Dirichlet series
| L(s) = 1 | + 1.68e3·7-s − 2.82e4·13-s + 4.22e4·19-s + 2.72e5·25-s + 1.64e5·31-s − 6.47e5·37-s + 1.34e6·43-s + 1.52e6·49-s − 4.31e6·61-s − 3.90e6·67-s + 6.47e6·73-s − 6.09e6·79-s − 4.75e7·91-s − 2.71e7·97-s − 2.84e7·103-s + 3.33e7·109-s + 6.28e7·121-s + 127-s + 131-s + 7.09e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.85·7-s − 3.57·13-s + 1.41·19-s + 3.48·25-s + 0.993·31-s − 2.10·37-s + 2.57·43-s + 1.85·49-s − 2.43·61-s − 1.58·67-s + 1.94·73-s − 1.39·79-s − 6.60·91-s − 3.01·97-s − 2.56·103-s + 2.46·109-s + 3.22·121-s + 2.61·133-s − 3.15·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.17496\times 10^{30}\) |
| Root analytic conductor: | \(8.87248\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [7/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(1.342192274\) |
| \(L(\frac12)\) | \(\approx\) | \(1.342192274\) |
| \(L(\frac{9}{2})\) | 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 | \( 1 \) | |
| 7 | \( ( 1 - 120 p T + 6026 p^{2} T^{2} + 228000 p^{4} T^{3} - 10473237 p^{6} T^{4} + 228000 p^{11} T^{5} + 6026 p^{16} T^{6} - 120 p^{22} T^{7} + p^{28} T^{8} )^{2} \) | |
| good | 5 | \( 1 - 54454 p T^{2} + 4638990991 p T^{4} - 68812974321826 p^{2} T^{6} + 15103040410818598721 p^{2} T^{8} - \)\(63\!\cdots\!72\)\( p^{4} T^{10} + \)\(61\!\cdots\!06\)\( p^{5} T^{12} - \)\(28\!\cdots\!08\)\( p^{7} T^{14} + \)\(70\!\cdots\!26\)\( p^{8} T^{16} - \)\(28\!\cdots\!08\)\( p^{21} T^{18} + \)\(61\!\cdots\!06\)\( p^{33} T^{20} - \)\(63\!\cdots\!72\)\( p^{46} T^{22} + 15103040410818598721 p^{58} T^{24} - 68812974321826 p^{72} T^{26} + 4638990991 p^{85} T^{28} - 54454 p^{99} T^{30} + p^{112} T^{32} \) |
| 11 | \( 1 - 62890118 T^{2} + 2248100321019611 T^{4} - \)\(37\!\cdots\!82\)\( T^{6} - \)\(43\!\cdots\!59\)\( T^{8} + \)\(16\!\cdots\!56\)\( T^{10} - \)\(33\!\cdots\!86\)\( T^{12} + \)\(15\!\cdots\!24\)\( T^{14} + \)\(33\!\cdots\!42\)\( T^{16} + \)\(15\!\cdots\!24\)\( p^{14} T^{18} - \)\(33\!\cdots\!86\)\( p^{28} T^{20} + \)\(16\!\cdots\!56\)\( p^{42} T^{22} - \)\(43\!\cdots\!59\)\( p^{56} T^{24} - \)\(37\!\cdots\!82\)\( p^{70} T^{26} + 2248100321019611 p^{84} T^{28} - 62890118 p^{98} T^{30} + p^{112} T^{32} \) | |
| 13 | \( ( 1 + 7070 T + 174407501 T^{2} + 803229292950 T^{3} + 13765673197267652 T^{4} + 803229292950 p^{7} T^{5} + 174407501 p^{14} T^{6} + 7070 p^{21} T^{7} + p^{28} T^{8} )^{4} \) | |
| 17 | \( 1 - 1751932304 T^{2} + 1577638554268391204 T^{4} - \)\(90\!\cdots\!84\)\( T^{6} + \)\(33\!\cdots\!86\)\( T^{8} - \)\(66\!\cdots\!72\)\( T^{10} - \)\(11\!\cdots\!64\)\( T^{12} + \)\(98\!\cdots\!64\)\( p T^{14} - \)\(85\!\cdots\!33\)\( T^{16} + \)\(98\!\cdots\!64\)\( p^{15} T^{18} - \)\(11\!\cdots\!64\)\( p^{28} T^{20} - \)\(66\!\cdots\!72\)\( p^{42} T^{22} + \)\(33\!\cdots\!86\)\( p^{56} T^{24} - \)\(90\!\cdots\!84\)\( p^{70} T^{26} + 1577638554268391204 p^{84} T^{28} - 1751932304 p^{98} T^{30} + p^{112} T^{32} \) | |
