2542 p^{11} T^{26} + 891 p^{12} T^{28} - 42 p^{14} T^{30} + p^{16} T^{32} \) | |
| 7 | \( 1 - 5 p T^{2} + 702 T^{4} - 9397 T^{6} + 91904 T^{8} - 647439 T^{10} + 3155554 T^{12} - 8792953 T^{14} + 20205630 T^{16} - 8792953 p^{2} T^{18} + 3155554 p^{4} T^{20} - 647439 p^{6} T^{22} + 91904 p^{8} T^{24} - 9397 p^{10} T^{26} + 702 p^{12} T^{28} - 5 p^{15} T^{30} + p^{16} T^{32} \) | |
| 13 | \( ( 1 + 8 T + 76 T^{2} + 440 T^{3} + 2560 T^{4} + 11678 T^{5} + 53270 T^{6} + 203314 T^{7} + 796698 T^{8} + 203314 p T^{9} + 53270 p^{2} T^{10} + 11678 p^{3} T^{11} + 2560 p^{4} T^{12} + 440 p^{5} T^{13} + 76 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 10 T + 6 p T^{2} - 704 T^{3} + 4972 T^{4} - 23474 T^{5} + 132880 T^{6} - 534612 T^{7} + 2731922 T^{8} - 534612 p T^{9} + 132880 p^{2} T^{10} - 23474 p^{3} T^{11} + 4972 p^{4} T^{12} - 704 p^{5} T^{13} + 6 p^{7} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 94 T^{2} + 6403 T^{4} - 318834 T^{6} + 13418985 T^{8} - 474658432 T^{10} + 14810371878 T^{12} - 404459224060 T^{14} + 9896517697314 T^{16} - 404459224060 p^{2} T^{18} + 14810371878 p^{4} T^{20} - 474658432 p^{6} T^{22} + 13418985 p^{8} T^{24} - 318834 p^{10} T^{26} + 6403 p^{12} T^{28} - 94 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( 1 - 275 T^{2} + 37330 T^{4} - 3317361 T^{6} + 216686716 T^{8} - 11111109655 T^{10} + 467842123902 T^{12} - 16720673667089 T^{14} + 518513547129086 T^{16} - 16720673667089 p^{2} T^{18} + 467842123902 p^{4} T^{20} - 11111109655 p^{6} T^{22} + 216686716 p^{8} T^{24} - 3317361 p^{10} T^{26} + 37330 p^{12} T^{28} - 275 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( 1 - 350 T^{2} + 59971 T^{4} - 6679930 T^{6} + 541861169 T^{8} - 33986308768 T^{10} + 1707438194222 T^{12} - 2263834049676 p T^{14} + 2387110276464738 T^{16} - 2263834049676 p^{3} T^{18} + 1707438194222 p^{4} T^{20} - 33986308768 p^{6} T^{22} + 541861169 p^{8} T^{24} - 6679930 p^{10} T^{26} + 59971 p^{12} T^{28} - 350 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( 1 - 458 T^{2} + 101875 T^{4} - 14597206 T^{6} + 40739213 p T^{8} - 118918487416 T^{10} + 7413468575246 T^{12} - 372504684002892 T^{14} + 15246011562036498 T^{16} - 372504684002892 p^{2} T^{18} + 7413468575246 p^{4} T^{20} - 118918487416 p^{6} T^{22} + 40739213 p^{9} T^{24} - 14597206 p^{10} T^{26} + 101875 p^{12} T^{28} - 458 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( 1 - 355 T^{2} + 60694 T^{4} - 6715569 T^{6} + 549601672 T^{8} - 36217873579 T^{10} + 2033466178122 T^{12} - 100172735856865 T^{14} + 4365696478573102 T^{16} - 100172735856865 p^{2} T^{18} + 2033466178122 p^{4} T^{20} - 36217873579 p^{6} T^{22} + 549601672 p^{8} T^{24} - 6715569 p^{10} T^{26} + 60694 p^{12} T^{28} - 355 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 4 T + 150 T^{2} - 436 T^{3} + 13604 T^{4} - 40460 T^{5} + 904866 T^{6} - 2287132 T^{7} + 43536710 T^{8} - 2287132 p T^{9} + 904866 p^{2} T^{10} - 40460 p^{3} T^{11} + 13604 p^{4} T^{12} - 436 p^{5} T^{13} + 150 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 - 7 T + 236 T^{2} - 1183 T^{3} + 24500 T^{4} - 89215 T^{5} + 1597908 T^{6} - 4522747 T^{7} + 81239638 T^{8} - 4522747 p T^{9} + 1597908 p^{2} T^{10} - 89215 p^{3} T^{11} + 24500 p^{4} T^{12} - 1183 p^{5} T^{13} + 236 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 13 T + 246 T^{2} - 2395 T^{3} + 24132 T^{4} - 164905 T^{5} + 1214178 T^{6} - 6144219 T^{7} + 50595742 T^{8} - 6144219 p T^{9} + 1214178 p^{2} T^{10} - 164905 p^{3} T^{11} + 24132 p^{4} T^{12} - 2395 p^{5} T^{13} + 246 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 + 15 T + 385 T^{2} + 3962 T^{3} + 60637 T^{4} + 481965 T^{5} + 5707058 T^{6} + 37723403 T^{7} + 384389162 T^{8} + 37723403 p T^{9} + 5707058 p^{2} T^{10} + 481965 p^{3} T^{11} + 60637 p^{4} T^{12} + 3962 p^{5} T^{13} + 385 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 452 T^{2} + 104908 T^{4} - 16703948 T^{6} + 2047916500 T^{8} - 205530629092 T^{10} + 17512781894132 T^{12} - 1295208961313868 T^{14} + 84142265912478102 T^{16} - 1295208961313868 p^{2} T^{18} + 17512781894132 p^{4} T^{20} - 205530629092 p^{6} T^{22} + 2047916500 p^{8} T^{24} - 16703948 p^{10} T^{26} + 104908 