Dirichlet series
| L(s) = 1 | − 4·11-s + 2·16-s + 4·19-s − 48·49-s − 40·59-s − 24·61-s + 104·71-s + 12·79-s + 2·81-s − 60·89-s + 12·101-s + 132·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 8·176-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1/2·16-s + 0.917·19-s − 6.85·49-s − 5.20·59-s − 3.07·61-s + 12.3·71-s + 1.35·79-s + 2/9·81-s − 6.35·89-s + 1.19·101-s + 12.6·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 0.603·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.96296\times 10^{14}\) |
| Root analytic conductor: | \(2.89556\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(38.27700965\) |
| \(L(\frac12)\) | \(\approx\) | \(38.27700965\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 3 | \( ( 1 - T^{4} + T^{8} )^{2} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + 24 T^{2} + 239 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| good | 11 | \( ( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 2902 p T^{9} - 168 p^{2} T^{10} + 26 p^{3} T^{11} - 58 p^{4} T^{12} - 48 p^{5} T^{13} - 14 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) |
| 13 | \( 1 + 50 T^{4} + 3061 p T^{8} + 11442962 T^{12} + 523271620 T^{16} + 11442962 p^{4} T^{20} + 3061 p^{9} T^{24} + 50 p^{12} T^{28} + p^{16} T^{32} \) | |
| 17 | \( 1 + 96 T^{2} + 15 p^{2} T^{4} + 121248 T^{6} + 2296193 T^{8} + 27677184 T^{10} + 58458882 T^{12} - 6525249024 T^{14} - 167337014754 T^{16} - 6525249024 p^{2} T^{18} + 58458882 p^{4} T^{20} + 27677184 p^{6} T^{22} + 2296193 p^{8} T^{24} + 121248 p^{10} T^{26} + 15 p^{14} T^{28} + 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 2 T - 43 T^{2} - 70 T^{3} + 1187 T^{4} + 3200 T^{5} - 13284 T^{6} - 38196 T^{7} + 120370 T^{8} - 38196 p T^{9} - 13284 p^{2} T^{10} + 3200 p^{3} T^{11} + 1187 p^{4} T^{12} - 70 p^{5} T^{13} - 43 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 96 T^{2} + 4119 T^{4} - 100512 T^{6} + 1547777 T^{8} - 13526016 T^{10} - 398667390 T^{12} + 32311398144 T^{14} - 1026908911218 T^{16} + 32311398144 p^{2} T^{18} - 398667390 p^{4} T^{20} - 13526016 p^{6} T^{22} + 1547777 p^{8} T^{24} - 100512 p^{10} T^{26} + 4119 p^{12} T^{28} - 96 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 - 64 T^{2} + 3856 T^{4} - 135376 T^{6} + 4752862 T^{8} - 135376 p^{2} T^{10} + 3856 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 99 T^{2} + 179 p T^{4} + 2448 T^{5} + 231102 T^{6} + 153072 T^{7} + 7750614 T^{8} + 153072 p T^{9} + 231102 p^{2} T^{10} + 2448 p^{3} T^{11} + 179 p^{5} T^{12} + 99 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( 1 + 36 T^{2} + 4499 T^{4} + 146412 T^{6} + 10243837 T^{8} + 329458176 T^{10} + 19288003070 T^{12} + 581067044496 T^{14} + 30324317162902 T^{16} + 581067044496 p^{2} T^{18} + 19288003070 p^{4} T^{20} + 329458176 p^{6} T^{22} + 10243837 p^{8} T^{24} + 146412 p^{10} T^{26} + 4499 p^{12} T^{28} + 36 p^{14} T^{30} + p^{16} T^{32} \) | |
| 41 | \( ( 1 - 158 T^{2} + 12905 T^{4} - 739782 T^{6} + 33449684 T^{8} - 739782 p^{2} T^{10} + 12905 p^{4} T^{12} - 158 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 43 | \( 1 - 1504 T^{4} - 258116 T^{8} - 4620736288 T^{12} + 14886307959622 T^{16} - 4620736288 p^{4} T^{20} - 258116 p^{8} T^{24} - 1504 p^{12} T^{28} + p^{16} T^{32} \) | |
| 47 | \( 1 + 72 T^{2} - 2097 T^{4} - 275400 T^{6} - 1208767 T^{8} + 434448 T^{10} - 23245686270 T^{12} + 566620331904 T^{14} + 118374567124638 T^{16} + 566620331904 p^{2} T^{18} - 23245686270 p^{4} T^{20} + 434448 p^{6} T^{22} - 1208767 p^{8} T^{24} - 275400 p^{10} T^{26} - 2097 p^{12} T^{28} + 72 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( 1 - 60 T^{2} + 2040 T^{4} - 50400 T^{6} + 8790530 T^{8} + 35607396 T^{10} - 20650486848 T^{12} + 1867209163140 T^{14} - 61066879873293 T^{16} + 1867209163140 p^{2} T^{18} - 20650486848 p^{4} T^{20} + 35607396 p^{6} T^{22} + 8790530 p^{8} T^{24} - 50400 p^{10} T^{26} + 2040 p^{12} T^{28} - 60 p^{14} T^{30} + p^{16} T^{32} \) | |
