Dirichlet series
| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 6·11-s − 2·13-s − 14-s − 5·17-s + 31·19-s + 21-s − 6·22-s − 23-s − 6·25-s − 2·26-s − 5·29-s + 3·31-s + 6·33-s − 5·34-s + 14·37-s + 31·38-s + 2·39-s − 28·41-s + 42-s + 9·43-s − 46-s − 44·47-s − 6·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 1.80·11-s − 0.554·13-s − 0.267·14-s − 1.21·17-s + 7.11·19-s + 0.218·21-s − 1.27·22-s − 0.208·23-s − 6/5·25-s − 0.392·26-s − 0.928·29-s + 0.538·31-s + 1.04·33-s − 0.857·34-s + 2.30·37-s + 5.02·38-s + 0.320·39-s − 4.37·41-s + 0.154·42-s + 1.37·43-s − 0.147·46-s − 6.41·47-s − 0.848·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.45662\times 10^{8}\) |
| Root analytic conductor: | \(2.77732\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6742263348\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6742263348\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) | |
| 7 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) | |
| 23 | \( 1 + T - 21 T^{2} - 219 T^{3} + 551 T^{4} + 4379 T^{5} + 551 p T^{6} - 219 p^{2} T^{7} - 21 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 + 6 T^{2} + p^{2} T^{4} - 44 T^{5} - 48 T^{6} + 33 T^{7} - 1146 T^{8} + 781 T^{9} - 5039 T^{10} + 781 p T^{11} - 1146 p^{2} T^{12} + 33 p^{3} T^{13} - 48 p^{4} T^{14} - 44 p^{5} T^{15} + p^{8} T^{16} + 6 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( 1 + 6 T + 3 T^{2} - 114 T^{3} - 431 T^{4} - 133 T^{5} + 5109 T^{6} + 21898 T^{7} + 33741 T^{8} - 174227 T^{9} - 1045835 T^{10} - 174227 p T^{11} + 33741 p^{2} T^{12} + 21898 p^{3} T^{13} + 5109 p^{4} T^{14} - 133 p^{5} T^{15} - 431 p^{6} T^{16} - 114 p^{7} T^{17} + 3 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 2 T - 31 T^{2} - 22 T^{3} + 359 T^{4} + 36 T^{5} - 1603 T^{6} - 3190 T^{7} + 60065 T^{8} + 48630 T^{9} - 1409849 T^{10} + 48630 p T^{11} + 60065 p^{2} T^{12} - 3190 p^{3} T^{13} - 1603 p^{4} T^{14} + 36 p^{5} T^{15} + 359 p^{6} T^{16} - 22 p^{7} T^{17} - 31 p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 5 T - 25 T^{2} - 177 T^{3} + 409 T^{4} + 3283 T^{5} - 6334 T^{6} - 31997 T^{7} + 180068 T^{8} + 13772 p T^{9} - 3178625 T^{10} + 13772 p^{2} T^{11} + 180068 p^{2} T^{12} - 31997 p^{3} T^{13} - 6334 p^{4} T^{14} + 3283 p^{5} T^{15} + 409 p^{6} T^{16} - 177 p^{7} T^{17} - 25 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 31 T + 469 T^{2} - 4633 T^{3} + 33743 T^{4} - 191471 T^{5} + 2350 p^{2} T^{6} - 2736579 T^{7} + 4660308 T^{8} + 9406276 T^{9} - 93546969 T^{10} + 9406276 p T^{11} + 4660308 p^{2} T^{12} - 2736579 p^{3} T^{13} + 2350 p^{6} T^{14} - 191471 p^{5} T^{15} + 33743 p^{6} T^{16} - 4633 p^{7} T^{17} + 469 p^{8} T^{18} - 31 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 5 T - 4 T^{2} + 66 T^{3} + 1238 T^{4} + 11822 T^{5} + 33328 T^{6} + 59928 T^{7} + 911969 T^{8} + 7353163 T^{9} + 55779448 T^{10} + 7353163 p T^{11} + 911969 p^{2} T^{12} + 59928 p^{3} T^{13} + 33328 p^{4} T^{14} + 11822 p^{5} T^{15} + 1238 p^{6} T^{16} + 66 p^{7} T^{17} - 4 p^{8} T^{18} + 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 3 T - 22 T^{2} - 226 T^{3} - 125 T^{4} + 2233 T^{5} + 77982 T^{6} + 111454 T^{7} + 1192972 T^{8} - 10798498 T^{9} - 68259687 T^{10} - 10798498 p T^{11} + 1192972 p^{2} T^{12} + 111454 p^{3} T^{13} + 77982 p^{4} T^{14} + 2233 p^{5} T^{15} - 125 p^{6} T^{16} - 226 