Dirichlet series
| L(s) = 1 | + 4·2-s + 15·4-s + 32·8-s + 24·9-s + 64·11-s + 61·16-s + 96·18-s − 32·19-s + 256·22-s + 60·32-s + 360·36-s − 128·38-s + 192·43-s + 960·44-s + 352·49-s + 128·59-s + 19·64-s + 64·67-s + 768·72-s + 160·73-s − 480·76-s + 324·81-s + 768·86-s + 2.04e3·88-s + 224·97-s + 1.40e3·98-s + 1.53e3·99-s + ⋯ |
| L(s) = 1 | + 2·2-s + 15/4·4-s + 4·8-s + 8/3·9-s + 5.81·11-s + 3.81·16-s + 16/3·18-s − 1.68·19-s + 11.6·22-s + 15/8·32-s + 10·36-s − 3.36·38-s + 4.46·43-s + 21.8·44-s + 7.18·49-s + 2.16·59-s + 0.296·64-s + 0.955·67-s + 32/3·72-s + 2.19·73-s − 6.31·76-s + 4·81-s + 8.93·86-s + 23.2·88-s + 2.30·97-s + 14.3·98-s + 15.5·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 3^{16} \cdot 5^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.60483\times 10^{19}\) |
| Root analytic conductor: | \(4.04336\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [1]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1077.368137\) |
| \(L(\frac12)\) | \(\approx\) | \(1077.368137\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p^{2} T + T^{2} + 3 p^{3} T^{3} - 11 p^{2} T^{4} - p^{5} T^{5} + 45 p^{2} T^{6} - p^{6} T^{7} - 11 p^{5} T^{8} - p^{8} T^{9} + 45 p^{6} T^{10} - p^{11} T^{11} - 11 p^{10} T^{12} + 3 p^{13} T^{13} + p^{12} T^{14} - p^{16} T^{15} + p^{16} T^{16} \) |
| 3 | \( ( 1 - p T^{2} )^{8} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 352 T^{2} + 66744 T^{4} - 8814880 T^{6} + 897822364 T^{8} - 74503987296 T^{10} + 5208388637960 T^{12} - 313326720029984 T^{14} + 16417814215112262 T^{16} - 313326720029984 p^{4} T^{18} + 5208388637960 p^{8} T^{20} - 74503987296 p^{12} T^{22} + 897822364 p^{16} T^{24} - 8814880 p^{20} T^{26} + 66744 p^{24} T^{28} - 352 p^{28} T^{30} + p^{32} T^{32} \) |
| 11 | \( ( 1 - 32 T + 856 T^{2} - 15968 T^{3} + 267260 T^{4} - 3656480 T^{5} + 47610984 T^{6} - 547518560 T^{7} + 571200402 p T^{8} - 547518560 p^{2} T^{9} + 47610984 p^{4} T^{10} - 3656480 p^{6} T^{11} + 267260 p^{8} T^{12} - 15968 p^{10} T^{13} + 856 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 13 | \( 1 - 1376 T^{2} + 948152 T^{4} - 435079968 T^{6} + 11515432844 p T^{8} - 41358213779552 T^{10} + 9609504049596680 T^{12} - 1939655993637720864 T^{14} + \)\(34\!\cdots\!34\)\( T^{16} - 1939655993637720864 p^{4} T^{18} + 9609504049596680 p^{8} T^{20} - 41358213779552 p^{12} T^{22} + 11515432844 p^{17} T^{24} - 435079968 p^{20} T^{26} + 948152 p^{24} T^{28} - 1376 p^{28} T^{30} + p^{32} T^{32} \) | |
| 17 | \( ( 1 + 1056 T^{2} - 128 p T^{3} + 630844 T^{4} - 1439872 T^{5} + 273288160 T^{6} - 701779712 T^{7} + 88961599622 T^{8} - 701779712 p^{2} T^{9} + 273288160 p^{4} T^{10} - 1439872 p^{6} T^{11} + 630844 p^{8} T^{12} - 128 p^{11} T^{13} + 1056 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 16 T + 1056 T^{2} + 16 p T^{3} + 201852 T^{4} - 5268848 T^{5} + 74005984 T^{6} + 35754512 p T^{7} + 68566421830 T^{8} + 35754512 p^{3} T^{9} + 74005984 p^{4} T^{10} - 5268848 p^{6} T^{11} + 201852 p^{8} T^{12} + 16 p^{11} T^{13} + 1056 p^{12} T^{14} + 16 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 23 | \( 1 - 3584 T^{2} + 6817272 T^{4} - 9239353856 T^{6} + 9917837759772 T^{8} - 8813985457167872 T^{10} + 6651683212554175432 T^{12} - \)\(43\!