Dirichlet series
| L(s) = 1 | + 4·3-s + 5-s − 3·9-s − 3·13-s + 4·15-s + 5·17-s + 12·19-s + 11·23-s − 16·25-s − 31·27-s − 15·29-s + 16·31-s + 3·37-s − 12·39-s + 28·41-s − 9·43-s − 3·45-s + 31·47-s + 20·51-s + 13·53-s + 48·57-s + 11·59-s − 19·61-s − 3·65-s + 19·67-s + 44·69-s − 5·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.447·5-s − 9-s − 0.832·13-s + 1.03·15-s + 1.21·17-s + 2.75·19-s + 2.29·23-s − 3.19·25-s − 5.96·27-s − 2.78·29-s + 2.87·31-s + 0.493·37-s − 1.92·39-s + 4.37·41-s − 1.37·43-s − 0.447·45-s + 4.52·47-s + 2.80·51-s + 1.78·53-s + 6.35·57-s + 1.43·59-s − 2.43·61-s − 0.372·65-s + 2.32·67-s + 5.29·69-s − 0.593·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{22} \cdot 23^{11}\right)^{s/2} \, \Gamma_{\C}(s)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{33} \cdot 7^{22} \cdot 23^{11}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{11} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(22\) |
| Conductor: | \(2^{33} \cdot 7^{22} \cdot 23^{11}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.69278\times 10^{20}\) |
| Root analytic conductor: | \(8.48487\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((22,\ 2^{33} \cdot 7^{22} \cdot 23^{11} ,\ ( \ : [1/2]^{11} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(229.4910848\) |
| \(L(\frac12)\) | \(\approx\) | \(229.4910848\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) | |
| 23 | \( ( 1 - T )^{11} \) | |
| good | 3 | \( 1 - 4 T + 19 T^{2} - 19 p T^{3} + 56 p T^{4} - 413 T^{5} + 1006 T^{6} - 715 p T^{7} + 1541 p T^{8} - 2951 p T^{9} + 5650 p T^{10} - 9841 p T^{11} + 5650 p^{2} T^{12} - 2951 p^{3} T^{13} + 1541 p^{4} T^{14} - 715 p^{5} T^{15} + 1006 p^{5} T^{16} - 413 p^{6} T^{17} + 56 p^{8} T^{18} - 19 p^{9} T^{19} + 19 p^{9} T^{20} - 4 p^{10} T^{21} + p^{11} T^{22} \) |
| 5 | \( 1 - T + 17 T^{2} - 22 T^{3} + 163 T^{4} - 311 T^{5} + 1146 T^{6} - 113 p^{2} T^{7} + 1356 p T^{8} - 19193 T^{9} + 36909 T^{10} - 104993 T^{11} + 36909 p T^{12} - 19193 p^{2} T^{13} + 1356 p^{4} T^{14} - 113 p^{6} T^{15} + 1146 p^{5} T^{16} - 311 p^{6} T^{17} + 163 p^{7} T^{18} - 22 p^{8} T^{19} + 17 p^{9} T^{20} - p^{10} T^{21} + p^{11} T^{22} \) | |
| 11 | \( 1 + 50 T^{2} - 49 T^{3} + 1338 T^{4} - 2437 T^{5} + 25519 T^{6} - 64756 T^{7} + 383523 T^{8} - 1154002 T^{9} + 4840259 T^{10} - 14814325 T^{11} + 4840259 p T^{12} - 1154002 p^{2} T^{13} + 383523 p^{3} T^{14} - 64756 p^{4} T^{15} + 25519 p^{5} T^{16} - 2437 p^{6} T^{17} + 1338 p^{7} T^{18} - 49 p^{8} T^{19} + 50 p^{9} T^{20} + p^{11} T^{22} \) | |
| 13 | \( 1 + 3 T + 83 T^{2} + 365 T^{3} + 3587 T^{4} + 18029 T^{5} + 8395 p T^{6} + 40906 p T^{7} + 2473307 T^{8} + 10925418 T^{9} + 42164045 T^{10} + 164912689 T^{11} + 42164045 p T^{12} + 10925418 p^{2} T^{13} + 2473307 p^{3} T^{14} + 40906 p^{5} T^{15} + 8395 p^{6} T^{16} + 18029 p^{6} T^{17} + 3587 p^{7} T^{18} + 365 p^{8} T^{19} + 83 p^{9} T^{20} + 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 17 | \( 1 - 5 T + 129 T^{2} - 648 T^{3} + 473 p T^{4} - 39569 T^{5} + 323727 T^{6} - 1517042 T^{7} + 9408642 T^{8} - 40735433 T^{9} + 207243759 T^{10} - 802914875 T^{11} + 207243759 p T^{12} - 40735433 p^{2} T^{13} + 