Dirichlet series
| L(s) = 1 | + 4·4-s + 2·7-s + 12·11-s + 6·16-s + 48·23-s + 16·25-s + 8·28-s + 12·29-s − 8·37-s + 4·43-s + 48·44-s − 2·49-s − 28·67-s + 24·77-s − 4·79-s + 192·92-s + 64·100-s − 56·109-s + 12·112-s + 48·116-s + 38·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.755·7-s + 3.61·11-s + 3/2·16-s + 10.0·23-s + 16/5·25-s + 1.51·28-s + 2.22·29-s − 1.31·37-s + 0.609·43-s + 7.23·44-s − 2/7·49-s − 3.42·67-s + 2.73·77-s − 0.450·79-s + 20.0·92-s + 32/5·100-s − 5.36·109-s + 1.13·112-s + 4.45·116-s + 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{16} \cdot 3^{48} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.74572\times 10^{7}\) |
| Root analytic conductor: | \(1.73733\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{16} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(72.36208561\) |
| \(L(\frac12)\) | \(\approx\) | \(72.36208561\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 2 T + 6 T^{2} + 8 T^{3} - 58 T^{4} + 222 T^{5} - 104 T^{6} - 662 T^{7} + 3483 T^{8} - 662 p T^{9} - 104 p^{2} T^{10} + 222 p^{3} T^{11} - 58 p^{4} T^{12} + 8 p^{5} T^{13} + 6 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 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 | \( ( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 124198 p^{2} T^{10} + 4285 p^{4} T^{12} + 94 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 19 | \( ( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 134984 p^{2} T^{10} + 4546 p^{4} T^{12} - 98 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 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 | \( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 70884338648 p^{2} T^{18} + 1870813504 p^{4} T^{20} + 148421160 p^{6} T^{22} + 4455362 p^{8} T^{24} + 96880 p^{10} T^{26} + 3888 p^{12} T^{28} + 104 p^{14} T^{30} + p^{16} T^{32} \) | |
| 37 | \( ( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 38 p T^{5} + 46 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 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 | \( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 1486359308360 p^{2} T^{18} - 30253322048 p^{4} T^{20} + 489880632 p^{6} T^{22} - 1533310 p^{8} T^{24} - 225584 p^{10} T^{26} + 8016 p^{12} T^{28} - 136 p^{14} T^{30} + p^{16} T^{32} \) | |
| 53 | \( ( 1 - p T^{2} )^{16} \) | |
| 59 | \( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 13219132050040 p^{2} T^{18} + 180649052698 p^{4} T^{20} - 3515943660 p^{6} T^{22} + 80456981 p^{8} T^{24} - 1496150 p^{10} T^{26} + 20643 p^{12} T^{28} - 178 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 32095759051208 p^{2} T^{18} + 457346362462 p^{4} T^{20} + 7198089144 p^{6} T^{22} + 122525357 p^{8} T^{24} + 2052448 p^{10} T^{26} + 28191 p^{12} T^{28} + 248 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 14 T + 39 T^{2} - 110 T^{3} - 2659 T^{4} - 53760 T^{5} - 92054 T^{6} + 2669060 T^{7} + 22240746 T^{8} + 2669060 p T^{9} - 92054 p^{2} T^{10} - 53760 p^{3} T^{11} - 2659 p^{4} T^{12} - 110 p^{5} T^{13} + 39 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 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 | \( ( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 7316714 p^{2} T^{10} + 64045 p^{4} T^{12} - 362 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 2 T - 183 T^{2} + 982 T^{3} + 19715 T^{4} - 144312 T^{5} - 491612 T^{6} + 7480148 T^{7} - 8945118 T^{8} + 7480148 p T^{9} - 491612 p^{2} T^{10} - 144312 p^{3} T^{11} + 19715 p^{4} T^{12} + 982 p^{5} T^{13} - 183 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 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 | \( ( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 18272464 p^{2} T^{10} + 119404 p^{4} T^{12} + 496 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 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
−3.19497013275602365911151996660, −3.08304445779108362641185327857, −2.95922330103898370400592643816, −2.94087104530900475085702836742, −2.83660499431385554635982477316, −2.79943301808077527856410302836, −2.62631417185724535138000997670, −2.52070856610407831100853057955, −2.50878354181396843450313673911, −2.39672813774276824402764573634, −2.27726906169723886926226184739, −2.25681828416401951661592498097, −2.04020815881690928892311926232, −1.87604717406807959069813666674, −1.69797315859040558244111900352, −1.46945971028880029227670026913, −1.45906329277505815785414400063, −1.32212933553603649494884239653, −1.28767326149428325860116780692, −1.23523868223938226474163272854, −1.19745041269591203298089414923, −1.16040691465158839983115760871, −0.863861593254812052036981271544, −0.63712773894055741352745792873, −0.58552571975631065600301525442, 0.58552571975631065600301525442, 0.63712773894055741352745792873, 0.863861593254812052036981271544, 1.16040691465158839983115760871, 1.19745041269591203298089414923, 1.23523868223938226474163272854, 1.28767326149428325860116780692, 1.32212933553603649494884239653, 1.45906329277505815785414400063, 1.46945971028880029227670026913, 1.69797315859040558244111900352, 1.87604717406807959069813666674, 2.04020815881690928892311926232, 2.25681828416401951661592498097, 2.27726906169723886926226184739, 2.39672813774276824402764573634, 2.50878354181396843450313673911, 2.52070856610407831100853057955, 2.62631417185724535138000997670, 2.79943301808077527856410302836, 2.83660499431385554635982477316, 2.94087104530900475085702836742, 2.95922330103898370400592643816, 3.08304445779108362641185327857, 3.19497013275602365911151996660
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.