Dirichlet series
| L(s) = 1 | + 8·4-s − 12·9-s − 8·11-s + 24·16-s − 144·19-s + 48·29-s + 192·31-s − 96·36-s − 64·44-s + 264·49-s − 624·59-s − 408·61-s − 128·71-s − 1.15e3·76-s + 288·79-s + 54·81-s + 672·89-s + 96·99-s − 384·101-s + 384·116-s + 900·121-s + 1.53e3·124-s + 127-s + 131-s + 137-s + 139-s − 288·144-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4/3·9-s − 0.727·11-s + 3/2·16-s − 7.57·19-s + 1.65·29-s + 6.19·31-s − 8/3·36-s − 1.45·44-s + 5.38·49-s − 10.5·59-s − 6.68·61-s − 1.80·71-s − 15.1·76-s + 3.64·79-s + 2/3·81-s + 7.55·89-s + 0.969·99-s − 3.80·101-s + 3.31·116-s + 7.43·121-s + 12.3·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.01553\times 10^{23}\) |
| Root analytic conductor: | \(5.34887\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(9.793500060\) |
| \(L(\frac12)\) | \(\approx\) | \(9.793500060\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) | |
| 5 | \( 1 \) | |
| 7 | \( ( 1 - 132 T^{2} + 167 p^{2} T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} )^{2} \) | |
| good | 11 | \( ( 1 + 4 T - 426 T^{2} - 952 T^{3} + 112064 T^{4} + 142800 T^{5} - 20280116 T^{6} - 6609908 T^{7} + 2820384747 T^{8} - 6609908 p^{2} T^{9} - 20280116 p^{4} T^{10} + 142800 p^{6} T^{11} + 112064 p^{8} T^{12} - 952 p^{10} T^{13} - 426 p^{12} T^{14} + 4 p^{14} T^{15} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 + 812 T^{2} + 291354 T^{4} + 64773520 T^{6} + 11461503587 T^{8} + 64773520 p^{4} T^{10} + 291354 p^{8} T^{12} + 812 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 17 | \( 1 - 724 T^{2} + 215268 T^{4} - 37401992 T^{6} + 493270406 T^{8} + 3289059610284 T^{10} - 1147207980658208 T^{12} + 261527361667335764 T^{14} - 71056954604749704669 T^{16} + 261527361667335764 p^{4} T^{18} - 1147207980658208 p^{8} T^{20} + 3289059610284 p^{12} T^{22} + 493270406 p^{16} T^{24} - 37401992 p^{20} T^{26} + 215268 p^{24} T^{28} - 724 p^{28} T^{30} + p^{32} T^{32} \) | |
| 19 | \( ( 1 + 72 T + 3028 T^{2} + 93600 T^{3} + 2298945 T^{4} + 49353840 T^{5} + 974880260 T^{6} + 18709788168 T^{7} + 358360390448 T^{8} + 18709788168 p^{2} T^{9} + 974880260 p^{4} T^{10} + 49353840 p^{6} T^{11} + 2298945 p^{8} T^{12} + 93600 p^{10} T^{13} + 3028 p^{12} T^{14} + 72 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 + 2736 T^{2} + 4106324 T^{4} + 4020863904 T^{6} + 2745829377322 T^{8} + 1207456985627856 T^{10} + 196165114417119440 T^{12} - \)\(16\!\cdots\!36\)\( T^{14} - \)\(15\!\cdots\!25\)\( T^{16} - \)\(16\!\cdots\!36\)\( p^{4} T^{18} + 196165114417119440 p^{8} T^{20} + 1207456985627856 p^{12} T^{22} + 2745829377322 p^{16} T^{24} + 4020863904 p^{20} T^{26} + 4106324 p^{24} T^{28} + 2736 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( ( 1 - 12 T + 2040 T^{2} - 5892 T^{3} + 1907222 T^{4} - 5892 p^{2} T^{5} + 2040 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 31 | \( ( 1 - 96 T + 7300 T^{2} - 405888 T^{3} + 20288250 T^{4} - 28540512 p T^{5} + 34750935632 T^{6} - 39793390752 p T^{7} + 39789694494419 T^{8} - 39793390752 p^{3} T^{9} + 34750935632 p^{4} T^{10} - 28540512 p^{7} T^{11} + 20288250 p^{8} T^{12} - 405888 p^{10} T^{13} + 7300 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 37 | \( 1 + 6876 T^{2} + 22398470 T^{4} + 55535727672 T^{6} + 127886289913873 T^{8} + 254012021588743272 T^{10} + \)\(43\!