Dirichlet series
| L(s) = 1 | − 4·2-s − 8·3-s + 8·4-s + 32·6-s − 12·8-s + 32·9-s − 64·12-s + 4·13-s + 22·16-s − 128·18-s + 96·24-s − 6·25-s − 16·26-s − 92·27-s − 4·31-s − 40·32-s + 256·36-s − 32·39-s − 36·41-s − 8·47-s − 176·48-s + 24·50-s + 32·52-s + 368·54-s + 16·62-s + 56·64-s + 44·71-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 4.61·3-s + 4·4-s + 13.0·6-s − 4.24·8-s + 32/3·9-s − 18.4·12-s + 1.10·13-s + 11/2·16-s − 30.1·18-s + 19.5·24-s − 6/5·25-s − 3.13·26-s − 17.7·27-s − 0.718·31-s − 7.07·32-s + 42.6·36-s − 5.12·39-s − 5.62·41-s − 1.16·47-s − 25.4·48-s + 3.39·50-s + 4.43·52-s + 50.0·54-s + 2.03·62-s + 7·64-s + 5.22·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 23^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 23^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.181760\) |
| Root analytic conductor: | \(0.958269\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 5^{20} \cdot 23^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.008955220876\) |
| \(L(\frac12)\) | \(\approx\) | \(0.008955220876\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 6 T^{2} + 3 T^{4} - 16 p T^{6} - 24 p^{2} T^{8} - 28 p^{3} T^{10} - 24 p^{4} T^{12} - 16 p^{5} T^{14} + 3 p^{6} T^{16} + 6 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( 1 + 112 T^{3} + 1429 T^{4} + 1424 T^{5} + 6272 T^{6} + 100048 T^{7} + 1214218 T^{8} + 2254896 T^{9} + 3256576 T^{10} + 2254896 p T^{11} + 1214218 p^{2} T^{12} + 100048 p^{3} T^{13} + 6272 p^{4} T^{14} + 1424 p^{5} T^{15} + 1429 p^{6} T^{16} + 112 p^{7} T^{17} + p^{10} T^{20} \) | |
| good | 2 | \( ( 1 + p T + p T^{2} + p T^{3} - T^{4} - 3 p T^{5} - p^{3} T^{6} - p^{4} T^{7} - 11 T^{8} + p^{2} T^{9} + p^{3} T^{10} + p^{3} T^{11} - 11 p^{2} T^{12} - p^{7} T^{13} - p^{7} T^{14} - 3 p^{6} T^{15} - p^{6} T^{16} + p^{8} T^{17} + p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} )^{2} \) |
| 3 | \( ( 1 + 4 T + 8 T^{2} + 14 T^{3} + 11 T^{4} - 8 T^{5} - 22 T^{6} - 38 T^{7} - 11 T^{8} + 82 T^{9} + 158 T^{10} + 82 p T^{11} - 11 p^{2} T^{12} - 38 p^{3} T^{13} - 22 p^{4} T^{14} - 8 p^{5} T^{15} + 11 p^{6} T^{16} + 14 p^{7} T^{17} + 8 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 7 | \( 1 - 237 T^{4} + 23161 T^{8} - 1134076 T^{12} + 24086022 T^{16} - 196683742 T^{20} + 24086022 p^{4} T^{24} - 1134076 p^{8} T^{28} + 23161 p^{12} T^{32} - 237 p^{16} T^{36} + p^{20} T^{40} \) | |
| 11 | \( ( 1 - 42 T^{2} + 1081 T^{4} - 20956 T^{6} + 316762 T^{8} - 3842292 T^{10} + 316762 p^{2} T^{12} - 20956 p^{4} T^{14} + 1081 p^{6} T^{16} - 42 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 13 | \( ( 1 - 2 T + 2 T^{2} - 60 T^{3} + 339 T^{4} + 508 T^{5} + 106 T^{6} - 3702 T^{7} - 35227 T^{8} + 234016 T^{9} - 82360 T^{10} + 234016 p T^{11} - 35227 p^{2} T^{12} - 3702 p^{3} T^{13} + 106 p^{4} T^{14} + 508 p^{5} T^{15} + 339 p^{6} T^{16} - 60 p^{7} T^{17} + 2 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 17 | \( 1 + 519 T^{4} + 89433 T^{8} - 14268808 T^{12} - 425916202 p T^{16} - 2204663931022 T^{20} - 425916202 p^{5} T^{24} - 14268808 p^{8} T^{28} + 89433 p^{12} T^{32} + 519 p^{16} T^{36} + p^{20} T^{40} \) | |
