Dirichlet series
| L(s) = 1 | − 6·3-s + 5·4-s − 6·7-s + 17·9-s − 30·12-s + 10·16-s + 36·19-s + 36·21-s + 19·25-s − 30·27-s − 30·28-s + 36·31-s + 85·36-s + 16·37-s + 32·43-s − 60·48-s − 3·49-s − 216·57-s + 42·61-s − 102·63-s + 5·64-s − 10·67-s + 12·73-s − 114·75-s + 180·76-s + 6·79-s + 47·81-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 5/2·4-s − 2.26·7-s + 17/3·9-s − 8.66·12-s + 5/2·16-s + 8.25·19-s + 7.85·21-s + 19/5·25-s − 5.77·27-s − 5.66·28-s + 6.46·31-s + 85/6·36-s + 2.63·37-s + 4.87·43-s − 8.66·48-s − 3/7·49-s − 28.6·57-s + 5.37·61-s − 12.8·63-s + 5/8·64-s − 1.22·67-s + 1.40·73-s − 13.1·75-s + 20.6·76-s + 0.675·79-s + 47/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 11^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 11^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.17962\times 10^{11}\) |
| Root analytic conductor: | \(1.92070\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 7^{20} \cdot 11^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(15.95909192\) |
| \(L(\frac12)\) | \(\approx\) | \(15.95909192\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + 2 p T + 19 T^{2} + 14 p T^{3} + 62 T^{4} + 14 p T^{5} - 25 T^{6} - 2 p T^{7} + 445 T^{8} + 196 p^{2} T^{9} + 1288 p T^{10} + 196 p^{3} T^{11} + 445 p^{2} T^{12} - 2 p^{4} T^{13} - 25 p^{4} T^{14} + 14 p^{6} T^{15} + 62 p^{6} T^{16} + 14 p^{8} T^{17} + 19 p^{8} T^{18} + 2 p^{10} T^{19} + p^{10} T^{20} \) | |
| 7 | \( ( 1 + 3 T + 15 T^{2} + 60 T^{3} + 159 T^{4} + 521 T^{5} + 159 p T^{6} + 60 p^{2} T^{7} + 15 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 11 | \( ( 1 - T^{2} + T^{4} )^{5} \) | |
| good | 5 | \( 1 - 19 T^{2} + 192 T^{4} - 931 T^{6} - 613 T^{8} + 49994 T^{10} - 409248 T^{12} + 1645154 T^{14} + 267001 T^{16} - 49627917 T^{18} + 356517984 T^{20} - 49627917 p^{2} T^{22} + 267001 p^{4} T^{24} + 1645154 p^{6} T^{26} - 409248 p^{8} T^{28} + 49994 p^{10} T^{30} - 613 p^{12} T^{32} - 931 p^{14} T^{34} + 192 p^{16} T^{36} - 19 p^{18} T^{38} + p^{20} T^{40} \) |
| 13 | \( ( 1 - 62 T^{2} + 2227 T^{4} - 55704 T^{6} + 1047485 T^{8} - 15383710 T^{10} + 1047485 p^{2} T^{12} - 55704 p^{4} T^{14} + 2227 p^{6} T^{16} - 62 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 17 | \( 1 - 111 T^{2} + 6420 T^{4} - 254715 T^{6} + 7690347 T^{8} - 183628642 T^{10} + 3496830180 T^{12} - 52376343078 T^{14} + 592010599101 T^{16} - 4802750928237 T^{18} + 43780635109536 T^{20} - 4802750928237 p^{2} T^{22} + 592010599101 p^{4} T^{24} - 52376343078 p^{6} T^{26} + 3496830180 p^{8} T^{28} - 183628642 p^{10} T^{30} + 7690347 p^{12} T^{32} - 254715 p^{14} T^{34} + 6420 p^{16} T^{36} - 111 p^{18} T^{38} + p^{20} T^{40} \) | |
| 19 | \( ( 1 - 18 T + 201 T^{2} - 1674 T^{3} + 11379 T^{4} - 69372 T^{5} + 20598 p T^{6} - 110736 p T^{7} + 10730643 T^{8} - 2706858 p T^{9} + 232025581 T^{10} - 2706858 p^{2} T^{11} + 10730643 p^{2} T^{12} - 110736 p^{4} T^{13} + 20598 p^{5} T^{14} - 69372 p^{5} T^{15} + 11379 p^{6} T^{16} - 1674 p^{7} T^{17} + 201 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 23 | \( 1 + 89 T^{2} + 3400 T^{4} + 61737 T^{6} + 42627 T^{8} - 30170334 T^{10} - 910332264 T^{12} - 9881110014 T^{14} + 193586725689 T^{16} + 447515608873 p T^{18} + 496256158288 p^{2} T^{20} + 447515608873 p^{3} T^{22} + 193586725689 p^{4} T^{24} - 9881110014 p^{6} T^{26} - 910332264 p^{8} T^{28} - 30170334 p^{10} T^{30} + 42627 p^{12} T^{32} + 61737 p^{14} T^{34} + 3400 p^{16} T^{36} + 89 p^{18} T^{38} + p^{20} T^{40} \) | |