\cdots\!78\)\( T^{12} + \)\(68\!\cdots\!20\)\( T^{14} + \)\(10\!\cdots\!52\)\( T^{16} + \)\(68\!\cdots\!20\)\( p^{4} T^{18} + \)\(43\!\cdots\!78\)\( p^{8} T^{20} + 254012021588743272 p^{12} T^{22} + 127886289913873 p^{16} T^{24} + 55535727672 p^{20} T^{26} + 22398470 p^{24} T^{28} + 6876 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 - 8684 T^{2} + 37350972 T^{4} - 104578299436 T^{6} + 123169986458 p^{2} T^{8} - 104578299436 p^{4} T^{10} + 37350972 p^{8} T^{12} - 8684 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 2648 T^{2} + 9575196 T^{4} - 13352111848 T^{6} + 35579464808198 T^{8} - 13352111848 p^{4} T^{10} + 9575196 p^{8} T^{12} - 2648 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 7436 T^{2} + 27004756 T^{4} - 42730994488 T^{6} - 31141212484666 T^{8} + 316091068382193364 T^{10} - \)\(55\!\cdots\!60\)\( T^{12} - \)\(18\!\cdots\!16\)\( T^{14} + \)\(22\!\cdots\!95\)\( T^{16} - \)\(18\!\cdots\!16\)\( p^{4} T^{18} - \)\(55\!\cdots\!60\)\( p^{8} T^{20} + 316091068382193364 p^{12} T^{22} - 31141212484666 p^{16} T^{24} - 42730994488 p^{20} T^{26} + 27004756 p^{24} T^{28} - 7436 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 + 14324 T^{2} + 99484420 T^{4} + 503925801160 T^{6} + 2257390850497478 T^{8} + 9237264425135993204 T^{10} + \)\(33\!\cdots\!48\)\( T^{12} + \)\(10\!\cdots\!44\)\( T^{14} + \)\(31\!\cdots\!31\)\( T^{16} + \)\(10\!\cdots\!44\)\( p^{4} T^{18} + \)\(33\!\cdots\!48\)\( p^{8} T^{20} + 9237264425135993204 p^{12} T^{22} + 2257390850497478 p^{16} T^{24} + 503925801160 p^{20} T^{26} + 99484420 p^{24} T^{28} + 14324 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 + 312 T + 52130 T^{2} + 6140784 T^{3} + 573933472 T^{4} + 46040844252 T^{5} + 3331577620436 T^{6} + 222192183834504 T^{7} + 13673835884421787 T^{8} + 222192183834504 p^{2} T^{9} + 3331577620436 p^{4} T^{10} + 46040844252 p^{6} T^{11} + 573933472 p^{8} T^{12} + 6140784 p^{10} T^{13} + 52130 p^{12} T^{14} + 312 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( ( 1 + 204 T + 29950 T^{2} + 3279912 T^{3} + 303356793 T^{4} + 24117206328 T^{5} + 1743285323678 T^{6} + 115555686856116 T^{7} + 7243930491862772 T^{8} + 115555686856116 p^{2} T^{9} + 1743285323678 p^{4} T^{10} + 24117206328 p^{6} T^{11} + 303356793 p^{8} T^{12} + 3279912 p^{10} T^{13} + 29950 p^{12} T^{14} + 204 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 67 | \( 1 + 29432 T^{2} + 466745634 T^{4} + 5198646124048 T^{6} + 45213437117540465 T^{8} + \)\(32\!\cdots\!04\)\( T^{10} + \)\(20\!\cdots\!14\)\( T^{12} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(51\!\cdots\!00\)\( T^{16} + \)\(10\!\cdots\!20\)\( p^{4} T^{18} + \)\(20\!\cdots\!14\)\( p^{8} T^{20} + \)\(32\!\cdots\!04\)\( p^{12} T^{22} + 45213437117540465 p^{16} T^{24} + 5198646124048 p^{20} T^{26} + 466745634 p^{24} T^{28} + 29432 p^{28} T^{30} + p^{32} T^{32} \) | |
| 71 | \( ( 1 + 32 T + 8676 T^{2} + 266080 T^{3} + 63690374 T^{4} + 266080 p^{2} T^{5} + 8676 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \) | |
