Dirichlet series
| L(s) = 1 | + 3·3-s + 6·7-s − 18·19-s + 18·21-s − 9·27-s − 18·31-s − 6·37-s + 4·43-s + 9·49-s − 54·57-s + 36·61-s + 30·67-s + 54·73-s − 5·81-s − 54·93-s − 72·103-s − 42·109-s − 18·111-s − 14·121-s + 127-s + 12·129-s + 131-s − 108·133-s + 137-s + 139-s + 27·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.26·7-s − 4.12·19-s + 3.92·21-s − 1.73·27-s − 3.23·31-s − 0.986·37-s + 0.609·43-s + 9/7·49-s − 7.15·57-s + 4.60·61-s + 3.66·67-s + 6.32·73-s − 5/9·81-s − 5.59·93-s − 7.09·103-s − 4.02·109-s − 1.70·111-s − 1.27·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s − 9.36·133-s + 0.0854·137-s + 0.0848·139-s + 2.22·147-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.90789\times 10^{19}\) |
| Root analytic conductor: | \(4.09494\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(6.802552266\) |
| \(L(\frac12)\) | \(\approx\) | \(6.802552266\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T + p^{2} T^{2} - 2 p^{2} T^{3} + 32 T^{4} - 4 p^{2} T^{5} + 17 p T^{7} - 167 T^{8} + 17 p^{2} T^{9} - 4 p^{5} T^{11} + 32 p^{4} T^{12} - 2 p^{7} T^{13} + p^{8} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 - 3 T + 9 T^{2} - 39 T^{3} + 95 T^{4} - 39 p T^{5} + 9 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| good | 11 | \( 1 + 14 T^{2} - 69 T^{4} - 2576 T^{6} - 25195 T^{8} - 226506 T^{10} - 1311833 T^{12} + 39166148 T^{14} + 900447201 T^{16} + 39166148 p^{2} T^{18} - 1311833 p^{4} T^{20} - 226506 p^{6} T^{22} - 25195 p^{8} T^{24} - 2576 p^{10} T^{26} - 69 p^{12} T^{28} + 14 p^{14} T^{30} + p^{16} T^{32} \) |
| 13 | \( ( 1 - 83 T^{2} + 3165 T^{4} - 5647 p T^{6} + 1146521 T^{8} - 5647 p^{3} T^{10} + 3165 p^{4} T^{12} - 83 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 14 T^{2} + 267 T^{4} + 10052 T^{6} - 224695 T^{8} + 3669918 T^{10} + 2136955 T^{12} - 1194075764 T^{14} + 23802887097 T^{16} - 1194075764 p^{2} T^{18} + 2136955 p^{4} T^{20} + 3669918 p^{6} T^{22} - 224695 p^{8} T^{24} + 10052 p^{10} T^{26} + 267 p^{12} T^{28} - 14 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 + 9 T + 64 T^{2} + 333 T^{3} + 1299 T^{4} + 3186 T^{5} - 217 p T^{6} - 75567 T^{7} - 435298 T^{8} - 75567 p T^{9} - 217 p^{3} T^{10} + 3186 p^{3} T^{11} + 1299 p^{4} T^{12} + 333 p^{5} T^{13} + 64 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 + 105 T^{2} + 5189 T^{4} + 177324 T^{6} + 5348524 T^{8} + 157563000 T^{10} + 4443717224 T^{12} + 116142079557 T^{14} + 2787310422721 T^{16} + 116142079557 p^{2} T^{18} + 4443717224 p^{4} T^{20} + 157563000 p^{6} T^{22} + 5348524 p^{8} T^{24} + 177324 p^{10} T^{26} + 5189 p^{12} T^{28} + 105 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 90 T^{2} + 101 p T^{4} - 16011 T^{6} - 944082 T^{8} - 16011 p^{2} T^{10} + 101 p^{5} T^{12} - 90 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 9 T + 97 T^{2} + 630 T^{3} + 3786 T^{4} + 11934 T^{5} + 10454 T^{6} - 251757 T^{7} - 2144539 T^{8} - 251757 p T^{9} + 10454 p^{2} T^{10} + 11934 p^{3} T^{11} + 3786 p^{4} T^{12} + 630 p^{5} T^{13} + 97 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 3 T + 93 T^{3} - 1331 T^{4} - 10584 T^{5} + 15795 T^{6} + 161037 T^{7} + 1169532 T^{8} + 161037 p T^{9} + 15795 p^{2} T^{10} - 10584 p^{3} T^{11} - 1331 p^{4} T^{12} + 93 p^{5} T^{13} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 175 T^{2} + 13716 T^{4} + 685412 T^{6} + 28562204 T^{8} + 685412 p^{2} T^{10} + 13716 p^{4} T^{12} + 175 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( ( 1 - T + 150 T^{2} - 110 T^{3} + 9290 T^{4} - 110 p T^{5} + 150 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 47 | \( 1 - 34 T^{2} - 1025 T^{4} + 3440 p T^{6} - 9103783 T^{8} + 152237258 T^{10} + 4847483963 T^{12} - 753776826296 T^{14} + 44233333645801 T^{16} - 753776826296 p^{2} T^{18} + 4847483963 p^{4} T^{20} + 152237258 p^{6} T^{22} - 9103783 p^{8} T^{24} + 3440 p^{11} T^{26} - 1025 p^{12} T^{28} - 34 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 + 262 T^{2} + 35395 T^{4} + 3231368 T^{6} + 221761433 T^{8} + 11879858878 T^{10} + 500403165131 T^{12} + 17219962264736 T^{14} + 683285105805889 T^{16} + 17219962264736 p^{2} T^{18} + 500403165131 p^{4} T^{20} + 11879858878 p^{6} T^{22} + 221761433 p^{8} T^{24} + 3231368 p^{10} T^{26} + 35395 