Dirichlet series
| L(s) = 1 | − 2·3-s + 14·5-s + 10·7-s + 2·9-s + 32·13-s − 28·15-s + 44·17-s − 80·19-s − 20·21-s + 230·23-s + 76·25-s − 46·27-s + 140·35-s + 292·37-s − 64·39-s + 932·41-s − 458·43-s + 28·45-s − 766·47-s + 50·49-s − 88·51-s + 892·53-s + 160·57-s − 16·59-s − 2.56e3·61-s + 20·63-s + 448·65-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 1.25·5-s + 0.539·7-s + 2/27·9-s + 0.682·13-s − 0.481·15-s + 0.627·17-s − 0.965·19-s − 0.207·21-s + 2.08·23-s + 0.607·25-s − 0.327·27-s + 0.676·35-s + 1.29·37-s − 0.262·39-s + 3.55·41-s − 1.62·43-s + 0.0927·45-s − 2.37·47-s + 0.145·49-s − 0.241·51-s + 2.31·53-s + 0.371·57-s − 0.0353·59-s − 5.38·61-s + 0.0399·63-s + 0.854·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.61483\times 10^{10}\) |
| Root analytic conductor: | \(4.34518\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 5^{8} ,\ ( \ : [3/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.6718359557\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6718359557\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 - 14 T + 24 p T^{2} - 162 p T^{3} + 238 p^{2} T^{4} - 162 p^{4} T^{5} + 24 p^{7} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \) | |
| good | 3 | \( 1 + 2 T + 2 T^{2} + 46 T^{3} - 1360 T^{4} - 2602 T^{5} - 1426 T^{6} - 4886 T^{7} + 998398 T^{8} - 4886 p^{3} T^{9} - 1426 p^{6} T^{10} - 2602 p^{9} T^{11} - 1360 p^{12} T^{12} + 46 p^{15} T^{13} + 2 p^{18} T^{14} + 2 p^{21} T^{15} + p^{24} T^{16} \) |
| 7 | \( 1 - 10 T + 50 T^{2} + 4258 T^{3} + 38960 T^{4} - 3628286 T^{5} + 43400142 T^{6} - 434163690 T^{7} - 11482214498 T^{8} - 434163690 p^{3} T^{9} + 43400142 p^{6} T^{10} - 3628286 p^{9} T^{11} + 38960 p^{12} T^{12} + 4258 p^{15} T^{13} + 50 p^{18} T^{14} - 10 p^{21} T^{15} + p^{24} T^{16} \) | |
| 11 | \( 1 - 8524 T^{2} + 34127828 T^{4} - 83173817556 T^{6} + 134411171777782 T^{8} - 83173817556 p^{6} T^{10} + 34127828 p^{12} T^{12} - 8524 p^{18} T^{14} + p^{24} T^{16} \) | |
| 13 | \( 1 - 32 T + 512 T^{2} - 6528 T^{3} - 3879012 T^{4} + 26493888 T^{5} + 1159557120 T^{6} - 122631557024 T^{7} + 16941340000550 T^{8} - 122631557024 p^{3} T^{9} + 1159557120 p^{6} T^{10} + 26493888 p^{9} T^{11} - 3879012 p^{12} T^{12} - 6528 p^{15} T^{13} + 512 p^{18} T^{14} - 32 p^{21} T^{15} + p^{24} T^{16} \) | |
| 17 | \( 1 - 44 T + 968 T^{2} - 129572 T^{3} - 44831476 T^{4} + 1708038260 T^{5} - 23362363080 T^{6} - 35298913700 T^{7} + 1020085916893030 T^{8} - 35298913700 p^{3} T^{9} - 23362363080 p^{6} T^{10} + 1708038260 p^{9} T^{11} - 44831476 p^{12} T^{12} - 129572 p^{15} T^{13} + 968 p^{18} T^{14} - 44 p^{21} T^{15} + p^{24} T^{16} \) | |
| 19 | \( ( 1 + 40 T + 20204 T^{2} + 682920 T^{3} + 191471446 T^{4} + 682920 p^{3} T^{5} + 20204 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 23 | \( 1 - 10 p T + 50 p^{2} T^{2} - 5650466 T^{3} + 653081040 T^{4} - 21285292418 T^{5} + 3585506756718 T^{6} - 135360139109510 T^{7} - 56795360851107362 T^{8} - 135360139109510 p^{3} T^{9} + 3585506756718 p^{6} T^{10} - 21285292418 p^{9} T^{11} + 653081040 p^{12} T^{12} - 5650466 p^{15} T^{13} + 50 p^{20} T^{14} - 10 p^{22} T^{15} + p^{24} T^{16} \) | |