| 29 | \( ( 1 - 94 T^{2} + 7091 T^{4} - 347992 T^{6} + 14421373 T^{8} - 452173374 T^{10} + 14421373 p^{2} T^{12} - 347992 p^{4} T^{14} + 7091 p^{6} T^{16} - 94 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 31 | \( ( 1 - 18 T + 147 T^{2} - 702 T^{3} + 723 T^{4} + 17964 T^{5} - 158778 T^{6} + 727044 T^{7} - 1347555 T^{8} - 5315526 T^{9} + 50915713 T^{10} - 5315526 p T^{11} - 1347555 p^{2} T^{12} + 727044 p^{3} T^{13} - 158778 p^{4} T^{14} + 17964 p^{5} T^{15} + 723 p^{6} T^{16} - 702 p^{7} T^{17} + 147 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 37 | \( ( 1 - 8 T - 75 T^{2} + 904 T^{3} + 2531 T^{4} - 53168 T^{5} + 27498 T^{6} + 1884160 T^{7} - 6789653 T^{8} - 26878968 T^{9} + 334097121 T^{10} - 26878968 p T^{11} - 6789653 p^{2} T^{12} + 1884160 p^{3} T^{13} + 27498 p^{4} T^{14} - 53168 p^{5} T^{15} + 2531 p^{6} T^{16} + 904 p^{7} T^{17} - 75 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 41 | \( ( 1 + 379 T^{2} + 65761 T^{4} + 6880596 T^{6} + 480812858 T^{8} + 23448503918 T^{10} + 480812858 p^{2} T^{12} + 6880596 p^{4} T^{14} + 65761 p^{6} T^{16} + 379 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 131 T^{2} - 528 T^{3} + 6718 T^{4} - 19536 T^{5} + 6718 p T^{6} - 528 p^{2} T^{7} + 131 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{4} \) | |
| 47 | \( 1 + 29 T^{2} - 4140 T^{4} - 66007 T^{6} + 5373875 T^{8} - 62873050 T^{10} - 8687712324 T^{12} - 332025992134 T^{14} + 26041090134253 T^{16} + 1069119520723071 T^{18} - 52828130382498024 T^{20} + 1069119520723071 p^{2} T^{22} + 26041090134253 p^{4} T^{24} - 332025992134 p^{6} T^{26} - 8687712324 p^{8} T^{28} - 62873050 p^{10} T^{30} + 5373875 p^{12} T^{32} - 66007 p^{14} T^{34} - 4140 p^{16} T^{36} + 29 p^{18} T^{38} + p^{20} T^{40} \) | |
| 53 | \( 1 + 130 T^{2} - 385 T^{4} - 618390 T^{6} + 7110947 T^{8} + 3357689580 T^{10} + 43055000374 T^{12} - 6566150768460 T^{14} - 45839867658151 T^{16} + 12920652761343390 T^{18} + 597130170364840945 T^{20} + 12920652761343390 p^{2} T^{22} - 45839867658151 p^{4} T^{24} - 6566150768460 p^{6} T^{26} + 43055000374 p^{8} T^{28} + 3357689580 p^{10} T^{30} + 7110947 p^{12} T^{32} - 618390 p^{14} T^{34} - 385 p^{16} T^{36} + 130 p^{18} T^{38} + p^{20} T^{40} \) | |
| 59 | \( 1 - 418 T^{2} + 88269 T^{4} - 13237606 T^{6} + 1619563400 T^{8} - 169695635974 T^{10} + 15567898011183 T^{12} - 1277174193111382 T^{14} + 94892351970598027 T^{16} - 6409403040247655580 T^{18} + \)\(39\!\cdots\!04\)\( T^{20} - 6409403040247655580 p^{2} T^{22} + 94892351970598027 p^{4} T^{24} - 1277174193111382 p^{6} T^{26} + 15567898011183 p^{8} T^{28} - 169695635974 p^{10} T^{30} + 1619563400 p^{12} T^{32} - 13237606 p^{14} T^{34} + 88269 p^{16} T^{36} - 418 p^{18} T^{38} + p^{20} T^{40} \) | |
| 61 | \( ( 1 - 21 T + 6 p T^{2} - 4599 T^{3} + 49662 T^{4} - 480303 T^{5} + 4201020 T^{6} - 35448507 T^{7} + 285032553 T^{8} - 2251615554 T^{9} + 17661732796 T^{10} - 2251615554 p T^{11} + 285032553 p^{2} T^{12} - 35448507 p^{3} T^{13} + 4201020 p^{4} T^{14} - 480303 p^{5} T^{15} + 49662 p^{6} T^{16} - 4599 p^{7} T^{17} + 6 p^{9} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 67 | \( ( 1 + 5 T - 166 T^{2} + 267 T^{3} + 19418 T^{4} - 84027 T^{5} - 774710 T^{6} + 10549095 T^{7} - 4212751 T^{8} - 4027380 p T^{9} + 3800526688 T^{10} - 4027380 p^{2} T^{11} - 4212751 p^{2} T^{12} + 10549095 p^{3} T^{13} - 774710 p^{4} T^{14} - 84027 p^{5} T^{15} + 19418 p^{6} T^{16} + 267 p^{7} T^{17} - 166 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 71 | \( ( 1 - 410 T^{2} + 80493 T^{4} - 10355376 T^{6} + 1004280390 T^{8} - 78592775460 T^{10} + 1004280390 p^{2} T^{12} - 10355376 p^{4} T^{14} + 80493 p^{6} T^{16} - 410 