p^{12} T^{28} + 262 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( 1 - 179 T^{2} + 16761 T^{4} - 465136 T^{6} - 46752400 T^{8} + 6296484144 T^{10} - 226286292944 T^{12} - 7423721157083 T^{14} + 1182150698554881 T^{16} - 7423721157083 p^{2} T^{18} - 226286292944 p^{4} T^{20} + 6296484144 p^{6} T^{22} - 46752400 p^{8} T^{24} - 465136 p^{10} T^{26} + 16761 p^{12} T^{28} - 179 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( ( 1 - 18 T + 325 T^{2} - 3906 T^{3} + 44196 T^{4} - 437580 T^{5} + 4261022 T^{6} - 36703131 T^{7} + 310887998 T^{8} - 36703131 p T^{9} + 4261022 p^{2} T^{10} - 437580 p^{3} T^{11} + 44196 p^{4} T^{12} - 3906 p^{5} T^{13} + 325 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 15 T - 19 T^{2} + 1578 T^{3} - 3827 T^{4} - 92169 T^{5} + 542666 T^{6} + 1629357 T^{7} - 31878578 T^{8} + 1629357 p T^{9} + 542666 p^{2} T^{10} - 92169 p^{3} T^{11} - 3827 p^{4} T^{12} + 1578 p^{5} T^{13} - 19 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 139 T^{2} + 17517 T^{4} - 1749386 T^{6} + 123496010 T^{8} - 1749386 p^{2} T^{10} + 17517 p^{4} T^{12} - 139 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 27 T + 513 T^{2} - 7290 T^{3} + 85003 T^{4} - 866997 T^{5} + 8113770 T^{6} - 71702037 T^{7} + 621942924 T^{8} - 71702037 p T^{9} + 8113770 p^{2} T^{10} - 866997 p^{3} T^{11} + 85003 p^{4} T^{12} - 7290 p^{5} T^{13} + 513 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 195 T^{2} - 582 T^{3} + 18208 T^{4} + 79734 T^{5} - 1345644 T^{6} - 3198963 T^{7} + 109759584 T^{8} - 3198963 p T^{9} - 1345644 p^{2} T^{10} + 79734 p^{3} T^{11} + 18208 p^{4} T^{12} - 582 p^{5} T^{13} - 195 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 406 T^{2} + 83877 T^{4} + 11459777 T^{6} + 1116323102 T^{8} + 11459777 p^{2} T^{10} + 83877 p^{4} T^{12} + 406 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 118 T^{2} + 3595 T^{4} + 1013020 T^{6} - 177688255 T^{8} + 15671126798 T^{10} - 78833677741 T^{12} - 122913712595300 T^{14} + 16662250146905929 T^{16} - 122913712595300 p^{2} T^{18} - 78833677741 p^{4} T^{20} + 15671126798 p^{6} T^{22} - 177688255 p^{8} T^{24} + 1013020 p^{10} T^{26} + 3595 p^{12} T^{28} - 118 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( ( 1 - 402 T^{2} + 92431 T^{4} - 14249709 T^{6} + 1608196287 T^{8} - 14249709 p^{2} T^{10} + 92431 p^{4} T^{12} - 402 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
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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
−2.25097299891011895541942925912, −2.22694765401982920210420471320, −2.13590031087996033628166289171, −2.06804466300406547384058116458, −1.98139497217744201106566022773, −1.93329794779777052691863968850, −1.88382587281802677246406961678, −1.76609121591930164421487173516, −1.73101509046596379099111390904, −1.64926459338405509452904119077, −1.62738967340307133392922856176, −1.61093622894084109361367360472, −1.49045001189378951719265864930, −1.20279904139983686856081306433, −1.14476895622670369217387091889, −1.05958391657571817420252573517, −1.00994753087139864947668159934, −0.913361695777543002527270735476, −0.844724425353209600631603986764, −0.74238630826277402857386002496, −0.61615990160134194830465873973, −0.34488428611032886543218435358, −0.31283718505233351980448939468, −0.31212192618537857961906157540, −0.081430459580887542773749383370, 0.081430459580887542773749383370, 0.31212192618537857961906157540, 0.31283718505233351980448939468, 0.34488428611032886543218435358, 0.61615990160134194830465873973, 0.74238630826277402857386002496, 0.844724425353209600631603986764, 0.913361695777543002527270735476, 1.00994753087139864947668159934, 1.05958391657571817420252573517, 1.14476895622670369217387091889, 1.20279904139983686856081306433, 1.49045001189378951719265864930, 1.61093622894084109361367360472, 1.62738967340307133392922856176, 1.64926459338405509452904119077, 1.73101509046596379099111390904, 1.76609121591930164421487173516, 1.88382587281802677246406961678, 1.93329794779777052691863968850, 1.98139497217744201106566022773, 2.06804466300406547384058116458, 2.13590031087996033628166289171, 2.22694765401982920210420471320, 2.25097299891011895541942925912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.