| 29 | \( 1 - 80472 T^{2} + 3106511452 T^{4} - 80137039768488 T^{6} + 1842648666818932134 T^{8} - 80137039768488 p^{6} T^{10} + 3106511452 p^{12} T^{12} - 80472 p^{18} T^{14} + p^{24} T^{16} \) | |
| 31 | \( 1 - 62748 T^{2} + 3477417012 T^{4} - 135905662590692 T^{6} + 4689277571795603094 T^{8} - 135905662590692 p^{6} T^{10} + 3477417012 p^{12} T^{12} - 62748 p^{18} T^{14} + p^{24} T^{16} \) | |
| 37 | \( 1 - 292 T + 42632 T^{2} - 5877852 T^{3} - 4509181140 T^{4} + 1091770393116 T^{5} - 109286972362440 T^{6} - 2156121499050556 T^{7} + 10042945903134047558 T^{8} - 2156121499050556 p^{3} T^{9} - 109286972362440 p^{6} T^{10} + 1091770393116 p^{9} T^{11} - 4509181140 p^{12} T^{12} - 5877852 p^{15} T^{13} + 42632 p^{18} T^{14} - 292 p^{21} T^{15} + p^{24} T^{16} \) | |
| 41 | \( ( 1 - 466 T + 230064 T^{2} - 58858398 T^{3} + 19118241406 T^{4} - 58858398 p^{3} T^{5} + 230064 p^{6} T^{6} - 466 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 43 | \( 1 + 458 T + 104882 T^{2} + 38116054 T^{3} + 1478176560 T^{4} - 2685524362738 T^{5} - 356185768434 p^{2} T^{6} - 5926476758433338 p T^{7} - 97733452598943785282 T^{8} - 5926476758433338 p^{4} T^{9} - 356185768434 p^{8} T^{10} - 2685524362738 p^{9} T^{11} + 1478176560 p^{12} T^{12} + 38116054 p^{15} T^{13} + 104882 p^{18} T^{14} + 458 p^{21} T^{15} + p^{24} T^{16} \) | |
| 47 | \( 1 + 766 T + 293378 T^{2} + 102474810 T^{3} + 21852951824 T^{4} - 2185479617782 T^{5} - 2834709345174434 T^{6} - 1436973169139510898 T^{7} - \)\(56\!\cdots\!10\)\( T^{8} - 1436973169139510898 p^{3} T^{9} - 2834709345174434 p^{6} T^{10} - 2185479617782 p^{9} T^{11} + 21852951824 p^{12} T^{12} + 102474810 p^{15} T^{13} + 293378 p^{18} T^{14} + 766 p^{21} T^{15} + p^{24} T^{16} \) | |
| 53 | \( 1 - 892 T + 397832 T^{2} - 226500196 T^{3} + 82564790060 T^{4} - 2638376352348 T^{5} - 4842314458836296 T^{6} + 4720656590664156636 T^{7} - \)\(29\!\cdots\!62\)\( T^{8} + 4720656590664156636 p^{3} T^{9} - 4842314458836296 p^{6} T^{10} - 2638376352348 p^{9} T^{11} + 82564790060 p^{12} T^{12} - 226500196 p^{15} T^{13} + 397832 p^{18} T^{14} - 892 p^{21} T^{15} + p^{24} T^{16} \) | |
| 59 | \( ( 1 + 8 T + 437068 T^{2} + 18916040 T^{3} + 111203970486 T^{4} + 18916040 p^{3} T^{5} + 437068 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 61 | \( ( 1 + 1282 T + 1358232 T^{2} + 909520246 T^{3} + 513483644510 T^{4} + 909520246 p^{3} T^{5} + 1358232 p^{6} T^{6} + 1282 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 67 | \( 1 - 194 T + 18818 T^{2} - 124825806 T^{3} + 834573936 T^{4} + 32376168442698 T^{5} + 1494059231563758 T^{6} + 3650767611352235158 T^{7} - \)\(14\!\cdots\!34\)\( T^{8} + 3650767611352235158 p^{3} T^{9} + 1494059231563758 p^{6} T^{10} + 32376168442698 p^{9} T^{11} + 834573936 p^{12} T^{12} - 124825806 p^{15} T^{13} + 18818 p^{18} T^{14} - 194 p^{21} T^{15} + p^{24} T^{16} \) | |
| 71 | \( 1 - 1114300 T^{2} + 598375465460 T^{4} - 290099172812677764 T^{6} + \)\(12\!\cdots\!90\)\( T^{8} - 290099172812677764 p^{6} T^{10} + 598375465460 p^{12} T^{12} - 1114300 p^{18} T^{14} + p^{24} T^{16} \) | |
| 73 | \( 1 + 1744 T + 1520768 T^{2} + 833868432 T^{3} + 276194091196 T^{4} + 94291850811248 T^{5} + 92086133077927296 T^{6} + 87931311923302359856 T^{7} + \)\(64\!\cdots\!54\)\( T^{8} + 87931311923302359856 p^{3} T^{9} + 92086133077927296 p^{6} T^{10} + 94291850811248 p^{9} T^{11} + 276194091196 p^{12} T^{12} + 833868432 p^{15} T^{13} + 1520768 p^{18} T^{14} + 1744 p^{21} T^{15} + p^{24} T^{16} \) | |