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 73 | \( ( 1 - 6 T + 151 T^{2} - 834 T^{3} + 13543 T^{4} - 50460 T^{5} + 491034 T^{6} + 980904 T^{7} - 21146605 T^{8} + 416825838 T^{9} - 4667757361 T^{10} + 416825838 p T^{11} - 21146605 p^{2} T^{12} + 980904 p^{3} T^{13} + 491034 p^{4} T^{14} - 50460 p^{5} T^{15} + 13543 p^{6} T^{16} - 834 p^{7} T^{17} + 151 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 3 T - 282 T^{2} + 327 T^{3} + 42522 T^{4} + 461 T^{5} - 5209026 T^{6} + 1523043 T^{7} + 539773281 T^{8} - 168174348 T^{9} - 46349886960 T^{10} - 168174348 p T^{11} + 539773281 p^{2} T^{12} + 1523043 p^{3} T^{13} - 5209026 p^{4} T^{14} + 461 p^{5} T^{15} + 42522 p^{6} T^{16} + 327 p^{7} T^{17} - 282 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} )^{2} \) | |
| 83 | \( ( 1 + 163 T^{2} + 14965 T^{4} + 834084 T^{6} + 73107890 T^{8} + 6785333138 T^{10} + 73107890 p^{2} T^{12} + 834084 p^{4} T^{14} + 14965 p^{6} T^{16} + 163 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
| 89 | \( 1 - 394 T^{2} + 68319 T^{4} - 6706402 T^{6} + 428982287 T^{8} - 25623147964 T^{10} + 2250843903162 T^{12} - 249200551523236 T^{14} + 34654480387751053 T^{16} - 5039456614966637886 T^{18} + \)\(54\!\cdots\!01\)\( T^{20} - 5039456614966637886 p^{2} T^{22} + 34654480387751053 p^{4} T^{24} - 249200551523236 p^{6} T^{26} + 2250843903162 p^{8} T^{28} - 25623147964 p^{10} T^{30} + 428982287 p^{12} T^{32} - 6706402 p^{14} T^{34} + 68319 p^{16} T^{36} - 394 p^{18} T^{38} + p^{20} T^{40} \) | |
| 97 | \( ( 1 - 707 T^{2} + 241423 T^{4} - 52507794 T^{6} + 8040820589 T^{8} - 904502071045 T^{10} + 8040820589 p^{2} T^{12} - 52507794 p^{4} T^{14} + 241423 p^{6} T^{16} - 707 p^{8} T^{18} + p^{10} T^{20} )^{2} \) | |
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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
−2.64587516636778916173118427263, −2.62544259351409991953451153178, −2.49448408557846160602665570334, −2.48579803947393596345511728340, −2.41694431670313631251557840071, −2.35390646510984031941839632665, −2.29517193137737427037287295169, −2.23790310177273693211716805509, −1.86500362640960582241221410405, −1.73713505650390112629153251776, −1.72317194341736921576048991713, −1.70369184049386184264485536363, −1.65827291992550441399966302957, −1.59423554295649071864622374238, −1.24679011934949238860738493392, −1.18634558499619859679573545454, −1.01204336707433453669615679974, −1.00040661563405330195834675942, −0.954510364384563022112775575844, −0.951577705562741792895519416639, −0.891144248472479829539645437816, −0.78623276514770520153541056580, −0.72369094950020359962664232489, −0.61105991229685101048697447851, −0.24575454677872382642398416254, 0.24575454677872382642398416254, 0.61105991229685101048697447851, 0.72369094950020359962664232489, 0.78623276514770520153541056580, 0.891144248472479829539645437816, 0.951577705562741792895519416639, 0.954510364384563022112775575844, 1.00040661563405330195834675942, 1.01204336707433453669615679974, 1.18634558499619859679573545454, 1.24679011934949238860738493392, 1.59423554295649071864622374238, 1.65827291992550441399966302957, 1.70369184049386184264485536363, 1.72317194341736921576048991713, 1.73713505650390112629153251776, 1.86500362640960582241221410405, 2.23790310177273693211716805509, 2.29517193137737427037287295169, 2.35390646510984031941839632665, 2.41694431670313631251557840071, 2.48579803947393596345511728340, 2.49448408557846160602665570334, 2.62544259351409991953451153178, 2.64587516636778916173118427263
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.