9408642 p^{3} T^{14} - 1517042 p^{4} T^{15} + 323727 p^{5} T^{16} - 39569 p^{6} T^{17} + 473 p^{8} T^{18} - 648 p^{8} T^{19} + 129 p^{9} T^{20} - 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 19 | \( 1 - 12 T + 158 T^{2} - 1128 T^{3} + 8272 T^{4} - 38991 T^{5} + 193557 T^{6} - 534996 T^{7} + 1736691 T^{8} + 1775952 T^{9} - 11520582 T^{10} + 153482439 T^{11} - 11520582 p T^{12} + 1775952 p^{2} T^{13} + 1736691 p^{3} T^{14} - 534996 p^{4} T^{15} + 193557 p^{5} T^{16} - 38991 p^{6} T^{17} + 8272 p^{7} T^{18} - 1128 p^{8} T^{19} + 158 p^{9} T^{20} - 12 p^{10} T^{21} + p^{11} T^{22} \) | |
| 29 | \( 1 + 15 T + 250 T^{2} + 2489 T^{3} + 25865 T^{4} + 207932 T^{5} + 1712810 T^{6} + 11890229 T^{7} + 83023851 T^{8} + 505703486 T^{9} + 3073923103 T^{10} + 16574465297 T^{11} + 3073923103 p T^{12} + 505703486 p^{2} T^{13} + 83023851 p^{3} T^{14} + 11890229 p^{4} T^{15} + 1712810 p^{5} T^{16} + 207932 p^{6} T^{17} + 25865 p^{7} T^{18} + 2489 p^{8} T^{19} + 250 p^{9} T^{20} + 15 p^{10} T^{21} + p^{11} T^{22} \) | |
| 31 | \( 1 - 16 T + 327 T^{2} - 3719 T^{3} + 45099 T^{4} - 407875 T^{5} + 3721818 T^{6} - 28222102 T^{7} + 210834851 T^{8} - 1374404711 T^{9} + 8727240518 T^{10} - 49320577685 T^{11} + 8727240518 p T^{12} - 1374404711 p^{2} T^{13} + 210834851 p^{3} T^{14} - 28222102 p^{4} T^{15} + 3721818 p^{5} T^{16} - 407875 p^{6} T^{17} + 45099 p^{7} T^{18} - 3719 p^{8} T^{19} + 327 p^{9} T^{20} - 16 p^{10} T^{21} + p^{11} T^{22} \) | |
| 37 | \( 1 - 3 T + 171 T^{2} - 473 T^{3} + 16050 T^{4} - 37278 T^{5} + 1076947 T^{6} - 2088290 T^{7} + 56840745 T^{8} - 94296008 T^{9} + 2483868298 T^{10} - 3678277269 T^{11} + 2483868298 p T^{12} - 94296008 p^{2} T^{13} + 56840745 p^{3} T^{14} - 2088290 p^{4} T^{15} + 1076947 p^{5} T^{16} - 37278 p^{6} T^{17} + 16050 p^{7} T^{18} - 473 p^{8} T^{19} + 171 p^{9} T^{20} - 3 p^{10} T^{21} + p^{11} T^{22} \) | |
| 41 | \( 1 - 28 T + 636 T^{2} - 10100 T^{3} + 140052 T^{4} - 1621342 T^{5} + 16980844 T^{6} - 157379982 T^{7} + 1345164453 T^{8} - 10435530768 T^{9} + 75396431243 T^{10} - 499502243005 T^{11} + 75396431243 p T^{12} - 10435530768 p^{2} T^{13} + 1345164453 p^{3} T^{14} - 157379982 p^{4} T^{15} + 16980844 p^{5} T^{16} - 1621342 p^{6} T^{17} + 140052 p^{7} T^{18} - 10100 p^{8} T^{19} + 636 p^{9} T^{20} - 28 p^{10} T^{21} + p^{11} T^{22} \) | |
| 43 | \( 1 + 9 T + 298 T^{2} + 2284 T^{3} + 41540 T^{4} + 6207 p T^{5} + 3568505 T^{6} + 19251818 T^{7} + 5047386 p T^{8} + 1009701258 T^{9} + 10565479525 T^{10} + 45109212581 T^{11} + 10565479525 p T^{12} + 1009701258 p^{2} T^{13} + 5047386 p^{4} T^{14} + 19251818 p^{4} T^{15} + 3568505 p^{5} T^{16} + 6207 p^{7} T^{17} + 41540 p^{7} T^{18} + 2284 p^{8} T^{19} + 298 p^{9} T^{20} + 9 p^{10} T^{21} + p^{11} T^{22} \) | |
| 47 | \( 1 - 31 T + 745 T^{2} - 12444 T^{3} + 179301 T^{4} - 2137679 T^{5} + 23014026 T^{6} - 218122181 T^{7} + 1920048340 T^{8} - 15376897025 T^{9} + 116597912461 T^{10} - 818388851633 T^{11} + 116597912461 p T^{12} - 15376897025 p^{2} T^{13} + 1920048340 p^{3} T^{14} - 218122181 p^{4} T^{15} + 23014026 p^{5} T^{16} - 2137679 p^{6} T^{17} + 179301 p^{7} T^{18} - 12444 p^{8} T^{19} + 745 p^{9} T^{20} - 31 p^{10} T^{21} + p^{11} T^{22} \) | |