\cdots\!20\)\( T^{14} + \)\(24\!\cdots\!66\)\( T^{16} - \)\(43\!\cdots\!20\)\( p^{4} T^{18} + 6651683212554175432 p^{8} T^{20} - 8813985457167872 p^{12} T^{22} + 9917837759772 p^{16} T^{24} - 9239353856 p^{20} T^{26} + 6817272 p^{24} T^{28} - 3584 p^{28} T^{30} + p^{32} T^{32} \) | |
| 29 | \( 1 - 7856 T^{2} + 28200056 T^{4} - 61477872912 T^{6} + 92026975658396 T^{8} - 103218432062453168 T^{10} + 95167863584482755656 T^{12} - \)\(80\!\cdots\!04\)\( T^{14} + \)\(66\!\cdots\!66\)\( T^{16} - \)\(80\!\cdots\!04\)\( p^{4} T^{18} + 95167863584482755656 p^{8} T^{20} - 103218432062453168 p^{12} T^{22} + 92026975658396 p^{16} T^{24} - 61477872912 p^{20} T^{26} + 28200056 p^{24} T^{28} - 7856 p^{28} T^{30} + p^{32} T^{32} \) | |
| 31 | \( 1 - 7392 T^{2} + 28548408 T^{4} - 76827506080 T^{6} + 159768299137180 T^{8} - 270152168295227872 T^{10} + \)\(38\!\cdots\!52\)\( T^{12} - \)\(46\!\cdots\!56\)\( T^{14} + \)\(47\!\cdots\!42\)\( T^{16} - \)\(46\!\cdots\!56\)\( p^{4} T^{18} + \)\(38\!\cdots\!52\)\( p^{8} T^{20} - 270152168295227872 p^{12} T^{22} + 159768299137180 p^{16} T^{24} - 76827506080 p^{20} T^{26} + 28548408 p^{24} T^{28} - 7392 p^{28} T^{30} + p^{32} T^{32} \) | |
| 37 | \( 1 - 8992 T^{2} + 42469432 T^{4} - 139214683744 T^{6} + 353514248671772 T^{8} - 740172918210452512 T^{10} + \)\(13\!\cdots\!40\)\( T^{12} - \)\(21\!\cdots\!12\)\( T^{14} + \)\(30\!\cdots\!14\)\( T^{16} - \)\(21\!\cdots\!12\)\( p^{4} T^{18} + \)\(13\!\cdots\!40\)\( p^{8} T^{20} - 740172918210452512 p^{12} T^{22} + 353514248671772 p^{16} T^{24} - 139214683744 p^{20} T^{26} + 42469432 p^{24} T^{28} - 8992 p^{28} T^{30} + p^{32} T^{32} \) | |
| 41 | \( ( 1 + 3960 T^{2} + 38656 T^{3} + 8721820 T^{4} + 152824576 T^{5} + 20050984264 T^{6} + 123966287360 T^{7} + 39604369264070 T^{8} + 123966287360 p^{2} T^{9} + 20050984264 p^{4} T^{10} + 152824576 p^{6} T^{11} + 8721820 p^{8} T^{12} + 38656 p^{10} T^{13} + 3960 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 96 T + 11240 T^{2} - 666016 T^{3} + 47413020 T^{4} - 2087417312 T^{5} + 118890718424 T^{6} - 4422377409056 T^{7} + 233507673291526 T^{8} - 4422377409056 p^{2} T^{9} + 118890718424 p^{4} T^{10} - 2087417312 p^{6} T^{11} + 47413020 p^{8} T^{12} - 666016 p^{10} T^{13} + 11240 p^{12} T^{14} - 96 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 47 | \( 1 - 12992 T^{2} + 93000312 T^{4} - 439845859904 T^{6} + 1518786378685212 T^{8} - 3937610752009472192 T^{10} + \)\(79\!\cdots\!20\)\( T^{12} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(25\!\cdots\!34\)\( T^{16} - \)\(13\!\cdots\!32\)\( p^{4} T^{18} + \)\(79\!\cdots\!20\)\( p^{8} T^{20} - 3937610752009472192 p^{12} T^{22} + 1518786378685212 p^{16} T^{24} - 439845859904 p^{20} T^{26} + 93000312 p^{24} T^{28} - 12992 p^{28} T^{30} + p^{32} T^{32} \) | |