| 19 | \( ( 1 - 21112 T - 1573523151 T^{2} - 1564937846696 T^{3} + 1596143042179258997 T^{4} + \)\(14\!\cdots\!52\)\( p T^{5} - \)\(87\!\cdots\!02\)\( T^{6} - \)\(17\!\cdots\!56\)\( T^{7} + \)\(55\!\cdots\!26\)\( T^{8} - \)\(17\!\cdots\!56\)\( p^{7} T^{9} - \)\(87\!\cdots\!02\)\( p^{14} T^{10} + \)\(14\!\cdots\!52\)\( p^{22} T^{11} + 1596143042179258997 p^{28} T^{12} - 1564937846696 p^{35} T^{13} - 1573523151 p^{42} T^{14} - 21112 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 23 | \( 1 - 7611111776 T^{2} + 15330428710089781124 T^{4} + \)\(74\!\cdots\!24\)\( T^{6} - \)\(42\!\cdots\!54\)\( T^{8} + \)\(16\!\cdots\!72\)\( T^{10} + \)\(28\!\cdots\!36\)\( T^{12} - \)\(21\!\cdots\!08\)\( T^{14} - \)\(22\!\cdots\!73\)\( T^{16} - \)\(21\!\cdots\!08\)\( p^{14} T^{18} + \)\(28\!\cdots\!36\)\( p^{28} T^{20} + \)\(16\!\cdots\!72\)\( p^{42} T^{22} - \)\(42\!\cdots\!54\)\( p^{56} T^{24} + \)\(74\!\cdots\!24\)\( p^{70} T^{26} + 15330428710089781124 p^{84} T^{28} - 7611111776 p^{98} T^{30} + p^{112} T^{32} \) | |
| 29 | \( ( 1 + 19278091022 T^{2} + \)\(48\!\cdots\!93\)\( T^{4} + \)\(10\!\cdots\!74\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{8} + \)\(10\!\cdots\!74\)\( p^{14} T^{10} + \)\(48\!\cdots\!93\)\( p^{28} T^{12} + 19278091022 p^{42} T^{14} + p^{56} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 82376 T - 102289038874 T^{2} + 134688852009488 p T^{3} + \)\(68\!\cdots\!17\)\( T^{4} - \)\(16\!\cdots\!16\)\( T^{5} - \)\(29\!\cdots\!58\)\( T^{6} + \)\(14\!\cdots\!28\)\( T^{7} + \)\(96\!\cdots\!76\)\( T^{8} + \)\(14\!\cdots\!28\)\( p^{7} T^{9} - \)\(29\!\cdots\!58\)\( p^{14} T^{10} - \)\(16\!\cdots\!16\)\( p^{21} T^{11} + \)\(68\!\cdots\!17\)\( p^{28} T^{12} + 134688852009488 p^{36} T^{13} - 102289038874 p^{42} T^{14} - 82376 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 323990 T - 201771957249 T^{2} - 36039171093768590 T^{3} + \)\(88\!\cdots\!17\)\( p T^{4} + \)\(15\!\cdots\!40\)\( T^{5} - \)\(41\!\cdots\!06\)\( T^{6} - \)\(89\!\cdots\!80\)\( T^{7} + \)\(38\!\cdots\!74\)\( T^{8} - \)\(89\!\cdots\!80\)\( p^{7} T^{9} - \)\(41\!\cdots\!06\)\( p^{14} T^{10} + \)\(15\!\cdots\!40\)\( p^{21} T^{11} + \)\(88\!\cdots\!17\)\( p^{29} T^{12} - 36039171093768590 p^{35} T^{13} - 201771957249 p^{42} T^{14} + 323990 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 41 | \( ( 1 + 1109357565448 T^{2} + \)\(61\!\cdots\!48\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!56\)\( p^{14} T^{10} + \)\(61\!\cdots\!48\)\( p^{28} T^{12} + 1109357565448 p^{42} T^{14} + p^{56} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 335360 T + 1003871014891 T^{2} - 272708680844108000 T^{3} + \)\(39\!\cdots\!92\)\( T^{4} - 272708680844108000 p^{7} T^{5} + 1003871014891 p^{14} T^{6} - 335360 p^{21} T^{7} + p^{28} T^{8} )^{4} \) | |
| 47 | \( 1 - 1876826056264 T^{2} + \)\(14\!\cdots\!44\)\( T^{4} - \)\(68\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!06\)\( T^{8} - \)\(21\!\cdots\!12\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{12} - \)\(62\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!27\)\( T^{16} - \)\(62\!\cdots\!32\)\( p^{14} T^{18} + \)\(13\!\cdots\!36\)\( p^{28} T^{20} - \)\(21\!\cdots\!12\)\( p^{42} T^{22} + \)\(30\!\cdots\!06\)\( p^{56} T^{24} - \)\(68\!\cdots\!04\)\( p^{70} T^{26} + \)\(14\!\cdots\!44\)\( p^{84} T^{28} - 1876826056264 p^{98} T^{30} + p^{112} T^{32} \) | |
