Dirichlet series
| L(s) = 1 | − 2·7-s − 12·11-s − 48·23-s + 16·25-s + 12·29-s − 8·37-s − 4·43-s − 2·49-s + 28·67-s + 24·77-s + 4·79-s − 56·109-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 96·161-s + 163-s + 167-s − 68·169-s + 173-s − 32·175-s + 179-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 3.61·11-s − 10.0·23-s + 16/5·25-s + 2.22·29-s − 1.31·37-s − 0.609·43-s − 2/7·49-s + 3.42·67-s + 2.73·77-s + 0.450·79-s − 5.36·109-s + 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 7.56·161-s + 0.0783·163-s + 0.0773·167-s − 5.23·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{48} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{64} \cdot 3^{48} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.33580\times 10^{22}\) |
| Root analytic conductor: | \(4.91393\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{64} \cdot 3^{48} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2433655261\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2433655261\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 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
−2.14956284193353438639065977019, −2.08729356707212144061518106666, −1.93381153651408915581846367631, −1.92898227771708857488957879302, −1.80011734228524007183759093562, −1.78819954516053344322758082449, −1.77732936110195526991086183092, −1.70125389670026418769538438799, −1.59669708982763601941658103714, −1.53368837545615769247711270293, −1.36431360993767287238165440278, −1.36271619458429914144653747618, −1.33674123627428007854371030844, −1.20277363312199472398141707590, −0.980019921759895930650159884963, −0.940668564392303221984320349764, −0.913536091751455614350517560345, −0.838194328422581196328556067859, −0.57844109585857873780466125014, −0.56498261337776937636888816072, −0.48468326016984530984498979122, −0.24898569664169838462018123366, −0.20304761126139529250283161600, −0.15665898566804663943202883330, −0.084836595438975286689586533003, 0.084836595438975286689586533003, 0.15665898566804663943202883330, 0.20304761126139529250283161600, 0.24898569664169838462018123366, 0.48468326016984530984498979122, 0.56498261337776937636888816072, 0.57844109585857873780466125014, 0.838194328422581196328556067859, 0.913536091751455614350517560345, 0.940668564392303221984320349764, 0.980019921759895930650159884963, 1.20277363312199472398141707590, 1.33674123627428007854371030844, 1.36271619458429914144653747618, 1.36431360993767287238165440278, 1.53368837545615769247711270293, 1.59669708982763601941658103714, 1.70125389670026418769538438799, 1.77732936110195526991086183092, 1.78819954516053344322758082449, 1.80011734228524007183759093562, 1.92898227771708857488957879302, 1.93381153651408915581846367631, 2.08729356707212144061518106666, 2.14956284193353438639065977019
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.