p^{7} T^{17} - 22 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 - 14 T + 49 T^{2} - 245 T^{3} + 3509 T^{4} - 10152 T^{5} - 1257 T^{6} + 140189 T^{7} - 6140236 T^{8} + 36558830 T^{9} - 88307999 T^{10} + 36558830 p T^{11} - 6140236 p^{2} T^{12} + 140189 p^{3} T^{13} - 1257 p^{4} T^{14} - 10152 p^{5} T^{15} + 3509 p^{6} T^{16} - 245 p^{7} T^{17} + 49 p^{8} T^{18} - 14 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 28 T + 347 T^{2} + 2496 T^{3} + 14609 T^{4} + 116614 T^{5} + 1059475 T^{6} + 7803084 T^{7} + 51549359 T^{8} + 363314798 T^{9} + 2493536827 T^{10} + 363314798 p T^{11} + 51549359 p^{2} T^{12} + 7803084 p^{3} T^{13} + 1059475 p^{4} T^{14} + 116614 p^{5} T^{15} + 14609 p^{6} T^{16} + 2496 p^{7} T^{17} + 347 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 9 T + 71 T^{2} - 813 T^{3} + 1756 T^{4} - 3637 T^{5} + 30221 T^{6} + 1144518 T^{7} - 2262980 T^{8} - 1288307 T^{9} - 264458239 T^{10} - 1288307 p T^{11} - 2262980 p^{2} T^{12} + 1144518 p^{3} T^{13} + 30221 p^{4} T^{14} - 3637 p^{5} T^{15} + 1756 p^{6} T^{16} - 813 p^{7} T^{17} + 71 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 22 T + 367 T^{2} + 4059 T^{3} + 38172 T^{4} + 279741 T^{5} + 38172 p T^{6} + 4059 p^{2} T^{7} + 367 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 - T - 85 T^{2} + 1260 T^{3} + 6215 T^{4} - 111407 T^{5} + 474814 T^{6} + 8660028 T^{7} - 52430933 T^{8} - 117852658 T^{9} + 4785199673 T^{10} - 117852658 p T^{11} - 52430933 p^{2} T^{12} + 8660028 p^{3} T^{13} + 474814 p^{4} T^{14} - 111407 p^{5} T^{15} + 6215 p^{6} T^{16} + 1260 p^{7} T^{17} - 85 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 11 T + 106 T^{2} - 99 T^{3} - 7783 T^{4} - 138754 T^{5} - 964159 T^{6} - 4647137 T^{7} + 25024464 T^{8} + 479597877 T^{9} + 5543005181 T^{10} + 479597877 p T^{11} + 25024464 p^{2} T^{12} - 4647137 p^{3} T^{13} - 964159 p^{4} T^{14} - 138754 p^{5} T^{15} - 7783 p^{6} T^{16} - 99 p^{7} T^{17} + 106 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 41 T + 751 T^{2} + 7423 T^{3} + 30205 T^{4} - 237189 T^{5} - 5099012 T^{6} - 42108611 T^{7} - 125648436 T^{8} + 1288308236 T^{9} + 18531518543 T^{10} + 1288308236 p T^{11} - 125648436 p^{2} T^{12} - 42108611 p^{3} T^{13} - 5099012 p^{4} T^{14} - 237189 p^{5} T^{15} + 30205 p^{6} T^{16} + 7423 p^{7} T^{17} + 751 p^{8} T^{18} + 41 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 12 T + 77 T^{2} + 21 T^{3} - 2443 T^{4} - 17380 T^{5} - 111561 T^{6} - 116269 T^{7} - 3554166 T^{8} - 13008898 T^{9} - 379723785 T^{10} - 13008898 p T^{11} - 3554166 p^{2} T^{12} - 116269 p^{3} T^{13} - 111561 p^{4} T^{14} - 17380 p^{5} T^{15} - 2443 p^{6} T^{16} + 21 p^{7} T^{17} + 77 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 22 T + 457 T^{2} + 7359 T^{3} + 111180 T^{4} + 1405481 T^{5} + 16944154 T^{6} + 182290801 T^{7} + 1860902503 T^{8} + 17210151629 T^{9} + 152018252385 T^{10} + 17210151629 p T^{11} + 1860902503 p^{2} T^{12} + 182290801 p^{3} T^{13} + 16944154 p^{4} T^{14} + 1405481 p^{5} T^{15} + 111180 p^{6} T^{16} + 7359 p^{7} T^{17} + 457 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 19 T + 134 T^{2} - 1082 T^{3} + 16430 T^{4} - 173949 T^{5} + 1983464 T^{6} - 22053421 T^{7} + 176804697 T^{8} - 1404393155 T^{9} + 12833366129 T^{10} - 1404393155 p T^{11} + 176804697 p^{2} T^{12} - 22053421 p^{3} T^{13} + 1983464 