Dirichlet series
| L(s) = 1 | + 4·7-s − 16·9-s + 8·19-s − 8·25-s − 16·27-s − 8·29-s − 16·37-s + 8·47-s + 6·49-s − 16·53-s + 8·59-s − 64·63-s + 126·81-s − 64·83-s − 24·103-s + 64·109-s + 76·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 5.33·9-s + 1.83·19-s − 8/5·25-s − 3.07·27-s − 1.48·29-s − 2.63·37-s + 1.16·47-s + 6/7·49-s − 2.19·53-s + 1.04·59-s − 8.06·63-s + 14·81-s − 7.02·83-s − 2.36·103-s + 6.13·109-s + 6.90·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-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 + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{96} \cdot 5^{16} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09749\times 10^{20}\) |
| Root analytic conductor: | \(4.22924\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{96} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.01657112147\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01657112147\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{8} \) | |
| 7 | \( 1 - 4 T + 10 T^{2} - 52 T^{3} + 128 T^{4} - 164 T^{5} + 598 T^{6} - 1076 T^{7} + 190 T^{8} - 1076 p T^{9} + 598 p^{2} T^{10} - 164 p^{3} T^{11} + 128 p^{4} T^{12} - 52 p^{5} T^{13} + 10 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( ( 1 + 8 T^{2} + 8 T^{3} + 11 p T^{4} + 56 T^{5} + 130 T^{6} + 200 T^{7} + 448 T^{8} + 200 p T^{9} + 130 p^{2} T^{10} + 56 p^{3} T^{11} + 11 p^{5} T^{12} + 8 p^{5} T^{13} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( 1 - 76 T^{2} + 2 p^{3} T^{4} - 5168 p T^{6} + 839761 T^{8} - 9730232 T^{10} + 9669666 p T^{12} - 1242923724 T^{14} + 14328067076 T^{16} - 1242923724 p^{2} T^{18} + 9669666 p^{5} T^{20} - 9730232 p^{6} T^{22} + 839761 p^{8} T^{24} - 5168 p^{11} T^{26} + 2 p^{15} T^{28} - 76 p^{14} T^{30} + p^{16} T^{32} \) | |
| 13 | \( 1 - 108 T^{2} + 5654 T^{4} - 196696 T^{6} + 5239105 T^{8} - 114756296 T^{10} + 164017246 p T^{12} - 34097224916 T^{14} + 474211456356 T^{16} - 34097224916 p^{2} T^{18} + 164017246 p^{5} T^{20} - 114756296 p^{6} T^{22} + 5239105 p^{8} T^{24} - 196696 p^{10} T^{26} + 5654 p^{12} T^{28} - 108 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 148 T^{2} + 11302 T^{4} - 585840 T^{6} + 22963569 T^{8} - 719314600 T^{10} + 18585980982 T^{12} - 403542086356 T^{14} + 7435491841764 T^{16} - 403542086356 p^{2} T^{18} + 18585980982 p^{4} T^{20} - 719314600 p^{6} T^{22} + 22963569 p^{8} T^{24} - 585840 p^{10} T^{26} + 11302 p^{12} T^{28} - 148 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( ( 1 - 4 T + 92 T^{2} - 276 T^{3} + 3652 T^{4} - 7796 T^{5} + 87236 T^{6} - 135172 T^{7} + 1681686 T^{8} - 135172 p T^{9} + 87236 p^{2} T^{10} - 7796 p^{3} T^{11} + 3652 p^{4} T^{12} - 276 p^{5} T^{13} + 92 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 23 | \( 1 - 252 T^{2} + 31076 T^{4} - 2496644 T^{6} + 146688948 T^{8} - 6700022476 T^{10} + 246528216092 T^{12} - 7461953111860 T^{14} + 187859376139030 T^{16} - 7461953111860 p^{2} T^{18} + 246528216092 p^{4} T^{20} - 6700022476 p^{6} T^{22} + 146688948 p^{8} T^{24} - 2496644 p^{10} T^{26} + 31076 p^{12} T^{28} - 252 p^{14} T^{30} + p^{16} T^{32} \) | |
