Dirichlet series
| L(s) = 1 | + 5·4-s + 5·9-s + 12·11-s + 10·16-s + 6·19-s + 7·25-s + 8·29-s + 40·31-s + 25·36-s − 14·41-s + 60·44-s + 66·49-s + 8·59-s + 16·61-s + 5·64-s − 4·71-s + 30·76-s + 8·79-s + 10·81-s − 2·89-s + 60·99-s + 35·100-s − 52·101-s + 24·109-s + 40·116-s − 78·121-s + 200·124-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 5/3·9-s + 3.61·11-s + 5/2·16-s + 1.37·19-s + 7/5·25-s + 1.48·29-s + 7.18·31-s + 25/6·36-s − 2.18·41-s + 9.04·44-s + 66/7·49-s + 1.04·59-s + 2.04·61-s + 5/8·64-s − 0.474·71-s + 3.44·76-s + 0.900·79-s + 10/9·81-s − 0.211·89-s + 6.03·99-s + 7/2·100-s − 5.17·101-s + 2.29·109-s + 3.71·116-s − 7.09·121-s + 17.9·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.45559\times 10^{13}\) |
| Root analytic conductor: | \(2.13341\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 19^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(59.52719398\) |
| \(L(\frac12)\) | \(\approx\) | \(59.52719398\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{5} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{5} \) | |
| 5 | \( 1 - 7 T^{2} + p T^{4} - 32 T^{5} + 132 T^{6} + 448 T^{7} - 479 T^{8} - 1408 T^{9} + 469 T^{10} - 1408 p T^{11} - 479 p^{2} T^{12} + 448 p^{3} T^{13} + 132 p^{4} T^{14} - 32 p^{5} T^{15} + p^{7} T^{16} - 7 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( ( 1 - 3 T - 10 T^{2} - 23 T^{3} + 241 T^{4} + 416 T^{5} + 241 p T^{6} - 23 p^{2} T^{7} - 10 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| good | 7 | \( ( 1 - 33 T^{2} + 619 T^{4} - 7974 T^{6} + 78413 T^{8} - 611071 T^{10} + 78413 p^{2} T^{12} - 7974 p^{4} T^{14} + 619 p^{6} T^{16} - 33 p^{8} T^{18} + p^{10} T^{20} )^{2} \) |
| 11 | \( ( 1 - 3 T + 42 T^{2} - 105 T^{3} + 821 T^{4} - 1604 T^{5} + 821 p T^{6} - 105 p^{2} T^{7} + 42 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 13 | \( 1 + 32 T^{2} + 10 p T^{4} - 8004 T^{6} - 82813 T^{8} + 1759688 T^{10} + 31044440 T^{12} - 237698572 T^{14} - 7560426707 T^{16} + 60941060 p^{2} T^{18} + 1300755776802 T^{20} + 60941060 p^{4} T^{22} - 7560426707 p^{4} T^{24} - 237698572 p^{6} T^{26} + 31044440 p^{8} T^{28} + 1759688 p^{10} T^{30} - 82813 p^{12} T^{32} - 8004 p^{14} T^{34} + 10 p^{17} T^{36} + 32 p^{18} T^{38} + p^{20} T^{40} \) | |
| 17 | \( 1 + 78 T^{2} + 3119 T^{4} + 81086 T^{6} + 1467111 T^{8} + 19222440 T^{10} + 154702394 T^{12} - 1227955216 T^{14} - 106876033875 T^{16} - 3294779997130 T^{18} - 66895695535959 T^{20} - 3294779997130 p^{2} T^{22} - 106876033875 p^{4} T^{24} - 1227955216 p^{6} T^{26} + 154702394 p^{8} T^{28} + 19222440 p^{10} T^{30} + 1467111 p^{12} T^{32} + 81086 p^{14} T^{34} + 3119 p^{16} T^{36} + 78 p^{18} T^{38} + p^{20} T^{40} \) | |
| 23 | \( 1 + 103 T^{2} + 5609 T^{4} + 222758 T^{6} + 6927294 T^{8} + 168812642 T^{10} + 3176325635 T^{12} + 39279770505 T^{14} - 59237949885 T^{16} - 18945834259160 T^{18} - 581888878638780 T^{20} - 18945834259160 p^{2} T^{22} - 59237949885 p^{4} T^{24} + 39279770505 p^{6} T^{26} + 3176325635 p^{8} T^{28} + 168812642 p^{10} T^{30} + 6927294 p^{12} T^{32} + 222758 p^{14} T^{34} + 5609 p^{16} T^{36} + 103 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 4 T - 75 T^{2} - 248 T^{3} + 5273 T^{4} + 21002 T^{5} - 71956 T^{6} - 1272906 T^{7} - 1212687 T^{8} + 12984190 T^{9} + 162375929 T^{10} + 12984190 p T^{11} - 1212687 p^{2} T^{12} - 1272906 p^{3} T^{13} - 71956 p^{4} T^{14} + 21002 p^{5} T^{15} + 5273 p^{6} T^{16} - 248 p^{7} T^{17} - 75 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 10 T + 92 T^{2} - 360 T^{3} + 1975 T^{4} - 4748 T^{5} + 1975 p T^{6} - 360 p^{2} T^{7} + 92 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 37 | \( ( 1 - 289 