Dirichlet series
| L(s) = 1 | − 8·2-s + 8·4-s − 96·8-s + 560·16-s + 2.01e4·25-s − 2.65e4·29-s + 1.53e3·32-s − 2.62e4·37-s + 4.42e3·49-s − 1.60e5·50-s + 4.18e4·53-s + 2.12e5·58-s + 2.22e4·64-s + 2.10e5·74-s − 3.53e4·98-s + 1.60e5·100-s − 3.35e5·106-s − 6.04e5·109-s + 3.84e5·113-s − 2.12e5·116-s + 1.99e6·121-s + 127-s − 1.52e5·128-s + 131-s + 137-s + 139-s − 2.10e5·148-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/4·4-s − 0.530·8-s + 0.546·16-s + 6.43·25-s − 5.87·29-s + 0.265·32-s − 3.15·37-s + 0.263·49-s − 9.10·50-s + 2.04·53-s + 8.30·58-s + 0.679·64-s + 4.46·74-s − 0.372·98-s + 1.60·100-s − 2.89·106-s − 4.87·109-s + 2.83·113-s − 1.46·116-s + 12.4·121-s + 5.50e−6·127-s − 0.823·128-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 0.788·148-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32} \cdot 3^{32} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.06964\times 10^{25}\) |
| Root analytic conductor: | \(6.35741\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [5/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(8.469074091\) |
| \(L(\frac12)\) | \(\approx\) | \(8.469074091\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T + 5 p^{2} T^{2} + 5 p^{5} T^{3} + 15 p^{6} T^{4} + 5 p^{10} T^{5} + 5 p^{12} T^{6} + p^{17} T^{7} + p^{20} T^{8} )^{2} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 632 p T^{2} + 816668 p^{2} T^{4} + 14200112312 p^{3} T^{6} + 13778850090 p^{7} T^{8} + 14200112312 p^{13} T^{10} + 816668 p^{22} T^{12} - 632 p^{31} T^{14} + p^{40} T^{16} \) | |
| good | 5 | \( ( 1 - 10056 T^{2} + 69598876 T^{4} - 325723131576 T^{6} + 1177095606611366 T^{8} - 325723131576 p^{10} T^{10} + 69598876 p^{20} T^{12} - 10056 p^{30} T^{14} + p^{40} T^{16} )^{2} \) |
| 11 | \( ( 1 - 999336 T^{2} + 476337829052 T^{4} - 139193321080762840 T^{6} + \)\(27\!\cdots\!94\)\( T^{8} - 139193321080762840 p^{10} T^{10} + 476337829052 p^{20} T^{12} - 999336 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 13 | \( ( 1 - 1300744 T^{2} + 992989290588 T^{4} - 546270224604567160 T^{6} + \)\(23\!\cdots\!74\)\( T^{8} - 546270224604567160 p^{10} T^{10} + 992989290588 p^{20} T^{12} - 1300744 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 17 | \( ( 1 - 3972232 T^{2} + 9137580611996 T^{4} - 14283035799546970808 T^{6} + \)\(20\!\cdots\!90\)\( T^{8} - 14283035799546970808 p^{10} T^{10} + 9137580611996 p^{20} T^{12} - 3972232 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 19 | \( ( 1 + 5276472 T^{2} + 22813716125148 T^{4} + 70922394429817060680 T^{6} + \)\(20\!\cdots\!06\)\( T^{8} + 70922394429817060680 p^{10} T^{10} + 22813716125148 p^{20} T^{12} + 5276472 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 23 | \( ( 1 - 41555208 T^{2} + 806551576509788 T^{4} - \)\(95\!\cdots\!00\)\( T^{6} + \)\(74\!\cdots\!94\)\( T^{8} - \)\(95\!\cdots\!00\)\( p^{10} T^{10} + 806551576509788 p^{20} T^{12} - 41555208 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 6648 T + 65607500 T^{2} + 297803740072 T^{3} + 1802505132079350 T^{4} + 297803740072 p^{5} T^{5} + 65607500 p^{10} T^{6} + 6648 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 31 | \( ( 1 + 127872760 T^{2} + 8202165018300700 T^{4} + \)\(36\!\cdots\!16\)\( T^{6} + \)\(12\!\cdots\!22\)\( T^{8} + \)\(36\!\cdots\!16\)\( p^{10} T^{10} + 8202165018300700 p^{20} T^{12} + 127872760 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 6568 T + 134356268 T^{2} - 177462047240 T^{3} + 5035841506176470 T^{4} - 177462047240 p^{5} T^{5} + 134356268 p^{10} T^{6} + 6568 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 41 | \( ( 1 - 205660872 T^{2} + 9227623649772508 T^{4} - \)\(46\!\cdots\!96\)\( T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(46\!\cdots\!96\)\( p^{10} T^{10} + 9227623649772508 p^{20} T^{12} - 205660872 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 43 | \( ( 1 - 862444968 T^{2} + 358529584311367612 T^{4} - \)\(93\!\cdots\!76\)\( T^{6} + \)\(16\!\cdots\!98\)\( T^{8} - \)\(93\!\cdots\!76\)\( p^{10} T^{10} + 358529584311367612 p^{20} T^{12} - 862444968 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 1201717624 T^{2} + 747095089914362780 T^{4} + \)\(29\!\cdots\!48\)\( T^{6} + \)\(80\!\cdots\!22\)\( T^{8} + \)\(29\!\cdots\!48\)\( p^{10} T^{10} + 747095089914362780 p^{20} T^{12} + 1201717624 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 53 | \( ( 1 - 10472 T + 1412350316 T^{2} - 12319406665016 T^{3} + 840021664521794454 T^{4} - 12319406665016 p^{5} T^{5} + 1412350316 p^{10} T^{6} - 10472 p^{15} T^{7} + p^{20} T^{8} )^{4} \) | |