| 29 | \( ( 1 + 4 T + 94 T^{2} + 376 T^{3} + 4153 T^{4} + 16952 T^{5} + 129982 T^{6} + 572444 T^{7} + 3724452 T^{8} + 572444 p T^{9} + 129982 p^{2} T^{10} + 16952 p^{3} T^{11} + 4153 p^{4} T^{12} + 376 p^{5} T^{13} + 94 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 120 T^{2} + 32 T^{3} + 7420 T^{4} + 1376 T^{5} + 332488 T^{6} - 17600 T^{7} + 11688582 T^{8} - 17600 p T^{9} + 332488 p^{2} T^{10} + 1376 p^{3} T^{11} + 7420 p^{4} T^{12} + 32 p^{5} T^{13} + 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 8 T + 184 T^{2} + 1432 T^{3} + 17308 T^{4} + 124264 T^{5} + 1054024 T^{6} + 6781752 T^{7} + 45623078 T^{8} + 6781752 p T^{9} + 1054024 p^{2} T^{10} + 124264 p^{3} T^{11} + 17308 p^{4} T^{12} + 1432 p^{5} T^{13} + 184 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 312 T^{2} + 48312 T^{4} - 4930984 T^{6} + 372434588 T^{8} - 22267877944 T^{10} + 1114049870344 T^{12} - 49472555502568 T^{14} + 2065146899220422 T^{16} - 49472555502568 p^{2} T^{18} + 1114049870344 p^{4} T^{20} - 22267877944 p^{6} T^{22} + 372434588 p^{8} T^{24} - 4930984 p^{10} T^{26} + 48312 p^{12} T^{28} - 312 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( 1 - 348 T^{2} + 64356 T^{4} - 8202756 T^{6} + 797939252 T^{8} - 62427193260 T^{10} + 4047512545756 T^{12} - 221312127392564 T^{14} + 10299993559399254 T^{16} - 221312127392564 p^{2} T^{18} + 4047512545756 p^{4} T^{20} - 62427193260 p^{6} T^{22} + 797939252 p^{8} T^{24} - 8202756 p^{10} T^{26} + 64356 p^{12} T^{28} - 348 p^{14} T^{30} + p^{16} T^{32} \) | |
| 47 | \( ( 1 - 4 T + 172 T^{2} - 1092 T^{3} + 17905 T^{4} - 118448 T^{5} + 1320746 T^{6} - 8313984 T^{7} + 70770016 T^{8} - 8313984 p T^{9} + 1320746 p^{2} T^{10} - 118448 p^{3} T^{11} + 17905 p^{4} T^{12} - 1092 p^{5} T^{13} + 172 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 8 T + 144 T^{2} + 792 T^{3} + 10092 T^{4} + 24936 T^{5} + 417072 T^{6} - 280904 T^{7} + 13942534 T^{8} - 280904 p T^{9} + 417072 p^{2} T^{10} + 24936 p^{3} T^{11} + 10092 p^{4} T^{12} + 792 p^{5} T^{13} + 144 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 59 | \( ( 1 - 4 T + 204 T^{2} - 1204 T^{3} + 25428 T^{4} - 139060 T^{5} + 2258196 T^{6} - 11379300 T^{7} + 148324534 T^{8} - 11379300 p T^{9} + 2258196 p^{2} T^{10} - 139060 p^{3} T^{11} + 25428 p^{4} T^{12} - 1204 p^{5} T^{13} + 204 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 61 | \( 1 - 192 T^{2} + 33368 T^{4} - 3834816 T^{6} + 399024412 T^{8} - 34592908864 T^{10} + 2737227404776 T^{12} - 193544986547520 T^{14} + 12406661017328006 T^{16} - 193544986547520 p^{2} T^{18} + 2737227404776 p^{4} T^{20} - 34592908864 p^{6} T^{22} + 399024412 p^{8} T^{24} - 3834816 p^{10} T^{26} + 33368 p^{12} T^{28} - 192 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( 1 - 636 T^{2} + 187204 T^{4} - 34054052 T^{6} + 4328448628 T^{8} - 415023190092 T^{10} + 32149434867068 T^{12} - 2190206756199252 T^{14} + 145034343503529814 T^{16} - 2190206756199252 p^{2} T^{18} + 32149434867068 p^{4} T^{20} - 415023190092 p^{6} T^{22} + 4328448628 p^{8} T^{24} - 34054052 p^{10} T^{26} + 187204 p^{12} T^{28} - 636 p^{14} T^{30} + p^{16} T^{32} \) | |
| 71 | \( 1 - 248 T^{2} + 29944 T^{4} - 1969384 T^{6} + 78783516 T^{8} - 4702297848 T^{10} + 671988027336 T^{12} - 77772203862568 T^{14} + 6307498316443078 T^{16} - 77772203862568 p^{2} T^{18} + 671988027336 p^{4} T^{20} - 4702297848 p^{6} T^{22} + 78783516 p^{8} T^{24} - 1969384 p^{10} T^{26} + 29944 p^{12} T^{28} - 248 p^{14} T^{30} + p^{16} T^{32} \) | |
| 73 | \( 1 - 8 p T^{2} + 167480 T^{4} - 32168408 T^{6} + 4747021724 T^{8} - 577316218952 T^{10} + 59758367403144 T^{12} - 5344716084380312 T^{14} + 5711723963955606 p T^{16} - 5344716084380312 p^{2} T^{18} + 59758367403144 p^{4} T^{20} - 577316218952 p^{6} T^{22} + 4747021724 p^{8} T^{24} - 32168408 p^{10} T^{26} + 167480 p^{12} T^{28} - 8 p^{15} T^{30} + p^{16} T^{32} \) | |
