Dirichlet series
| L(s) = 1 | − 2·3-s + 4·5-s − 5·9-s − 8·15-s + 2·16-s + 46·23-s + 22·25-s + 2·27-s + 14·31-s − 12·37-s − 20·45-s + 16·47-s − 4·48-s + 7·49-s − 96·53-s + 48·59-s − 18·64-s − 22·67-s − 92·69-s + 68·71-s − 44·75-s + 8·80-s + 29·81-s − 16·89-s − 28·93-s − 4·97-s + 24·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 5/3·9-s − 2.06·15-s + 1/2·16-s + 9.59·23-s + 22/5·25-s + 0.384·27-s + 2.51·31-s − 1.97·37-s − 2.98·45-s + 2.33·47-s − 0.577·48-s + 49-s − 13.1·53-s + 6.24·59-s − 9/4·64-s − 2.68·67-s − 11.0·69-s + 8.07·71-s − 5.08·75-s + 0.894·80-s + 29/9·81-s − 1.69·89-s − 2.90·93-s − 0.406·97-s + 2.27·111-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 11^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{32} \cdot 11^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.06875\times 10^{15}\) |
| Root analytic conductor: | \(2.94884\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 3^{32} \cdot 11^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(10.01337737\) |
| \(L(\frac12)\) | \(\approx\) | \(10.01337737\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T + 4 T^{2} + p^{2} T^{3} + 5 p T^{4} + p^{3} T^{5} + 4 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 \) | |
| good | 2 | \( ( 1 - 3 T^{2} + p T^{4} + 3 p^{2} T^{6} - 39 T^{8} + 3 p^{4} T^{10} + p^{5} T^{12} - 3 p^{6} T^{14} + p^{8} T^{16} )( 1 + 3 T^{2} + 5 T^{4} + 15 T^{6} + 45 T^{8} + 15 p^{2} T^{10} + 5 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 5 | \( ( 1 - 2 T - p T^{2} - 6 T^{3} + 38 T^{4} + 8 p T^{5} + 147 T^{6} - 346 T^{7} - 869 T^{8} - 346 p T^{9} + 147 p^{2} T^{10} + 8 p^{4} T^{11} + 38 p^{4} T^{12} - 6 p^{5} T^{13} - p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 7 | \( 1 - p T^{2} - 58 T^{4} + 19 p^{2} T^{6} - 466 T^{8} - 950 p^{2} T^{10} + 285549 T^{12} + 159564 p T^{14} - 19877423 T^{16} + 159564 p^{3} T^{18} + 285549 p^{4} T^{20} - 950 p^{8} T^{22} - 466 p^{8} T^{24} + 19 p^{12} T^{26} - 58 p^{12} T^{28} - p^{15} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 68 T^{2} + 2502 T^{4} - 60646 T^{6} + 1046819 T^{8} - 12593835 T^{10} + 91300879 T^{12} - 46501493 T^{14} - 5919985413 T^{16} - 46501493 p^{2} T^{18} + 91300879 p^{4} T^{20} - 12593835 p^{6} T^{22} + 1046819 p^{8} T^{24} - 60646 p^{10} T^{26} + 2502 p^{12} T^{28} - 68 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( ( 1 + 54 T^{2} + 1331 T^{4} + 20148 T^{6} + 291597 T^{8} + 20148 p^{2} T^{10} + 1331 p^{4} T^{12} + 54 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 4 p T^{2} + 2180 T^{4} + 26196 T^{6} + 205718 T^{8} + 26196 p^{2} T^{10} + 2180 p^{4} T^{12} + 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) | |
| 23 | \( ( 1 - p T + 247 T^{2} - 1992 T^{3} + 15128 T^{4} - 100445 T^{5} + 573072 T^{6} - 3133456 T^{7} + 16068667 T^{8} - 3133456 p T^{9} + 573072 p^{2} T^{10} - 100445 p^{3} T^{11} + 15128 p^{4} T^{12} - 1992 p^{5} T^{13} + 247 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - 66 T^{2} + 2053 T^{4} - 1710 p T^{6} + 823564 T^{8} - 15682284 T^{10} + 502358035 T^{12} - 4781857164 T^{14} - 62685690833 T^{16} - 4781857164 p^{2} T^{18} + 502358035 p^{4} T^{20} - 15682284 p^{6} T^{22} + 823564 p^{8} T^{24} - 1710 p^{11} T^{26} + 2053 p^{12} T^{28} - 66 p^{14} T^{30} + p^{16} T^{32} \) | |
| 31 | \( ( 1 - 7 T - 79 T^{2} + 346 T^{3} + 194 p T^{4} - 14417 T^{5} - 274044 T^{6} + 118380 T^{7} + 10572631 T^{8} + 118380 p T^{9} - 274044 p^{2} T^{10} - 14417 p^{3} T^{11} + 194 p^{5} T^{12} + 346 p^{5} T^{13} - 79 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 3 T + 77 T^{2} - 36 T^{3} + 2595 T^{4} - 36 p T^{5} + 77 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 41 | \( 1 - 243 T^{2} + 31798 T^{4} - 2835873 T^{6} + 191144758 T^{8} - 10321101702 T^{10} + 472062936229 T^{12} - 19565071633296 T^{14} + 795062132797705 T^{16} - 19565071633296 p^{2} T^{18} + 472062936229 p^{4} T^{20} - 10321101702 p^{6} T^{22} + 191144758 p^{8} T^{24} - 2835873 p^{10} T^{26} + 31798 p^{12} T^{28} - 243 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 187 T^{2} + 15470 T^{4} - 910217 T^{6} + 55847702 T^{8} - 3571755902 T^{10} + 195432590901 T^{12} - 8765578837884 T^{14} + 367545452133793 T^{16} - 8765578837884 p^{2} T^{18} + 195432590901 p^{4} T^{20} - 3571755902 p^{6} T^{22} + 55847702 p^{8} T^{24} - 910217 p^{10} T^{26} + 15470 p^{12} T^{28} - 187 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 8 T - 86 T^{2} + 258 T^{3} + 7589 T^{4} + 1651 T^{5} - 441813 T^{6} + 178925 T^{7} + 16157677 