T^{2} + 38987 T^{4} - 3266658 T^{6} + 190492981 T^{8} - 8157392099 T^{10} + 190492981 p^{2} T^{12} - 3266658 p^{4} T^{14} + 38987 p^{6} T^{16} - 289 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 7 T - 51 T^{2} - 66 T^{3} + 1378 T^{4} - 21368 T^{5} - 124791 T^{6} - 76907 T^{7} + 4056005 T^{8} + 26132446 T^{9} - 17065384 T^{10} + 26132446 p T^{11} + 4056005 p^{2} T^{12} - 76907 p^{3} T^{13} - 124791 p^{4} T^{14} - 21368 p^{5} T^{15} + 1378 p^{6} T^{16} - 66 p^{7} T^{17} - 51 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 43 | \( 1 + 296 T^{2} + 44378 T^{4} + 4652748 T^{6} + 391103723 T^{8} + 28118455016 T^{10} + 1789090828752 T^{12} + 102981546416236 T^{14} + 5427435319638877 T^{16} + 263126944757437268 T^{18} + 11769683343689193842 T^{20} + 263126944757437268 p^{2} T^{22} + 5427435319638877 p^{4} T^{24} + 102981546416236 p^{6} T^{26} + 1789090828752 p^{8} T^{28} + 28118455016 p^{10} T^{30} + 391103723 p^{12} T^{32} + 4652748 p^{14} T^{34} + 44378 p^{16} T^{36} + 296 p^{18} T^{38} + p^{20} T^{40} \) | |
| 47 | \( 1 + 330 T^{2} + 55043 T^{4} + 138550 p T^{6} + 13331853 p T^{8} + 51366803268 T^{10} + 3670410305798 T^{12} + 234518358343748 T^{14} + 13621432336026117 T^{16} + 723256893872907374 T^{18} + 35345905781722306761 T^{20} + 723256893872907374 p^{2} T^{22} + 13621432336026117 p^{4} T^{24} + 234518358343748 p^{6} T^{26} + 3670410305798 p^{8} T^{28} + 51366803268 p^{10} T^{30} + 13331853 p^{13} T^{32} + 138550 p^{15} T^{34} + 55043 p^{16} T^{36} + 330 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 + 187 T^{2} + 10857 T^{4} + 187526 T^{6} + 18164622 T^{8} + 1782936710 T^{10} + 1930248759 T^{12} + 245939802605 T^{14} + 487877737630815 T^{16} + 418584463595508 p T^{18} + 111262864381332 p^{2} T^{20} + 418584463595508 p^{3} T^{22} + 487877737630815 p^{4} T^{24} + 245939802605 p^{6} T^{26} + 1930248759 p^{8} T^{28} + 1782936710 p^{10} T^{30} + 18164622 p^{12} T^{32} + 187526 p^{14} T^{34} + 10857 p^{16} T^{36} + 187 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( ( 1 - 4 T - 219 T^{2} + 844 T^{3} + 27227 T^{4} - 91508 T^{5} - 2371958 T^{6} + 5682068 T^{7} + 165702725 T^{8} - 143523552 T^{9} - 10227447425 T^{10} - 143523552 p T^{11} + 165702725 p^{2} T^{12} + 5682068 p^{3} T^{13} - 2371958 p^{4} T^{14} - 91508 p^{5} T^{15} + 27227 p^{6} T^{16} + 844 p^{7} T^{17} - 219 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 8 T - 2 T^{2} + 840 T^{3} - 6891 T^{4} + 5646 T^{5} + 458376 T^{6} - 3820944 T^{7} + 7381113 T^{8} + 175321114 T^{9} - 1797159854 T^{10} + 175321114 p T^{11} + 7381113 p^{2} T^{12} - 3820944 p^{3} T^{13} + 458376 p^{4} T^{14} + 5646 p^{5} T^{15} - 6891 p^{6} T^{16} + 840 p^{7} T^{17} - 2 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 288 T^{2} + 30906 T^{4} + 1995764 T^{6} + 211528563 T^{8} + 24504724644 T^{10} + 1746661616560 T^{12} + 124205353816172 T^{14} + 11458025705858037 T^{16} + 781602398189492280 T^{18} + 45201647957489487250 T^{20} + 781602398189492280 p^{2} T^{22} + 11458025705858037 p^{4} T^{24} + 124205353816172 p^{6} T^{26} + 1746661616560 p^{8} T^{28} + 24504724644 p^{10} T^{30} + 211528563 p^{12} T^{32} + 1995764 p^{14} T^{34} + 30906 p^{16} T^{36} + 288 p^{18} T^{38} + p^{20} T^{40} \) | |
| 71 | \( ( 1 + 2 T - 139 T^{2} - 894 T^{3} + 2411 T^{4} + 56182 T^{5} + 252754 T^{6} + 4090462 T^{7} + 50594417 T^{8} - 324762896 T^{9} - 7660356205 T^{10} - 324762896 p T^{11} + 50594417 p^{2} T^{12} + 4090462 p^{3} T^{13} + 252754 p^{4} T^{14} + 56182 p^{5} T^{15} + 2411 p^{6} T^{16} - 894 p^{7} T^{17} - 139 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 73 | \( 1 + 188 T^{2} + 4598 T^{4} - 781620 T^{6} - 28309 T^{8} + 2118014036 T^{10} - 491616189588 T^{12} - 22927336728728 T^{14} + 3216679965567997 T^{16} + 85610631285123104 T^{18} - 14020439874794901742 T^{20} + 85610631285123104 p^{2} T^{22} + 3216679965567997 p^{4} T^{24} - 22927336728728 p^{6} T^{26} - 491616189588 p^{8} T^{28} + 2118014036 p^{10} T^{30} - 28309 p^{12} T^{32} - 781620 p^{14} T^{34} + 4598 p^{16} T^{36} + 188 p^{18} T^{38} + p^{20} T^{40} \) | |