| 79 | \( 1 - 972 T^{2} + 456246 T^{4} - 137616352 T^{6} + 29914587041 T^{8} - 4976708515608 T^{10} + 656136410510662 T^{12} - 69974580058110652 T^{14} + 6102549953423995716 T^{16} - 69974580058110652 p^{2} T^{18} + 656136410510662 p^{4} T^{20} - 4976708515608 p^{6} T^{22} + 29914587041 p^{8} T^{24} - 137616352 p^{10} T^{26} + 456246 p^{12} T^{28} - 972 p^{14} T^{30} + p^{16} T^{32} \) | |
| 83 | \( ( 1 + 32 T + 766 T^{2} + 13016 T^{3} + 193632 T^{4} + 2456872 T^{5} + 28512818 T^{6} + 295713808 T^{7} + 2834055390 T^{8} + 295713808 p T^{9} + 28512818 p^{2} T^{10} + 2456872 p^{3} T^{11} + 193632 p^{4} T^{12} + 13016 p^{5} T^{13} + 766 p^{6} T^{14} + 32 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 720 T^{2} + 253592 T^{4} - 58648304 T^{6} + 10147445724 T^{8} - 1423316521808 T^{10} + 170653764495912 T^{12} - 18024065052269808 T^{14} + 1696978658673832006 T^{16} - 18024065052269808 p^{2} T^{18} + 170653764495912 p^{4} T^{20} - 1423316521808 p^{6} T^{22} + 10147445724 p^{8} T^{24} - 58648304 p^{10} T^{26} + 253592 p^{12} T^{28} - 720 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 820 T^{2} + 345606 T^{4} - 98035376 T^{6} + 20905996753 T^{8} - 3560811727272 T^{10} + 503484822323926 T^{12} - 60617374699024500 T^{14} + 6308457346533623716 T^{16} - 60617374699024500 p^{2} T^{18} + 503484822323926 p^{4} T^{20} - 3560811727272 p^{6} T^{22} + 20905996753 p^{8} T^{24} - 98035376 p^{10} T^{26} + 345606 p^{12} T^{28} - 820 p^{14} T^{30} + p^{16} T^{32} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.23648687447358413106497851463, −2.17255349557843358131238612787, −2.09389519222259167991622231345, −2.05024232297072719771146866925, −1.92563805449427215429308865066, −1.81711254969906229995782832940, −1.75821313359566376102934084216, −1.72050462108741900801446411131, −1.68922583224964391536238863893, −1.67757576557642177017544034555, −1.66468999862703774334859672078, −1.50605381457549831295443709093, −1.22820841844305307305441750108, −1.21852567423239041890190240755, −1.08482958284073209445000679647, −1.04691640973163927991721586281, −0.970496259337481081938523286514, −0.947254719831133989455729039327, −0.928765966831237855600611678153, −0.50649836434888355680315125757, −0.46456606692977132297619422273, −0.33580417947727745552800865030, −0.18477193530277770095371149575, −0.11405959280656554460455777061, −0.03384119450292908675721954011, 0.03384119450292908675721954011, 0.11405959280656554460455777061, 0.18477193530277770095371149575, 0.33580417947727745552800865030, 0.46456606692977132297619422273, 0.50649836434888355680315125757, 0.928765966831237855600611678153, 0.947254719831133989455729039327, 0.970496259337481081938523286514, 1.04691640973163927991721586281, 1.08482958284073209445000679647, 1.21852567423239041890190240755, 1.22820841844305307305441750108, 1.50605381457549831295443709093, 1.66468999862703774334859672078, 1.67757576557642177017544034555, 1.68922583224964391536238863893, 1.72050462108741900801446411131, 1.75821313359566376102934084216, 1.81711254969906229995782832940, 1.92563805449427215429308865066, 2.05024232297072719771146866925, 2.09389519222259167991622231345, 2.17255349557843358131238612787, 2.23648687447358413106497851463
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.