| 19 | \( ( 1 + 72 T^{2} + 2599 T^{4} + 72796 T^{6} + 1835388 T^{8} + 39000088 T^{10} + 1835388 p^{2} T^{12} + 72796 p^{4} T^{14} + 2599 p^{6} T^{16} + 72 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 29 | \( ( 1 - 193 T^{2} + 18883 T^{4} - 1197882 T^{6} + 54156853 T^{8} - 1813801707 T^{10} + 54156853 p^{2} T^{12} - 1197882 p^{4} T^{14} + 18883 p^{6} T^{16} - 193 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 + T + 97 T^{2} + 2 p T^{3} + 4277 T^{4} + 1947 T^{5} + 4277 p T^{6} + 2 p^{3} T^{7} + 97 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 37 | \( 1 - 9781 T^{4} + 44061273 T^{8} - 123293552508 T^{12} + 245670541820726 T^{16} - 377506395088955022 T^{20} + 245670541820726 p^{4} T^{24} - 123293552508 p^{8} T^{28} + 44061273 p^{12} T^{32} - 9781 p^{16} T^{36} + p^{20} T^{40} \) | |
| 41 | \( ( 1 + 9 T + 195 T^{2} + 1370 T^{3} + 15765 T^{4} + 82035 T^{5} + 15765 p T^{6} + 1370 p^{2} T^{7} + 195 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 43 | \( 1 - 62 p T^{4} - 1105447 T^{8} + 8937665552 T^{12} - 1848148296154 T^{16} - 20946707502982572 T^{20} - 1848148296154 p^{4} T^{24} + 8937665552 p^{8} T^{28} - 1105447 p^{12} T^{32} - 62 p^{17} T^{36} + p^{20} T^{40} \) | |
| 47 | \( ( 1 + 4 T + 8 T^{2} - 330 T^{3} + 435 T^{4} + 17576 T^{5} + 121274 T^{6} - 601990 T^{7} - 5812707 T^{8} + 11707642 T^{9} + 393692622 T^{10} + 11707642 p T^{11} - 5812707 p^{2} T^{12} - 601990 p^{3} T^{13} + 121274 p^{4} T^{14} + 17576 p^{5} T^{15} + 435 p^{6} T^{16} - 330 p^{7} T^{17} + 8 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 53 | \( 1 - 6865 T^{4} + 5634885 T^{8} + 55524537540 T^{12} - 66605380961110 T^{16} - 238481873986046502 T^{20} - 66605380961110 p^{4} T^{24} + 55524537540 p^{8} T^{28} + 5634885 p^{12} T^{32} - 6865 p^{16} T^{36} + p^{20} T^{40} \) | |
| 59 | \( ( 1 - 277 T^{2} + 38049 T^{4} - 3370504 T^{6} + 3836530 p T^{8} - 13536944758 T^{10} + 3836530 p^{3} T^{12} - 3370504 p^{4} T^{14} + 38049 p^{6} T^{16} - 277 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 61 | \( ( 1 - 288 T^{2} + 45599 T^{4} - 4912324 T^{6} + 406368068 T^{8} - 27198365512 T^{10} + 406368068 p^{2} T^{12} - 4912324 p^{4} T^{14} + 45599 p^{6} T^{16} - 288 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 67 | \( 1 + 12179 T^{4} + 21394393 T^{8} - 187118931868 T^{12} + 373223367469206 T^{16} + 8514442786702047778 T^{20} + 373223367469206 p^{4} T^{24} - 187118931868 p^{8} T^{28} + 21394393 p^{12} T^{32} + 12179 p^{16} T^{36} + p^{20} T^{40} \) | |