| 53 | \( 1 - 13 T + 358 T^{2} - 3896 T^{3} + 62447 T^{4} - 590500 T^{5} + 7139327 T^{6} - 59734832 T^{7} + 600753113 T^{8} - 4504227641 T^{9} + 39567917817 T^{10} - 267066343647 T^{11} + 39567917817 p T^{12} - 4504227641 p^{2} T^{13} + 600753113 p^{3} T^{14} - 59734832 p^{4} T^{15} + 7139327 p^{5} T^{16} - 590500 p^{6} T^{17} + 62447 p^{7} T^{18} - 3896 p^{8} T^{19} + 358 p^{9} T^{20} - 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 59 | \( 1 - 11 T + 550 T^{2} - 4688 T^{3} + 132412 T^{4} - 882653 T^{5} + 18894091 T^{6} - 99079454 T^{7} + 1842751982 T^{8} - 7802636740 T^{9} + 135561611869 T^{10} - 495592967901 T^{11} + 135561611869 p T^{12} - 7802636740 p^{2} T^{13} + 1842751982 p^{3} T^{14} - 99079454 p^{4} T^{15} + 18894091 p^{5} T^{16} - 882653 p^{6} T^{17} + 132412 p^{7} T^{18} - 4688 p^{8} T^{19} + 550 p^{9} T^{20} - 11 p^{10} T^{21} + p^{11} T^{22} \) | |
| 61 | \( 1 + 19 T + 551 T^{2} + 7948 T^{3} + 137285 T^{4} + 1614889 T^{5} + 21182931 T^{6} + 211644750 T^{7} + 2293122452 T^{8} + 19870255791 T^{9} + 184309341373 T^{10} + 1394163244673 T^{11} + 184309341373 p T^{12} + 19870255791 p^{2} T^{13} + 2293122452 p^{3} T^{14} + 211644750 p^{4} T^{15} + 21182931 p^{5} T^{16} + 1614889 p^{6} T^{17} + 137285 p^{7} T^{18} + 7948 p^{8} T^{19} + 551 p^{9} T^{20} + 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 67 | \( 1 - 19 T + 592 T^{2} - 7404 T^{3} + 133961 T^{4} - 1196445 T^{5} + 16316391 T^{6} - 102989631 T^{7} + 1241792748 T^{8} - 5214311667 T^{9} + 72952299790 T^{10} - 247013648897 T^{11} + 72952299790 p T^{12} - 5214311667 p^{2} T^{13} + 1241792748 p^{3} T^{14} - 102989631 p^{4} T^{15} + 16316391 p^{5} T^{16} - 1196445 p^{6} T^{17} + 133961 p^{7} T^{18} - 7404 p^{8} T^{19} + 592 p^{9} T^{20} - 19 p^{10} T^{21} + p^{11} T^{22} \) | |
| 71 | \( 1 + 5 T + 306 T^{2} + 1673 T^{3} + 58714 T^{4} + 284004 T^{5} + 7772263 T^{6} + 35270566 T^{7} + 812365038 T^{8} + 3289918933 T^{9} + 68952722585 T^{10} + 259728789423 T^{11} + 68952722585 p T^{12} + 3289918933 p^{2} T^{13} + 812365038 p^{3} T^{14} + 35270566 p^{4} T^{15} + 7772263 p^{5} T^{16} + 284004 p^{6} T^{17} + 58714 p^{7} T^{18} + 1673 p^{8} T^{19} + 306 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 73 | \( 1 + 5 T + 473 T^{2} + 1653 T^{3} + 109305 T^{4} + 301112 T^{5} + 16906307 T^{6} + 40509145 T^{7} + 1953299352 T^{8} + 4240786091 T^{9} + 177323674207 T^{10} + 348987600625 T^{11} + 177323674207 p T^{12} + 4240786091 p^{2} T^{13} + 1953299352 p^{3} T^{14} + 40509145 p^{4} T^{15} + 16906307 p^{5} T^{16} + 301112 p^{6} T^{17} + 109305 p^{7} T^{18} + 1653 p^{8} T^{19} + 473 p^{9} T^{20} + 5 p^{10} T^{21} + p^{11} T^{22} \) | |
| 79 | \( 1 + 13 T + 567 T^{2} + 6864 T^{3} + 160143 T^{4} + 1738990 T^{5} + 29293243 T^{6} + 283603607 T^{7} + 3860418730 T^{8} + 33393719311 T^{9} + 388529864760 T^{10} + 2996434097201 T^{11} + 388529864760 p T^{12} + 33393719311 p^{2} T^{13} + 3860418730 p^{3} T^{14} + 283603607 p^{4} T^{15} + 29293243 p^{5} T^{16} + 1738990 p^{6} T^{17} + 160143 p^{7} T^{18} + 6864 p^{8} T^{19} + 567 p^{9} T^{20} + 13 p^{10} T^{21} + p^{11} T^{22} \) | |