p^{12} T^{28} - 452 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 5 T + 285 T^{2} - 958 T^{3} + 44141 T^{4} - 127107 T^{5} + 4721614 T^{6} - 11224453 T^{7} + 363638154 T^{8} - 11224453 p T^{9} + 4721614 p^{2} T^{10} - 127107 p^{3} T^{11} + 44141 p^{4} T^{12} - 958 p^{5} T^{13} + 285 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 - 366 T^{2} + 60003 T^{4} - 5848518 T^{6} + 381135329 T^{8} - 16947384340 T^{10} + 282156043254 T^{12} + 41632718359680 T^{14} - 4897207768047590 T^{16} + 41632718359680 p^{2} T^{18} + 282156043254 p^{4} T^{20} - 16947384340 p^{6} T^{22} + 381135329 p^{8} T^{24} - 5848518 p^{10} T^{26} + 60003 p^{12} T^{28} - 366 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 751 T^{2} + 273826 T^{4} - 64183885 T^{6} + 10855577328 T^{8} - 1417028551475 T^{10} + 149977974027678 T^{12} - 13405042246588877 T^{14} + 1042828292770880022 T^{16} - 13405042246588877 p^{2} T^{18} + 149977974027678 p^{4} T^{20} - 1417028551475 p^{6} T^{22} + 10855577328 p^{8} T^{24} - 64183885 p^{10} T^{26} + 273826 p^{12} T^{28} - 751 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 200 T^{2} + 29688 T^{4} - 3405000 T^{6} + 4095404 p T^{8} - 25863326292 T^{10} + 1930576896740 T^{12} - 133959516529548 T^{14} + 10324895744277054 T^{16} - 133959516529548 p^{2} T^{18} + 1930576896740 p^{4} T^{20} - 25863326292 p^{6} T^{22} + 4095404 p^{9} T^{24} - 3405000 p^{10} T^{26} + 29688 p^{12} T^{28} - 200 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 4 T + 482 T^{2} + 1720 T^{3} + 112132 T^{4} + 356008 T^{5} + 16349846 T^{6} + 44929204 T^{7} + 1625104774 T^{8} + 44929204 p T^{9} + 16349846 p^{2} T^{10} + 356008 p^{3} T^{11} + 112132 p^{4} T^{12} + 1720 p^{5} T^{13} + 482 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 5 T + 443 T^{2} - 1672 T^{3} + 96627 T^{4} - 309145 T^{5} + 13959218 T^{6} - 39725405 T^{7} + 1453384656 T^{8} - 39725405 p T^{9} + 13959218 p^{2} T^{10} - 309145 p^{3} T^{11} + 96627 p^{4} T^{12} - 1672 p^{5} T^{13} + 443 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 - 846 T^{2} + 365803 T^{4} - 107477582 T^{6} + 23934470449 T^{8} - 4267264330604 T^{10} + 628345768004790 T^{12} - 77804305130138232 T^{14} + 8178778141651688666 T^{16} - 77804305130138232 p^{2} T^{18} + 628345768004790 p^{4} T^{20} - 4267264330604 p^{6} T^{22} + 23934470449 p^{8} T^{24} - 107477582 p^{10} T^{26} + 365803 p^{12} T^{28} - 846 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−3.84503081998584983701383291358, −3.82079442698140655105277042188, −3.44793969543204994629708454584, −3.40192191268803877853358618436, −3.35137442880925681546772360411, −3.27060073962150401101631053377, −3.09918058223883989266465641839, −3.05541027509517324807163679948, −3.02847815534086221393718072960, −2.83776191786203429437057917443, −2.72109713567822012604295537971, −2.58149509869780598166509732885, −2.48274393819265258792421661930, −2.38696967145402826179708175337, −2.21090102278387852087467831238, −2.15809550655155116269314537728, −1.97423549472464444989826093792, −1.51722658987686601780797286219, −1.44956151069080635527688480691, −1.22425880378127128720661022589, −1.02674550437905237275154851910, −1.02007923893496294045139688662, −1.01278856522464693889090598455, −0.69066020734626662715006410941, −0.60268238580361756095548510585, 0.60268238580361756095548510585, 0.69066020734626662715006410941, 1.01278856522464693889090598455, 1.02007923893496294045139688662, 1.02674550437905237275154851910, 1.22425880378127128720661022589, 1.44956151069080635527688480691, 1.51722658987686601780797286219, 1.97423549472464444989826093792, 2.15809550655155116269314537728, 2.21090102278387852087467831238, 2.38696967145402826179708175337, 2.48274393819265258792421661930, 2.58149509869780598166509732885, 2.72109713567822012604295537971, 2.83776191786203429437057917443, 3.02847815534086221393718072960, 3.05541027509517324807163679948, 3.09918058223883989266465641839, 3.27060073962150401101631053377, 3.35137442880925681546772360411, 3.40192191268803877853358618436, 3.44793969543204994629708454584, 3.82079442698140655105277042188, 3.84503081998584983701383291358
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.