Dirichlet series
| L(s) = 1 | − 8·4-s − 6·9-s + 12·11-s + 36·16-s + 48·23-s + 16·25-s − 12·29-s + 48·36-s + 4·37-s + 4·43-s − 96·44-s − 120·64-s + 56·67-s + 8·79-s + 18·81-s − 384·92-s − 72·99-s − 128·100-s − 132·107-s + 28·109-s + 96·116-s + 38·121-s + 127-s + 131-s + 137-s + 139-s − 216·144-s + ⋯ |
| L(s) = 1 | − 4·4-s − 2·9-s + 3.61·11-s + 9·16-s + 10.0·23-s + 16/5·25-s − 2.22·29-s + 8·36-s + 0.657·37-s + 0.609·43-s − 14.4·44-s − 15·64-s + 6.84·67-s + 0.900·79-s + 2·81-s − 40.0·92-s − 7.23·99-s − 12.7·100-s − 12.7·107-s + 2.68·109-s + 8.91·116-s + 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 18·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{32} \cdot 7^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.66379\times 10^{13}\) |
| Root analytic conductor: | \(2.65382\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03250122632\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03250122632\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 + 2 p T^{2} + 2 p^{2} T^{4} + 8 p^{2} T^{6} + 31 p^{2} T^{8} + 8 p^{4} T^{10} + 2 p^{6} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 576968 p^{2} T^{18} - 59414 p^{4} T^{20} + 1416 p^{6} T^{22} + 1481 p^{8} T^{24} - 584 p^{10} T^{26} + 123 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \) |
| 11 | \( ( 1 - 6 T + 35 T^{2} - 138 T^{3} + 481 T^{4} - 1512 T^{5} + 3854 T^{6} - 13116 T^{7} + 37618 T^{8} - 13116 p T^{9} + 3854 p^{2} T^{10} - 1512 p^{3} T^{11} + 481 p^{4} T^{12} - 138 p^{5} T^{13} + 35 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 13 | \( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 2987085740 p^{2} T^{18} + 232506496 p^{4} T^{20} + 16951644 p^{6} T^{22} + 1082018 p^{8} T^{24} + 57352 p^{10} T^{26} + 2376 p^{12} T^{28} + 68 p^{14} T^{30} + p^{16} T^{32} \) | |
| 17 | \( 1 - 94 T^{2} + 4551 T^{4} - 9082 p T^{6} + 4183433 T^{8} - 97483812 T^{10} + 2040829006 T^{12} - 39195392176 T^{14} + 694079215146 T^{16} - 39195392176 p^{2} T^{18} + 2040829006 p^{4} T^{20} - 97483812 p^{6} T^{22} + 4183433 p^{8} T^{24} - 9082 p^{11} T^{26} + 4551 p^{12} T^{28} - 94 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 + 98 T^{2} + 5058 T^{4} + 175540 T^{6} + 237119 p T^{8} + 87611580 T^{10} + 1299299878 T^{12} + 15453626030 T^{14} + 214883894484 T^{16} + 15453626030 p^{2} T^{18} + 1299299878 p^{4} T^{20} + 87611580 p^{6} T^{22} + 237119 p^{9} T^{24} + 175540 p^{10} T^{26} + 5058 p^{12} T^{28} + 98 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 24 T + 317 T^{2} - 3000 T^{3} + 22111 T^{4} - 134028 T^{5} + 704756 T^{6} - 3411156 T^{7} + 16228318 T^{8} - 3411156 p T^{9} + 704756 p^{2} T^{10} - 134028 p^{3} T^{11} + 22111 p^{4} T^{12} - 3000 p^{5} T^{13} + 317 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 + 6 T + 98 T^{2} + 516 T^{3} + 4846 T^{4} + 25650 T^{5} + 193448 T^{6} + 972210 T^{7} + 6347347 T^{8} + 972210 p T^{9} + 193448 p^{2} T^{10} + 25650 p^{3} T^{11} + 4846 p^{4} T^{12} + 516 p^{5} T^{13} + 98 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 - 104 T^{2} + 6928 T^{4} - 311816 T^{6} + 11112958 T^{8} - 311816 p^{2} T^{10} + 6928 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 - 2 T - 42 T^{2} - 16 T^{3} + 38 T^{4} + 4854 T^{5} + 32488 T^{6} - 173978 T^{7} - 411165 T^{8} - 173978 p T^{9} + 32488 p^{2} T^{10} + 4854 p^{3} T^{11} + 38 p^{4} T^{12} - 16 p^{5} T^{13} - 42 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 6221865664 p T^{14} + 9422146980954 T^{16} + 6221865664 p^{3} T^{18} - 8583026738 p^{4} T^{20} - 280356900 p^{6} T^{22} + 5171081 p^{8} T^{24} + 150526 p^{10} T^{26} - 1569 p^{12} T^{28} - 70 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 2 T - 3 p T^{2} - 46 T^{3} + 9833 T^{4} + 11184 T^{5} - 521114 T^{6} - 232628 T^{7} + 22298490 T^{8} - 232628 p T^{9} - 521114 p^{2} T^{10} + 11184 p^{3} T^{11} + 9833 p^{4} T^{12} - 46 p^{5} T^{13} - 3 p^{7} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 136 T^{2} + 10480 T^{4} + 599848 T^{6} + 29784382 T^{8} + 599848 p^{2} T^{10} + 10480 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) | |