| 83 | \( 1 - 17 T + 420 T^{2} - 6195 T^{3} + 95158 T^{4} - 1096354 T^{5} + 13912071 T^{6} - 138123669 T^{7} + 1519109286 T^{8} - 13981652946 T^{9} + 143006773665 T^{10} - 1219023699615 T^{11} + 143006773665 p T^{12} - 13981652946 p^{2} T^{13} + 1519109286 p^{3} T^{14} - 138123669 p^{4} T^{15} + 13912071 p^{5} T^{16} - 1096354 p^{6} T^{17} + 95158 p^{7} T^{18} - 6195 p^{8} T^{19} + 420 p^{9} T^{20} - 17 p^{10} T^{21} + p^{11} T^{22} \) | |
| 89 | \( 1 - 10 T + 749 T^{2} - 7437 T^{3} + 270566 T^{4} - 2567242 T^{5} + 62042298 T^{6} - 545225263 T^{7} + 10001563743 T^{8} - 79274897402 T^{9} + 1186884418841 T^{10} - 8271596800073 T^{11} + 1186884418841 p T^{12} - 79274897402 p^{2} T^{13} + 10001563743 p^{3} T^{14} - 545225263 p^{4} T^{15} + 62042298 p^{5} T^{16} - 2567242 p^{6} T^{17} + 270566 p^{7} T^{18} - 7437 p^{8} T^{19} + 749 p^{9} T^{20} - 10 p^{10} T^{21} + p^{11} T^{22} \) | |
| 97 | \( 1 - 35 T + 1210 T^{2} - 26369 T^{3} + 550815 T^{4} - 9020522 T^{5} + 142389388 T^{6} - 1900666731 T^{7} + 24688735693 T^{8} - 280916997270 T^{9} + 3135374988615 T^{10} - 31130119494289 T^{11} + 3135374988615 p T^{12} - 280916997270 p^{2} T^{13} + 24688735693 p^{3} T^{14} - 1900666731 p^{4} T^{15} + 142389388 p^{5} T^{16} - 9020522 p^{6} T^{17} + 550815 p^{7} T^{18} - 26369 p^{8} T^{19} + 1210 p^{9} T^{20} - 35 p^{10} T^{21} + p^{11} T^{22} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{22} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.57967789520958115214679446089, −2.52485324609763696943801798031, −2.37197722321271049182467924550, −2.19206911778933644191664279012, −2.04280650328974550075612124000, −2.00673365547415722459152064366, −1.98017212234095966715091721118, −1.94814040350850783146984090691, −1.83855707818759283710119936981, −1.76299711890003561666624921439, −1.57282404681281870755378194555, −1.52329969560520061137547742279, −1.44011570997897287556849796759, −1.39378913567255354818274828486, −1.21570888026817507460656444531, −1.01972023809861786744552649726, −0.942511784832934752247382491084, −0.69488230034167039341481078424, −0.69485990413596158048192105710, −0.65989505943509083022128677658, −0.63759512572313532036588909197, −0.62311173346806808920881804128, −0.54534951838396403364082192667, −0.39299692369410535603584922309, −0.18521519538263777324726135144, 0.18521519538263777324726135144, 0.39299692369410535603584922309, 0.54534951838396403364082192667, 0.62311173346806808920881804128, 0.63759512572313532036588909197, 0.65989505943509083022128677658, 0.69485990413596158048192105710, 0.69488230034167039341481078424, 0.942511784832934752247382491084, 1.01972023809861786744552649726, 1.21570888026817507460656444531, 1.39378913567255354818274828486, 1.44011570997897287556849796759, 1.52329969560520061137547742279, 1.57282404681281870755378194555, 1.76299711890003561666624921439, 1.83855707818759283710119936981, 1.94814040350850783146984090691, 1.98017212234095966715091721118, 2.00673365547415722459152064366, 2.04280650328974550075612124000, 2.19206911778933644191664279012, 2.37197722321271049182467924550, 2.52485324609763696943801798031, 2.57967789520958115214679446089
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.