| 59 | \( ( 1 + 20 T + 124 T^{2} - 312 T^{3} - 7894 T^{4} - 53140 T^{5} - 123312 T^{6} + 3441412 T^{7} + 48399379 T^{8} + 3441412 p T^{9} - 123312 p^{2} T^{10} - 53140 p^{3} T^{11} - 7894 p^{4} T^{12} - 312 p^{5} T^{13} + 124 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 12 T + 225 T^{2} + 2124 T^{3} + 22907 T^{4} + 153816 T^{5} + 1374816 T^{6} + 7573500 T^{7} + 71943462 T^{8} + 7573500 p T^{9} + 1374816 p^{2} T^{10} + 153816 p^{3} T^{11} + 22907 p^{4} T^{12} + 2124 p^{5} T^{13} + 225 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 67 | \( 1 + 144 T^{2} + 15479 T^{4} + 1233648 T^{6} + 82019857 T^{8} + 5129471520 T^{10} + 354430386290 T^{12} + 27207762471936 T^{14} + 1909973041583038 T^{16} + 27207762471936 p^{2} T^{18} + 354430386290 p^{4} T^{20} + 5129471520 p^{6} T^{22} + 82019857 p^{8} T^{24} + 1233648 p^{10} T^{26} + 15479 p^{12} T^{28} + 144 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( ( 1 - 26 T + 513 T^{2} - 6318 T^{3} + 63556 T^{4} - 6318 p T^{5} + 513 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 73 | \( 1 - 576 T^{2} + 168527 T^{4} - 33370560 T^{6} + 5044720033 T^{8} - 621838271232 T^{10} + 64869439165922 T^{12} - 5836679136073728 T^{14} + 456532218503037886 T^{16} - 5836679136073728 p^{2} T^{18} + 64869439165922 p^{4} T^{20} - 621838271232 p^{6} T^{22} + 5044720033 p^{8} T^{24} - 33370560 p^{10} T^{26} + 168527 p^{12} T^{28} - 576 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 6 T + 194 T^{2} - 1092 T^{3} + 17173 T^{4} - 108768 T^{5} + 1220606 T^{6} - 9301122 T^{7} + 98127316 T^{8} - 9301122 p T^{9} + 1220606 p^{2} T^{10} - 108768 p^{3} T^{11} + 17173 p^{4} T^{12} - 1092 p^{5} T^{13} + 194 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 83 | \( 1 + 2720 T^{4} - 86504132 T^{8} - 3906190240 T^{12} + 5664126472953286 T^{16} - 3906190240 p^{4} T^{20} - 86504132 p^{8} T^{24} + 2720 p^{12} T^{28} + p^{16} T^{32} \) | |
| 89 | \( ( 1 + 30 T + 439 T^{2} + 3930 T^{3} + 17353 T^{4} - 77580 T^{5} - 2924786 T^{6} - 44484000 T^{7} - 479997158 T^{8} - 44484000 p T^{9} - 2924786 p^{2} T^{10} - 77580 p^{3} T^{11} + 17353 p^{4} T^{12} + 3930 p^{5} T^{13} + 439 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 4700 T^{4} - 3632918 T^{8} + 688067520560 T^{12} + 9198193610860915 T^{16} + 688067520560 p^{4} T^{20} - 3632918 p^{8} T^{24} + 4700 p^{12} T^{28} + p^{16} T^{32} \) | |
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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
−2.56756818529853604154226583848, −2.46487036535931724204284371514, −2.40299373497103433902093520410, −2.26994971355922840723291174498, −2.20099463937572447255767113825, −2.12350804937858081007753595971, −2.07425643145534539418193563551, −2.04714174375230944595410531866, −1.88981064512488575826402419498, −1.79282370505397805040633406141, −1.75561956518681618516683034513, −1.65455521021482851020229349463, −1.56622395115109241635414996293, −1.51208748549246877187221045458, −1.49463465718163946857547612090, −1.39344580644312816297141941292, −1.29390611527076235331765320417, −1.09805364410008308933405730067, −0.71451695463501772149758708691, −0.68956279596837705559118222272, −0.62132031065853336761966040006, −0.54575216813679515097828070339, −0.47967691801338567979504145024, −0.40736101717178560589706915186, −0.39859934575295350060050044077, 0.39859934575295350060050044077, 0.40736101717178560589706915186, 0.47967691801338567979504145024, 0.54575216813679515097828070339, 0.62132031065853336761966040006, 0.68956279596837705559118222272, 0.71451695463501772149758708691, 1.09805364410008308933405730067, 1.29390611527076235331765320417, 1.39344580644312816297141941292, 1.49463465718163946857547612090, 1.51208748549246877187221045458, 1.56622395115109241635414996293, 1.65455521021482851020229349463, 1.75561956518681618516683034513, 1.79282370505397805040633406141, 1.88981064512488575826402419498, 2.04714174375230944595410531866, 2.07425643145534539418193563551, 2.12350804937858081007753595971, 2.20099463937572447255767113825, 2.26994971355922840723291174498, 2.40299373497103433902093520410, 2.46487036535931724204284371514, 2.56756818529853604154226583848
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.