| 73 | \( 1 - 8892 T^{2} + 9999014 T^{4} - 232436433528 T^{6} + 1899330239271025 T^{8} + 2276003707516310616 T^{10} + \)\(25\!\cdots\!98\)\( p T^{12} - \)\(17\!\cdots\!00\)\( T^{14} - \)\(53\!\cdots\!40\)\( T^{16} - \)\(17\!\cdots\!00\)\( p^{4} T^{18} + \)\(25\!\cdots\!98\)\( p^{9} T^{20} + 2276003707516310616 p^{12} T^{22} + 1899330239271025 p^{16} T^{24} - 232436433528 p^{20} T^{26} + 9999014 p^{24} T^{28} - 8892 p^{28} T^{30} + p^{32} T^{32} \) | |
| 79 | \( ( 1 - 144 T - 612 T^{2} + 13824 p T^{3} - 51094919 T^{4} + 2106377280 T^{5} - 285913987380 T^{6} - 30587071339440 T^{7} + 6524654383140480 T^{8} - 30587071339440 p^{2} T^{9} - 285913987380 p^{4} T^{10} + 2106377280 p^{6} T^{11} - 51094919 p^{8} T^{12} + 13824 p^{11} T^{13} - 612 p^{12} T^{14} - 144 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 44180 T^{2} + 901887852 T^{4} + 11200337241460 T^{6} + 93097977064556858 T^{8} + 11200337241460 p^{4} T^{10} + 901887852 p^{8} T^{12} + 44180 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 336 T + 78946 T^{2} - 13881504 T^{3} + 2085368016 T^{4} - 269643799980 T^{5} + 31112060435732 T^{6} - 3208326913520640 T^{7} + 300257635897974779 T^{8} - 3208326913520640 p^{2} T^{9} + 31112060435732 p^{4} T^{10} - 269643799980 p^{6} T^{11} + 2085368016 p^{8} T^{12} - 13881504 p^{10} T^{13} + 78946 p^{12} T^{14} - 336 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 + 46260 T^{2} + 993971866 T^{4} + 13674806149680 T^{6} + 142735721327901795 T^{8} + 13674806149680 p^{4} T^{10} + 993971866 p^{8} T^{12} + 46260 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
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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
−2.29218306631369685791643646717, −2.16638937465204474257653796497, −2.14483823096378007864088845838, −2.04318776990961335925221467856, −2.03310250993784674566164057659, −1.98927342138101932479785944537, −1.89644199537457691236138680619, −1.72055447486433305729950196607, −1.70663467123549663616412153087, −1.68221088019155216394889385055, −1.62933922293769788452726382371, −1.58040508197361957577634785982, −1.36806978218168958461799705975, −1.21533064598380487461773830785, −1.09000644025359673517613323248, −1.00384436264255268218561504514, −0.937998066113458779694047016295, −0.859644045037976033619192467890, −0.789546404636446001727360630459, −0.58275141318831633435328047507, −0.35202897984745051059788810677, −0.33989603321301049270679996507, −0.30059879567356664135722582713, −0.24348534115068906251376761850, −0.13169094320852044026516343736, 0.13169094320852044026516343736, 0.24348534115068906251376761850, 0.30059879567356664135722582713, 0.33989603321301049270679996507, 0.35202897984745051059788810677, 0.58275141318831633435328047507, 0.789546404636446001727360630459, 0.859644045037976033619192467890, 0.937998066113458779694047016295, 1.00384436264255268218561504514, 1.09000644025359673517613323248, 1.21533064598380487461773830785, 1.36806978218168958461799705975, 1.58040508197361957577634785982, 1.62933922293769788452726382371, 1.68221088019155216394889385055, 1.70663467123549663616412153087, 1.72055447486433305729950196607, 1.89644199537457691236138680619, 1.98927342138101932479785944537, 2.03310250993784674566164057659, 2.04318776990961335925221467856, 2.14483823096378007864088845838, 2.16638937465204474257653796497, 2.29218306631369685791643646717
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.