| 53 | \( 1 - 19248 T^{2} + 200282232 T^{4} - 1462177856656 T^{6} + 8295233466459292 T^{8} - 728669409787842416 p T^{10} + \)\(15\!\cdots\!00\)\( T^{12} - \)\(52\!\cdots\!68\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(52\!\cdots\!68\)\( p^{4} T^{18} + \)\(15\!\cdots\!00\)\( p^{8} T^{20} - 728669409787842416 p^{13} T^{22} + 8295233466459292 p^{16} T^{24} - 1462177856656 p^{20} T^{26} + 200282232 p^{24} T^{28} - 19248 p^{28} T^{30} + p^{32} T^{32} \) | |
| 59 | \( ( 1 - 64 T + 12440 T^{2} - 356160 T^{3} + 68403836 T^{4} - 1114496704 T^{5} + 345709855016 T^{6} - 6678479907264 T^{7} + 1475835157324614 T^{8} - 6678479907264 p^{2} T^{9} + 345709855016 p^{4} T^{10} - 1114496704 p^{6} T^{11} + 68403836 p^{8} T^{12} - 356160 p^{10} T^{13} + 12440 p^{12} T^{14} - 64 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 61 | \( 1 - 29072 T^{2} + 451446776 T^{4} - 4818504865200 T^{6} + 39233997104181404 T^{8} - \)\(25\!\cdots\!44\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{12} - \)\(65\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!78\)\( T^{16} - \)\(65\!\cdots\!32\)\( p^{4} T^{18} + \)\(14\!\cdots\!80\)\( p^{8} T^{20} - \)\(25\!\cdots\!44\)\( p^{12} T^{22} + 39233997104181404 p^{16} T^{24} - 4818504865200 p^{20} T^{26} + 451446776 p^{24} T^{28} - 29072 p^{28} T^{30} + p^{32} T^{32} \) | |
| 67 | \( ( 1 - 32 T + 14728 T^{2} - 605664 T^{3} + 128253532 T^{4} - 6428066464 T^{5} + 815977936440 T^{6} - 41913150669152 T^{7} + 4090672355748614 T^{8} - 41913150669152 p^{2} T^{9} + 815977936440 p^{4} T^{10} - 6428066464 p^{6} T^{11} + 128253532 p^{8} T^{12} - 605664 p^{10} T^{13} + 14728 p^{12} T^{14} - 32 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 71 | \( 1 - 37136 T^{2} + 757464440 T^{4} - 10784717006640 T^{6} + 118343601485433884 T^{8} - \)\(10\!\cdots\!80\)\( T^{10} + \)\(78\!\cdots\!20\)\( T^{12} - \)\(49\!\cdots\!44\)\( T^{14} + \)\(26\!\cdots\!74\)\( T^{16} - \)\(49\!\cdots\!44\)\( p^{4} T^{18} + \)\(78\!\cdots\!20\)\( p^{8} T^{20} - \)\(10\!\cdots\!80\)\( p^{12} T^{22} + 118343601485433884 p^{16} T^{24} - 10784717006640 p^{20} T^{26} + 757464440 p^{24} T^{28} - 37136 p^{28} T^{30} + p^{32} T^{32} \) | |
| 73 | \( ( 1 - 80 T + 23608 T^{2} - 1861872 T^{3} + 305981212 T^{4} - 21597410896 T^{5} + 36447212616 p T^{6} - 164093911185008 T^{7} + 16548291788162246 T^{8} - 164093911185008 p^{2} T^{9} + 36447212616 p^{5} T^{10} - 21597410896 p^{6} T^{11} + 305981212 p^{8} T^{12} - 1861872 p^{10} T^{13} + 23608 p^{12} T^{14} - 80 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
| 79 | \( 1 - 50272 T^{2} + 1307628088 T^{4} - 292044074464 p T^{6} + 308397489205548188 T^{8} - \)\(33\!\cdots\!16\)\( T^{10} + \)\(29\!\cdots\!28\)\( T^{12} - \)\(22\!\cdots\!24\)\( T^{14} + \)\(15\!\cdots\!10\)\( T^{16} - \)\(22\!\cdots\!24\)\( p^{4} T^{18} + \)\(29\!\cdots\!28\)\( p^{8} T^{20} - \)\(33\!\cdots\!16\)\( p^{12} T^{22} + 308397489205548188 p^{16} T^{24} - 292044074464 p^{21} T^{26} + 1307628088 p^{24} T^{28} - 50272 p^{28} T^{30} + p^{32} T^{32} \) | |