| 59 | \( ( 1 + 5184810104 T^{2} + 12117475555842987548 T^{4} + \)\(16\!\cdots\!12\)\( T^{6} + \)\(14\!\cdots\!78\)\( T^{8} + \)\(16\!\cdots\!12\)\( p^{10} T^{10} + 12117475555842987548 p^{20} T^{12} + 5184810104 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 4560728584 T^{2} + 9977688053227399644 T^{4} - \)\(13\!\cdots\!52\)\( T^{6} + \)\(22\!\cdots\!58\)\( p T^{8} - \)\(13\!\cdots\!52\)\( p^{10} T^{10} + 9977688053227399644 p^{20} T^{12} - 4560728584 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 1346295784 T^{2} + 4461779133789702012 T^{4} - \)\(78\!\cdots\!36\)\( T^{6} + \)\(99\!\cdots\!42\)\( T^{8} - \)\(78\!\cdots\!36\)\( p^{10} T^{10} + 4461779133789702012 p^{20} T^{12} - 1346295784 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 8192611400 T^{2} + 36558172219298943836 T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!54\)\( T^{8} - \)\(10\!\cdots\!20\)\( p^{10} T^{10} + 36558172219298943836 p^{20} T^{12} - 8192611400 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 73 | \( ( 1 - 8709786312 T^{2} + 41105121806495510748 T^{4} - \)\(13\!\cdots\!72\)\( T^{6} + \)\(32\!\cdots\!58\)\( T^{8} - \)\(13\!\cdots\!72\)\( p^{10} T^{10} + 41105121806495510748 p^{20} T^{12} - 8709786312 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 79 | \( ( 1 - 12003414280 T^{2} + 75037801601829695388 T^{4} - \)\(32\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!58\)\( T^{8} - \)\(32\!\cdots\!88\)\( p^{10} T^{10} + 75037801601829695388 p^{20} T^{12} - 12003414280 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 83 | \( ( 1 + 19610385592 T^{2} + \)\(20\!\cdots\!68\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{6} + \)\(63\!\cdots\!34\)\( T^{8} + \)\(13\!\cdots\!20\)\( p^{10} T^{10} + \)\(20\!\cdots\!68\)\( p^{20} T^{12} + 19610385592 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 89 | \( ( 1 - 26604331848 T^{2} + \)\(35\!\cdots\!52\)\( T^{4} - \)\(32\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!54\)\( T^{8} - \)\(32\!\cdots\!20\)\( p^{10} T^{10} + \)\(35\!\cdots\!52\)\( p^{20} T^{12} - 26604331848 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
| 97 | \( ( 1 - 33439016200 T^{2} + \)\(67\!\cdots\!28\)\( T^{4} - \)\(90\!\cdots\!84\)\( T^{6} + \)\(90\!\cdots\!14\)\( T^{8} - \)\(90\!\cdots\!84\)\( p^{10} T^{10} + \)\(67\!\cdots\!28\)\( p^{20} T^{12} - 33439016200 p^{30} T^{14} + p^{40} T^{16} )^{2} \) | |
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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
−2.46392757010289166992567203796, −2.26199950686547092983717381991, −2.23460468617812466588515656211, −1.94741363009424698957619623509, −1.92229382157638603171615523604, −1.87825341574913983212127478066, −1.87287889159671862750360792139, −1.82909948575347557491398866035, −1.69766895650432270909612951102, −1.68070191731307427774086418963, −1.62771156233417483838782049017, −1.49049673993786579122372646521, −1.08729701845617197238337405767, −1.02446971262623140827981214180, −1.02378147280044881677074118501, −0.926421247938411806525810199086, −0.905683507645311269394759260018, −0.75691647703727980523677952700, −0.61395838284860064557682466708, −0.56848071278173644061761948204, −0.38917033246074463312388263748, −0.34864164497427543660915369676, −0.32591163893731231819804058623, −0.28176860926525857296849364135, −0.12398442581004411852020720616, 0.12398442581004411852020720616, 0.28176860926525857296849364135, 0.32591163893731231819804058623, 0.34864164497427543660915369676, 0.38917033246074463312388263748, 0.56848071278173644061761948204, 0.61395838284860064557682466708, 0.75691647703727980523677952700, 0.905683507645311269394759260018, 0.926421247938411806525810199086, 1.02378147280044881677074118501, 1.02446971262623140827981214180, 1.08729701845617197238337405767, 1.49049673993786579122372646521, 1.62771156233417483838782049017, 1.68070191731307427774086418963, 1.69766895650432270909612951102, 1.82909948575347557491398866035, 1.87287889159671862750360792139, 1.87825341574913983212127478066, 1.92229382157638603171615523604, 1.94741363009424698957619623509, 2.23460468617812466588515656211, 2.26199950686547092983717381991, 2.46392757010289166992567203796
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.