| 79 | \( ( 1 - 1608 T + 2683644 T^{2} - 2441147816 T^{3} + 2160325296198 T^{4} - 2441147816 p^{3} T^{5} + 2683644 p^{6} T^{6} - 1608 p^{9} T^{7} + p^{12} T^{8} )^{2} \) | |
| 83 | \( 1 + 3762 T + 7076322 T^{2} + 9679307854 T^{3} + 10795655378256 T^{4} + 10021238795574134 T^{5} + 8150866957631680334 T^{6} + \)\(62\!\cdots\!98\)\( T^{7} + \)\(46\!\cdots\!06\)\( T^{8} + \)\(62\!\cdots\!98\)\( p^{3} T^{9} + 8150866957631680334 p^{6} T^{10} + 10021238795574134 p^{9} T^{11} + 10795655378256 p^{12} T^{12} + 9679307854 p^{15} T^{13} + 7076322 p^{18} T^{14} + 3762 p^{21} T^{15} + p^{24} T^{16} \) | |
| 89 | \( 1 - 1919560 T^{2} + 2619010310876 T^{4} - 2594659557541013880 T^{6} + \)\(21\!\cdots\!06\)\( T^{8} - 2594659557541013880 p^{6} T^{10} + 2619010310876 p^{12} T^{12} - 1919560 p^{18} T^{14} + p^{24} T^{16} \) | |
| 97 | \( 1 + 680 T + 231200 T^{2} - 398706440 T^{3} - 358700279364 T^{4} + 486641844960680 T^{5} + 493331371810956000 T^{6} + 87968718802581283000 T^{7} - \)\(41\!\cdots\!74\)\( T^{8} + 87968718802581283000 p^{3} T^{9} + 493331371810956000 p^{6} T^{10} + 486641844960680 p^{9} T^{11} - 358700279364 p^{12} T^{12} - 398706440 p^{15} T^{13} + 231200 p^{18} T^{14} + 680 p^{21} T^{15} + p^{24} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.61098757314837825748197202823, −4.60847307598451141760919951383, −4.52439774617222629719468338378, −4.32256637171250957127084868823, −4.05876493053585308572135264652, −4.05344799105831637905687632623, −3.82837906124257491404672667262, −3.60048823300896749537347893428, −3.40793375345303665960595511547, −3.13900147828126476632086683026, −3.08084423536727237046307236625, −2.99424324214342750249382118077, −2.72604492737869838093028411893, −2.50997027765869731723636884903, −2.35092724485817471671373349349, −2.22226651537680248961470705629, −2.15693127226471530534225131711, −1.54441271286045345041384570019, −1.54318779092595775357385688075, −1.25971688128552871518907501842, −1.24945466917599890059261417465, −1.01953024293096630154431601362, −0.960246755297020521937782363200, −0.16504988857285309458026749977, −0.13041507587875194659224240964, 0.13041507587875194659224240964, 0.16504988857285309458026749977, 0.960246755297020521937782363200, 1.01953024293096630154431601362, 1.24945466917599890059261417465, 1.25971688128552871518907501842, 1.54318779092595775357385688075, 1.54441271286045345041384570019, 2.15693127226471530534225131711, 2.22226651537680248961470705629, 2.35092724485817471671373349349, 2.50997027765869731723636884903, 2.72604492737869838093028411893, 2.99424324214342750249382118077, 3.08084423536727237046307236625, 3.13900147828126476632086683026, 3.40793375345303665960595511547, 3.60048823300896749537347893428, 3.82837906124257491404672667262, 4.05344799105831637905687632623, 4.05876493053585308572135264652, 4.32256637171250957127084868823, 4.52439774617222629719468338378, 4.60847307598451141760919951383, 4.61098757314837825748197202823
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.