| 79 | \( ( 1 - 4 T - 212 T^{2} - 372 T^{3} + 25325 T^{4} + 134092 T^{5} - 1753550 T^{6} - 12579484 T^{7} + 92199361 T^{8} + 339253840 T^{9} - 4502841482 T^{10} + 339253840 p T^{11} + 92199361 p^{2} T^{12} - 12579484 p^{3} T^{13} - 1753550 p^{4} T^{14} + 134092 p^{5} T^{15} + 25325 p^{6} T^{16} - 372 p^{7} T^{17} - 212 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 - 474 T^{2} + 118745 T^{4} - 19940352 T^{6} + 2460856750 T^{8} - 232261247628 T^{10} + 2460856750 p^{2} T^{12} - 19940352 p^{4} T^{14} + 118745 p^{6} T^{16} - 474 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( ( 1 + T - 329 T^{2} + 44 T^{3} + 58042 T^{4} - 40958 T^{5} - 7701513 T^{6} + 4132463 T^{7} + 850237985 T^{8} - 136355702 T^{9} - 80999117520 T^{10} - 136355702 p T^{11} + 850237985 p^{2} T^{12} + 4132463 p^{3} T^{13} - 7701513 p^{4} T^{14} - 40958 p^{5} T^{15} + 58042 p^{6} T^{16} + 44 p^{7} T^{17} - 329 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 + 602 T^{2} + 179371 T^{4} + 36646930 T^{6} + 5985397091 T^{8} + 848379234712 T^{10} + 109390450095662 T^{12} + 13235363573020984 T^{14} + 1519734504345201317 T^{16} + \)\(16\!\cdots\!38\)\( T^{18} + \)\(16\!\cdots\!85\)\( T^{20} + \)\(16\!\cdots\!38\)\( p^{2} T^{22} + 1519734504345201317 p^{4} T^{24} + 13235363573020984 p^{6} T^{26} + 109390450095662 p^{8} T^{28} + 848379234712 p^{10} T^{30} + 5985397091 p^{12} T^{32} + 36646930 p^{14} T^{34} + 179371 p^{16} T^{36} + 602 p^{18} T^{38} + p^{20} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.43598437746529874560303087356, −2.41973154541398015910941192412, −2.33807003843672439927529322718, −2.33194560803818436480698668085, −2.32564100932489144329559499395, −2.10513638000661952733863513143, −2.09256242134810151151352496589, −2.04577928983534869703768239162, −1.98323027957190965900444698109, −1.74742285424802915658289820323, −1.73559907115587974656910244690, −1.71735132474871084527421014886, −1.33117195892030511643172331287, −1.28627081245362057779957284341, −1.22035342901075430813145475333, −1.19169532427809235541209810735, −1.17014323794812588605781122369, −1.13223134423734512717517007884, −1.10135484412473183657276632379, −1.08554276390479888542018662878, −1.07846705924031717969005969352, −0.863000752122615408681113861171, −0.75813841244013705124521789867, −0.36213723560067244779896110422, −0.16990736490730590929846155585, 0.16990736490730590929846155585, 0.36213723560067244779896110422, 0.75813841244013705124521789867, 0.863000752122615408681113861171, 1.07846705924031717969005969352, 1.08554276390479888542018662878, 1.10135484412473183657276632379, 1.13223134423734512717517007884, 1.17014323794812588605781122369, 1.19169532427809235541209810735, 1.22035342901075430813145475333, 1.28627081245362057779957284341, 1.33117195892030511643172331287, 1.71735132474871084527421014886, 1.73559907115587974656910244690, 1.74742285424802915658289820323, 1.98323027957190965900444698109, 2.04577928983534869703768239162, 2.09256242134810151151352496589, 2.10513638000661952733863513143, 2.32564100932489144329559499395, 2.33194560803818436480698668085, 2.33807003843672439927529322718, 2.41973154541398015910941192412, 2.43598437746529874560303087356
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.