| 71 | \( ( 1 - 11 T + 235 T^{2} - 1020 T^{3} + 15325 T^{4} - 20035 T^{5} + 15325 p T^{6} - 1020 p^{2} T^{7} + 235 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 73 | \( ( 1 + 28 T + 392 T^{2} + 4930 T^{3} + 65419 T^{4} + 742868 T^{5} + 7308506 T^{6} + 74556498 T^{7} + 765085933 T^{8} + 6852665726 T^{9} + 57335768430 T^{10} + 6852665726 p T^{11} + 765085933 p^{2} T^{12} + 74556498 p^{3} T^{13} + 7308506 p^{4} T^{14} + 742868 p^{5} T^{15} + 65419 p^{6} T^{16} + 4930 p^{7} T^{17} + 392 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 + 332 T^{2} + 60799 T^{4} + 7894836 T^{6} + 804461788 T^{8} + 68672209888 T^{10} + 804461788 p^{2} T^{12} + 7894836 p^{4} T^{14} + 60799 p^{6} T^{16} + 332 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 83 | \( 1 + 14383 T^{4} + 96015001 T^{8} + 196390156024 T^{12} - 4395513412321338 T^{16} - 50701641519546426142 T^{20} - 4395513412321338 p^{4} T^{24} + 196390156024 p^{8} T^{28} + 96015001 p^{12} T^{32} + 14383 p^{16} T^{36} + p^{20} T^{40} \) | |
| 89 | \( ( 1 + 468 T^{2} + 124191 T^{4} + 22228084 T^{6} + 2951501052 T^{8} + 298597247008 T^{10} + 2951501052 p^{2} T^{12} + 22228084 p^{4} T^{14} + 124191 p^{6} T^{16} + 468 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 97 | \( 1 - 5386 T^{4} - 21562987 T^{8} - 1116224538328 T^{12} + 10456340019940986 T^{16} - 23761540774597810972 T^{20} + 10456340019940986 p^{4} T^{24} - 1116224538328 p^{8} T^{28} - 21562987 p^{12} T^{32} - 5386 p^{16} T^{36} + p^{20} T^{40} \) | |
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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
−3.61831608785286041555834371648, −3.53825669965460172542808693667, −3.52721067466873155351044636768, −3.51606841080895166037933950084, −3.42061059001975109197016555512, −3.24606370444421455850381099673, −3.14992461662657701232646384212, −3.10788967710628497797005283568, −2.93177684409792271680397832872, −2.87289013732522231965374798422, −2.73083379939639106893359615581, −2.44373488218025080121320440329, −2.44130488794554820901559658068, −2.35898699670303890040293250708, −2.29820057851878747187405259878, −1.91345111599671797539715026534, −1.79537354330328615713727568977, −1.75721139074267897435928259539, −1.63575709745771659985237943900, −1.56314230835396831370429900544, −1.45009208945290336375455119391, −1.33969364083138672952590430689, −0.869016560018669358046344868647, −0.73657114257203354887106223194, −0.24795590160750646434709525274, 0.24795590160750646434709525274, 0.73657114257203354887106223194, 0.869016560018669358046344868647, 1.33969364083138672952590430689, 1.45009208945290336375455119391, 1.56314230835396831370429900544, 1.63575709745771659985237943900, 1.75721139074267897435928259539, 1.79537354330328615713727568977, 1.91345111599671797539715026534, 2.29820057851878747187405259878, 2.35898699670303890040293250708, 2.44130488794554820901559658068, 2.44373488218025080121320440329, 2.73083379939639106893359615581, 2.87289013732522231965374798422, 2.93177684409792271680397832872, 3.10788967710628497797005283568, 3.14992461662657701232646384212, 3.24606370444421455850381099673, 3.42061059001975109197016555512, 3.51606841080895166037933950084, 3.52721067466873155351044636768, 3.53825669965460172542808693667, 3.61831608785286041555834371648
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.