p^{4} T^{14} - 173949 p^{5} T^{15} + 16430 p^{6} T^{16} - 1082 p^{7} T^{17} + 134 p^{8} T^{18} - 19 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 3 T + 7 T^{2} + 458 T^{3} - 5183 T^{4} + 79533 T^{5} - 62694 T^{6} + 1073543 T^{7} + 27018216 T^{8} - 249132895 T^{9} + 3447651967 T^{10} - 249132895 p T^{11} + 27018216 p^{2} T^{12} + 1073543 p^{3} T^{13} - 62694 p^{4} T^{14} + 79533 p^{5} T^{15} - 5183 p^{6} T^{16} + 458 p^{7} T^{17} + 7 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 39 T + 613 T^{2} + 3477 T^{3} - 31348 T^{4} - 643747 T^{5} - 2090262 T^{6} + 43985533 T^{7} + 485716757 T^{8} - 195758068 T^{9} - 29944421627 T^{10} - 195758068 p T^{11} + 485716757 p^{2} T^{12} + 43985533 p^{3} T^{13} - 2090262 p^{4} T^{14} - 643747 p^{5} T^{15} - 31348 p^{6} T^{16} + 3477 p^{7} T^{17} + 613 p^{8} T^{18} + 39 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 17 T + 310 T^{2} + 3405 T^{3} + 30339 T^{4} + 188276 T^{5} + 572153 T^{6} - 4582727 T^{7} - 90679052 T^{8} - 1436954965 T^{9} - 10131976361 T^{10} - 1436954965 p T^{11} - 90679052 p^{2} T^{12} - 4582727 p^{3} T^{13} + 572153 p^{4} T^{14} + 188276 p^{5} T^{15} + 30339 p^{6} T^{16} + 3405 p^{7} T^{17} + 310 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 12 T + 278 T^{2} + 1369 T^{3} + 31669 T^{4} + 2903 T^{5} + 3403773 T^{6} - 5878150 T^{7} + 359085403 T^{8} - 1554556916 T^{9} + 32945256213 T^{10} - 1554556916 p T^{11} + 359085403 p^{2} T^{12} - 5878150 p^{3} T^{13} + 3403773 p^{4} T^{14} + 2903 p^{5} T^{15} + 31669 p^{6} T^{16} + 1369 p^{7} T^{17} + 278 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.64066552275948603004954433549, −3.42157999402600728304126039430, −3.29799456704866539473843544719, −3.28620280318406598720068653769, −3.28406205716277659572634212386, −3.13012845360625262660901069750, −3.04779923232769956015805393418, −2.84123577224189617954891257791, −2.75602498487219923124739778943, −2.67796105194348686454185230880, −2.64181182550621425606749898586, −2.55783060164967216590306023609, −2.55146850707979701904757671855, −2.05806981585844232991910906710, −1.81724973309312619233118224303, −1.70810108687034217222820124547, −1.55558094419027538571023386187, −1.50771752845096912547844588969, −1.43954668933624380715257303040, −1.34791442380542821855832263092, −1.31548876609415637346250030222, −0.959432236716981169510426186805, −0.36494198617309845921639099201, −0.32645905527197898126913246221, −0.15973880528540755876048745978, 0.15973880528540755876048745978, 0.32645905527197898126913246221, 0.36494198617309845921639099201, 0.959432236716981169510426186805, 1.31548876609415637346250030222, 1.34791442380542821855832263092, 1.43954668933624380715257303040, 1.50771752845096912547844588969, 1.55558094419027538571023386187, 1.70810108687034217222820124547, 1.81724973309312619233118224303, 2.05806981585844232991910906710, 2.55146850707979701904757671855, 2.55783060164967216590306023609, 2.64181182550621425606749898586, 2.67796105194348686454185230880, 2.75602498487219923124739778943, 2.84123577224189617954891257791, 3.04779923232769956015805393418, 3.13012845360625262660901069750, 3.28406205716277659572634212386, 3.28620280318406598720068653769, 3.29799456704866539473843544719, 3.42157999402600728304126039430, 3.64066552275948603004954433549
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.