| 59 | \( ( 1 + 178 T^{2} + 11041 T^{4} + 234574 T^{6} - 307472 T^{8} + 234574 p^{2} T^{10} + 11041 p^{4} T^{12} + 178 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 61 | \( ( 1 - 248 T^{2} + 33313 T^{4} - 3104588 T^{6} + 217292788 T^{8} - 3104588 p^{2} T^{10} + 33313 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 67 | \( ( 1 - 14 T + 157 T^{2} - 1154 T^{3} + 11152 T^{4} - 1154 p T^{5} + 157 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 71 | \( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 13986142 p^{2} T^{10} + 105553 p^{4} T^{12} - 478 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 73 | \( 1 + 362 T^{2} + 66999 T^{4} + 8550862 T^{6} + 840900233 T^{8} + 64348718460 T^{10} + 3848101857934 T^{12} + 198223620850304 T^{14} + 11891998605677418 T^{16} + 198223620850304 p^{2} T^{18} + 3848101857934 p^{4} T^{20} + 64348718460 p^{6} T^{22} + 840900233 p^{8} T^{24} + 8550862 p^{10} T^{26} + 66999 p^{12} T^{28} + 362 p^{14} T^{30} + p^{16} T^{32} \) | |
| 79 | \( ( 1 - 2 T + 187 T^{2} + 304 T^{3} + 15862 T^{4} + 304 p T^{5} + 187 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 83 | \( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 11841264313180 p^{2} T^{18} - 160886956928 p^{4} T^{20} + 3596762100 p^{6} T^{22} + 87112226 p^{8} T^{24} - 595880 p^{10} T^{26} - 13368 p^{12} T^{28} + 44 p^{14} T^{30} + p^{16} T^{32} \) | |
| 89 | \( 1 - 496 T^{2} + 126612 T^{4} - 22679456 T^{6} + 3259241258 T^{8} - 407089119312 T^{10} + 46038542478544 T^{12} - 4754185780844944 T^{14} + 445332554542016595 T^{16} - 4754185780844944 p^{2} T^{18} + 46038542478544 p^{4} T^{20} - 407089119312 p^{6} T^{22} + 3259241258 p^{8} T^{24} - 22679456 p^{10} T^{26} + 126612 p^{12} T^{28} - 496 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 73367970993632 p^{2} T^{18} - 114467134418 p^{4} T^{20} - 25062425316 p^{6} T^{22} - 150210775 p^{8} T^{24} - 803858 p^{10} T^{26} + 11511 p^{12} T^{28} + 74 p^{14} T^{30} + p^{16} T^{32} \) | |
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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
−2.73796071969938767255466687906, −2.72976787249408411583587856489, −2.60325010098881176887055236681, −2.50251104155536634637322391672, −2.28956623355845193632810413868, −2.24954325368431915653955865672, −2.17914974178818764100338531620, −2.17523691815868291847584730992, −2.01892135632647361520354248230, −1.94628629956668636247032242094, −1.78732735211731380114650224022, −1.55619037073229275365053975842, −1.38361507077374396532515417267, −1.28489043551998518625079248510, −1.16321884396700978557865779389, −1.16284276674626848777552557581, −1.12076942665013579077635812358, −1.10172082220649300624825526206, −1.08008766164946601273802717641, −0.981096951439944518196630057216, −0.833071384143681845729544972047, −0.77088162248007306029638269857, −0.58759975036858138841270606990, −0.099705584474472892653758285990, −0.03648119600138499497867441989, 0.03648119600138499497867441989, 0.099705584474472892653758285990, 0.58759975036858138841270606990, 0.77088162248007306029638269857, 0.833071384143681845729544972047, 0.981096951439944518196630057216, 1.08008766164946601273802717641, 1.10172082220649300624825526206, 1.12076942665013579077635812358, 1.16284276674626848777552557581, 1.16321884396700978557865779389, 1.28489043551998518625079248510, 1.38361507077374396532515417267, 1.55619037073229275365053975842, 1.78732735211731380114650224022, 1.94628629956668636247032242094, 2.01892135632647361520354248230, 2.17523691815868291847584730992, 2.17914974178818764100338531620, 2.24954325368431915653955865672, 2.28956623355845193632810413868, 2.50251104155536634637322391672, 2.60325010098881176887055236681, 2.72976787249408411583587856489, 2.73796071969938767255466687906
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.