| 53 | \( 1 - 5688319346686 T^{2} + \)\(15\!\cdots\!19\)\( T^{4} - \)\(30\!\cdots\!46\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} - \)\(76\!\cdots\!88\)\( T^{10} + \)\(10\!\cdots\!86\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(18\!\cdots\!02\)\( T^{16} - \)\(14\!\cdots\!68\)\( p^{14} T^{18} + \)\(10\!\cdots\!86\)\( p^{28} T^{20} - \)\(76\!\cdots\!88\)\( p^{42} T^{22} + \)\(50\!\cdots\!81\)\( p^{56} T^{24} - \)\(30\!\cdots\!46\)\( p^{70} T^{26} + \)\(15\!\cdots\!19\)\( p^{84} T^{28} - 5688319346686 p^{98} T^{30} + p^{112} T^{32} \) | |
| 59 | \( 1 - 9343105737542 T^{2} + \)\(37\!\cdots\!31\)\( T^{4} - \)\(10\!\cdots\!98\)\( T^{6} + \)\(25\!\cdots\!81\)\( T^{8} - \)\(51\!\cdots\!96\)\( T^{10} + \)\(61\!\cdots\!14\)\( T^{12} - \)\(10\!\cdots\!04\)\( T^{14} + \)\(36\!\cdots\!62\)\( T^{16} - \)\(10\!\cdots\!04\)\( p^{14} T^{18} + \)\(61\!\cdots\!14\)\( p^{28} T^{20} - \)\(51\!\cdots\!96\)\( p^{42} T^{22} + \)\(25\!\cdots\!81\)\( p^{56} T^{24} - \)\(10\!\cdots\!98\)\( p^{70} T^{26} + \)\(37\!\cdots\!31\)\( p^{84} T^{28} - 9343105737542 p^{98} T^{30} + p^{112} T^{32} \) | |
| 61 | \( ( 1 + 2159668 T - 6040231341304 T^{2} - 12727224889326574648 T^{3} + \)\(27\!\cdots\!86\)\( T^{4} + \)\(39\!\cdots\!44\)\( T^{5} - \)\(10\!\cdots\!96\)\( T^{6} - \)\(42\!\cdots\!84\)\( T^{7} + \)\(39\!\cdots\!07\)\( T^{8} - \)\(42\!\cdots\!84\)\( p^{7} T^{9} - \)\(10\!\cdots\!96\)\( p^{14} T^{10} + \)\(39\!\cdots\!44\)\( p^{21} T^{11} + \)\(27\!\cdots\!86\)\( p^{28} T^{12} - 12727224889326574648 p^{35} T^{13} - 6040231341304 p^{42} T^{14} + 2159668 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 67 | \( ( 1 + 1952880 T - 4074135876479 T^{2} + 38904781874328107520 T^{3} + \)\(85\!\cdots\!29\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(86\!\cdots\!34\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} - \)\(27\!\cdots\!86\)\( T^{8} + \)\(21\!\cdots\!40\)\( p^{7} T^{9} + \)\(86\!\cdots\!34\)\( p^{14} T^{10} - \)\(16\!\cdots\!20\)\( p^{21} T^{11} + \)\(85\!\cdots\!29\)\( p^{28} T^{12} + 38904781874328107520 p^{35} T^{13} - 4074135876479 p^{42} T^{14} + 1952880 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 71 | \( ( 1 + 50110911624488 T^{2} + \)\(12\!\cdots\!68\)\( T^{4} + \)\(19\!\cdots\!36\)\( T^{6} + \)\(21\!\cdots\!10\)\( T^{8} + \)\(19\!\cdots\!36\)\( p^{14} T^{10} + \)\(12\!\cdots\!68\)\( p^{28} T^{12} + 50110911624488 p^{42} T^{14} + p^{56} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 3235890 T - 29397615639701 T^{2} + 43900547376750355410 T^{3} + \)\(68\!\cdots\!89\)\( T^{4} - \)\(37\!\cdots\!40\)\( T^{5} - \)\(10\!\cdots\!94\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!54\)\( T^{8} + \)\(21\!\cdots\!20\)\( p^{7} T^{9} - \)\(10\!\cdots\!94\)\( p^{14} T^{10} - \)\(37\!\cdots\!40\)\( p^{21} T^{11} + \)\(68\!\cdots\!89\)\( p^{28} T^{12} + 43900547376750355410 p^{35} T^{13} - 29397615639701 p^{42} T^{14} - 3235890 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 3046552 T - 46967931496306 T^{2} - \)\(15\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} + \)\(33\!\cdots\!12\)\( T^{5} - \)\(24\!\cdots\!02\)\( T^{6} - \)\(26\!\cdots\!44\)\( T^{7} + \)\(51\!\cdots\!96\)\( T^{8} - \)\(26\!\cdots\!44\)\( p^{7} T^{9} - \)\(24\!