T^{8} + 178925 p T^{9} - 441813 p^{2} T^{10} + 1651 p^{3} T^{11} + 7589 p^{4} T^{12} + 258 p^{5} T^{13} - 86 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 24 T + 356 T^{2} + 3735 T^{3} + 31065 T^{4} + 3735 p T^{5} + 356 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 59 | \( ( 1 - 24 T + 202 T^{2} - 966 T^{3} + 8989 T^{4} - 65349 T^{5} - 30125 T^{6} + 1332273 T^{7} - 46295 T^{8} + 1332273 p T^{9} - 30125 p^{2} T^{10} - 65349 p^{3} T^{11} + 8989 p^{4} T^{12} - 966 p^{5} T^{13} + 202 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 193 T^{2} + 16973 T^{4} - 831926 T^{6} + 14728058 T^{8} + 1257423403 T^{10} - 179400783930 T^{12} + 15766156767906 T^{14} - 1069559799784055 T^{16} + 15766156767906 p^{2} T^{18} - 179400783930 p^{4} T^{20} + 1257423403 p^{6} T^{22} + 14728058 p^{8} T^{24} - 831926 p^{10} T^{26} + 16973 p^{12} T^{28} - 193 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 11 T - 166 T^{2} - 1019 T^{3} + 30260 T^{4} + 98050 T^{5} - 2965893 T^{6} - 1973730 T^{7} + 239974117 T^{8} - 1973730 p T^{9} - 2965893 p^{2} T^{10} + 98050 p^{3} T^{11} + 30260 p^{4} T^{12} - 1019 p^{5} T^{13} - 166 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 17 T + 306 T^{2} - 2949 T^{3} + 31003 T^{4} - 2949 p T^{5} + 306 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( ( 1 + 457 T^{2} + 94028 T^{4} + 11761923 T^{6} + 1013855399 T^{8} + 11761923 p^{2} T^{10} + 94028 p^{4} T^{12} + 457 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( 1 - 518 T^{2} + 146001 T^{4} - 28480138 T^{6} + 4266348260 T^{8} - 520709117172 T^{10} + 54072666209671 T^{12} - 4948361973049124 T^{14} + 409224108771995319 T^{16} - 4948361973049124 p^{2} T^{18} + 54072666209671 p^{4} T^{20} - 520709117172 p^{6} T^{22} + 4266348260 p^{8} T^{24} - 28480138 p^{10} T^{26} + 146001 p^{12} T^{28} - 518 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( 1 - 321 T^{2} + 40717 T^{4} - 3828030 T^{6} + 506523742 T^{8} - 58337531937 T^{10} + 4772524815106 T^{12} - 427440725025378 T^{14} + 40285302216725329 T^{16} - 427440725025378 p^{2} T^{18} + 4772524815106 p^{4} T^{20} - 58337531937 p^{6} T^{22} + 506523742 p^{8} T^{24} - 3828030 p^{10} T^{26} + 40717 p^{12} T^{28} - 321 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 4 T + 141 T^{2} + 228 T^{3} + 14449 T^{4} + 228 p T^{5} + 141 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 97 | \( ( 1 + 2 T - 87 T^{2} + 1630 T^{3} + 3272 T^{4} - 161940 T^{5} + 1789483 T^{6} + 11644412 T^{7} - 168296049 T^{8} + 11644412 p T^{9} + 1789483 p^{2} T^{10} - 161940 p^{3} T^{11} + 3272 p^{4} T^{12} + 1630 p^{5} T^{13} - 87 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
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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
−2.51676448969263378899701272148, −2.47543935446113911294282045388, −2.43316283037037908980775723195, −2.35148513110430456866197294939, −2.29700743987777249020086293414, −2.25074642463027769646331143400, −2.20714458657491172084986621443, −2.01645736047230991775960782587, −1.91472893127888759148040635680, −1.69986634731633082299433121436, −1.63019876807497118867672348480, −1.51987774381627148329187886991, −1.43112134628616248566916059900, −1.38468802565201690438067881098, −1.29209415006475868811053957491, −1.25010809255322386054466661137, −1.24775694811176664472388189958, −1.21596672538402019128302614850, −0.891722309710676060534831480070, −0.842059646936194978795432258553, −0.72437537085161684738259282683, −0.61033049214406870271705753111, −0.48660651616835605669945217909, −0.37848825955452992173462183502, −0.13598443046436090190915980434, 0.13598443046436090190915980434, 0.37848825955452992173462183502, 0.48660651616835605669945217909, 0.61033049214406870271705753111, 0.72437537085161684738259282683, 0.842059646936194978795432258553, 0.891722309710676060534831480070, 1.21596672538402019128302614850, 1.24775694811176664472388189958, 1.25010809255322386054466661137, 1.29209415006475868811053957491, 1.38468802565201690438067881098, 1.43112134628616248566916059900, 1.51987774381627148329187886991, 1.63019876807497118867672348480, 1.69986634731633082299433121436, 1.91472893127888759148040635680, 2.01645736047230991775960782587, 2.20714458657491172084986621443, 2.25074642463027769646331143400, 2.29700743987777249020086293414, 2.35148513110430456866197294939, 2.43316283037037908980775723195, 2.47543935446113911294282045388, 2.51676448969263378899701272148
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.