| 83 | \( ( 1 + 38440 T^{2} - 160768 T^{3} + 717051420 T^{4} - 3993019392 T^{5} + 8499988428696 T^{6} - 45606981027840 T^{7} + 69678823700497030 T^{8} - 45606981027840 p^{2} T^{9} + 8499988428696 p^{4} T^{10} - 3993019392 p^{6} T^{11} + 717051420 p^{8} T^{12} - 160768 p^{10} T^{13} + 38440 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 89 | \( ( 1 + 50168 T^{2} - 106240 T^{3} + 1173067164 T^{4} - 3722720000 T^{5} + 16752895495880 T^{6} - 55866443578880 T^{7} + 160186201074675910 T^{8} - 55866443578880 p^{2} T^{9} + 16752895495880 p^{4} T^{10} - 3722720000 p^{6} T^{11} + 1173067164 p^{8} T^{12} - 106240 p^{10} T^{13} + 50168 p^{12} T^{14} + p^{16} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 112 T + 35672 T^{2} - 3556560 T^{3} + 704218076 T^{4} - 55395805936 T^{5} + 9405117820520 T^{6} - 652123681253328 T^{7} + 96575703350911686 T^{8} - 652123681253328 p^{2} T^{9} + 9405117820520 p^{4} T^{10} - 55395805936 p^{6} T^{11} + 704218076 p^{8} T^{12} - 3556560 p^{10} T^{13} + 35672 p^{12} T^{14} - 112 p^{14} T^{15} + p^{16} T^{16} )^{2} \) | |
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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
−2.54741081156776690472622270758, −2.52267108385296491372900602047, −2.50411043223635226666306542770, −2.49770143643113150265616904656, −2.32485514531030518757778545498, −2.15850726329211201807180481897, −2.12621881895350474758929723067, −2.06248450801277286065902016524, −1.99005081651632499401862803023, −1.79624727506018063538005592231, −1.65793797107970583510473814955, −1.56581785147300766744473672159, −1.53692878711008000284217101277, −1.51663268040191169222066832216, −1.30894211966084836272643528556, −1.23018160124045625084039222016, −1.15274474168327074414160987809, −1.00207394048118203336896302790, −0.962670356390322122461603569194, −0.904982266509399866924383356936, −0.856946931590515126391126298249, −0.54354686680435618966312197364, −0.51212481485065863306651969954, −0.40443143150353414432442392440, −0.20274911030744239054851206630, 0.20274911030744239054851206630, 0.40443143150353414432442392440, 0.51212481485065863306651969954, 0.54354686680435618966312197364, 0.856946931590515126391126298249, 0.904982266509399866924383356936, 0.962670356390322122461603569194, 1.00207394048118203336896302790, 1.15274474168327074414160987809, 1.23018160124045625084039222016, 1.30894211966084836272643528556, 1.51663268040191169222066832216, 1.53692878711008000284217101277, 1.56581785147300766744473672159, 1.65793797107970583510473814955, 1.79624727506018063538005592231, 1.99005081651632499401862803023, 2.06248450801277286065902016524, 2.12621881895350474758929723067, 2.15850726329211201807180481897, 2.32485514531030518757778545498, 2.49770143643113150265616904656, 2.50411043223635226666306542770, 2.52267108385296491372900602047, 2.54741081156776690472622270758
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.