\cdots\!02\)\( p^{14} T^{10} + \)\(33\!\cdots\!12\)\( p^{21} T^{11} + \)\(11\!\cdots\!77\)\( p^{28} T^{12} - \)\(15\!\cdots\!84\)\( p^{35} T^{13} - 46967931496306 p^{42} T^{14} + 3046552 p^{49} T^{15} + p^{56} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 23143013697446 T^{2} + \)\(42\!\cdots\!37\)\( T^{4} + \)\(11\!\cdots\!18\)\( T^{6} + \)\(91\!\cdots\!80\)\( T^{8} + \)\(11\!\cdots\!18\)\( p^{14} T^{10} + \)\(42\!\cdots\!37\)\( p^{28} T^{12} + 23143013697446 p^{42} T^{14} + p^{56} T^{16} )^{2} \) | |
| 89 | \( 1 - 102006157521952 T^{2} + \)\(37\!\cdots\!36\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{6} - \)\(18\!\cdots\!34\)\( T^{8} + \)\(69\!\cdots\!24\)\( T^{10} + \)\(25\!\cdots\!44\)\( T^{12} - \)\(11\!\cdots\!24\)\( T^{14} + \)\(70\!\cdots\!47\)\( T^{16} - \)\(11\!\cdots\!24\)\( p^{14} T^{18} + \)\(25\!\cdots\!44\)\( p^{28} T^{20} + \)\(69\!\cdots\!24\)\( p^{42} T^{22} - \)\(18\!\cdots\!34\)\( p^{56} T^{24} + \)\(14\!\cdots\!12\)\( p^{70} T^{26} + \)\(37\!\cdots\!36\)\( p^{84} T^{28} - 102006157521952 p^{98} T^{30} + p^{112} T^{32} \) | |
| 97 | \( ( 1 + 6785310 T + 143082320790377 T^{2} + \)\(93\!\cdots\!90\)\( T^{3} + \)\(11\!\cdots\!64\)\( T^{4} + \)\(93\!\cdots\!90\)\( p^{7} T^{5} + 143082320790377 p^{14} T^{6} + 6785310 p^{21} T^{7} + p^{28} T^{8} )^{4} \) | |
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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
−2.10852900048182298097533728272, −2.10753301742885692228728690664, −1.99293402345879507107371153901, −1.89495337260695133117491748178, −1.78588238399791664907258739073, −1.56360838039804702958263333052, −1.55093840107216416976266542117, −1.53828735351925056179386566235, −1.45082065343894091580955085589, −1.35730523348083663826886043989, −1.31604002762442467840688181910, −1.31405453113246736353126097017, −1.22954042517745053427139224459, −1.19370101748091007767074527603, −0.910766372001103737422720415968, −0.69987957666320816424478386156, −0.65816637014139211596068865218, −0.60147936039658374280973889753, −0.55113192819840708759356741865, −0.55008896058878543023097529060, −0.52240100222114391536864137158, −0.43940009785767311320573301581, −0.31136013973892167230369183641, −0.07796492368229624761206507996, −0.03238511865991817837331678201, 0.03238511865991817837331678201, 0.07796492368229624761206507996, 0.31136013973892167230369183641, 0.43940009785767311320573301581, 0.52240100222114391536864137158, 0.55008896058878543023097529060, 0.55113192819840708759356741865, 0.60147936039658374280973889753, 0.65816637014139211596068865218, 0.69987957666320816424478386156, 0.910766372001103737422720415968, 1.19370101748091007767074527603, 1.22954042517745053427139224459, 1.31405453113246736353126097017, 1.31604002762442467840688181910, 1.35730523348083663826886043989, 1.45082065343894091580955085589, 1.53828735351925056179386566235, 1.55093840107216416976266542117, 1.56360838039804702958263333052, 1.78588238399791664907258739073, 1.89495337260695133117491748178, 1.99293402345879507107371153901, 2.10753301742885692228728690664, 2.10852900048182298097533728272
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.