Dirichlet series
| L(s) = 1 | + 2·2-s − 8·3-s + 5·4-s − 6·5-s − 16·6-s + 2·7-s + 6·8-s + 22·9-s − 12·10-s + 2·11-s − 40·12-s + 4·14-s + 48·15-s + 12·16-s − 6·17-s + 44·18-s − 30·20-s − 16·21-s + 4·22-s + 2·23-s − 48·24-s − 25-s − 14·27-s + 10·28-s − 8·29-s + 96·30-s + 12·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 4.61·3-s + 5/2·4-s − 2.68·5-s − 6.53·6-s + 0.755·7-s + 2.12·8-s + 22/3·9-s − 3.79·10-s + 0.603·11-s − 11.5·12-s + 1.06·14-s + 12.3·15-s + 3·16-s − 1.45·17-s + 10.3·18-s − 6.70·20-s − 3.49·21-s + 0.852·22-s + 0.417·23-s − 9.79·24-s − 1/5·25-s − 2.69·27-s + 1.88·28-s − 1.48·29-s + 17.5·30-s + 2.15·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(67^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(67^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00192096\) |
| Root analytic conductor: | \(0.731435\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 67^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.07742943059\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07742943059\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 67 | \( 1 + 10 T + 24 T^{2} - 43 T^{3} - 994 T^{4} - 30801 T^{5} - 994 p T^{6} - 43 p^{2} T^{7} + 24 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 2 | \( 1 - p T - T^{2} + 3 p T^{3} - 7 T^{4} + 3 T^{5} + 3 p T^{6} - 7 p T^{7} + p^{4} T^{8} - 23 T^{10} + p^{6} T^{12} - 7 p^{4} T^{13} + 3 p^{5} T^{14} + 3 p^{5} T^{15} - 7 p^{6} T^{16} + 3 p^{8} T^{17} - p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( ( 1 + 4 T + 13 T^{2} + 35 T^{3} + 26 p T^{4} + 139 T^{5} + 26 p^{2} T^{6} + 35 p^{2} T^{7} + 13 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 5 | \( ( 1 + 3 T + 14 T^{2} + 38 T^{3} + 118 T^{4} + 221 T^{5} + 118 p T^{6} + 38 p^{2} T^{7} + 14 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 7 | \( 1 - 2 T - 18 T^{2} + 30 T^{3} + 192 T^{4} - 265 T^{5} - 1021 T^{6} + 183 p T^{7} + 1707 T^{8} - 2864 T^{9} + 10193 T^{10} - 2864 p T^{11} + 1707 p^{2} T^{12} + 183 p^{4} T^{13} - 1021 p^{4} T^{14} - 265 p^{5} T^{15} + 192 p^{6} T^{16} + 30 p^{7} T^{17} - 18 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 2 T - 42 T^{2} + 72 T^{3} + 1037 T^{4} - 1433 T^{5} - 17878 T^{6} + 16450 T^{7} + 244375 T^{8} - 78355 T^{9} - 2864861 T^{10} - 78355 p T^{11} + 244375 p^{2} T^{12} + 16450 p^{3} T^{13} - 17878 p^{4} T^{14} - 1433 p^{5} T^{15} + 1037 p^{6} T^{16} + 72 p^{7} T^{17} - 42 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - 29 T^{2} - 34 T^{3} + 343 T^{4} + 1104 T^{5} - 1205 T^{6} - 24099 T^{7} - 13842 T^{8} + 991 p^{2} T^{9} + 32653 p T^{10} + 991 p^{3} T^{11} - 13842 p^{2} T^{12} - 24099 p^{3} T^{13} - 1205 p^{4} T^{14} + 1104 p^{5} T^{15} + 343 p^{6} T^{16} - 34 p^{7} T^{17} - 29 p^{8} T^{18} + p^{10} T^{20} \) | |
| 17 | \( 1 + 6 T - 6 T^{2} - 120 T^{3} - 419 T^{4} - 933 T^{5} + 1706 T^{6} + 1182 p T^{7} + 60379 T^{8} + 76737 T^{9} + 29375 T^{10} + 76737 p T^{11} + 60379 p^{2} T^{12} + 1182 p^{4} T^{13} + 1706 p^{4} T^{14} - 933 p^{5} T^{15} - 419 p^{6} T^{16} - 120 p^{7} T^{17} - 6 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 - 62 T^{2} + 52 T^{3} + 1987 T^{4} - 2517 T^{5} - 43892 T^{6} + 54552 T^{7} + 812115 T^{8} - 427801 T^{9} - 14689649 T^{10} - 427801 p T^{11} + 812115 p^{2} T^{12} + 54552 p^{3} T^{13} - 43892 p^{4} T^{14} - 2517 p^{5} T^{15} + 1987 p^{6} T^{16} + 52 p^{7} T^{17} - 62 p^{8} T^{18} + p^{10} T^{20} \) | |
| 23 | \( 1 - 2 T - 42 T^{2} + 160 T^{3} + 913 T^{4} - 5369 T^{5} + 9122 T^{6} + 80458 T^{7} - 820325 T^{8} - 765615 T^{9} + 26139539 T^{10} - 765615 p T^{11} - 820325 p^{2} T^{12} + 80458 p^{3} T^{13} + 9122 p^{4} T^{14} - 5369 p^{5} T^{15} + 913 p^{6} T^{16} + 160 p^{7} T^{17} - 42 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 8 T - 85 T^{2} - 618 T^{3} + 6215 T^{4} + 33948 T^{5} - 289173 T^{6} - 928315 T^{7} + 11914474 T^{8} + 13517391 T^{9} - 365896223 T^{10} + 13517391 p T^{11} + 11914474 p^{2} T^{12} - 928315 p^{3} T^{13} - 289173 p^{4} T^{14} + 33948 p^{5} T^{15} + 6215 p^{6} T^{16} - 618 p^{7} T^{17} - 85 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 - 12 T + 10 T^{2} + 446 T^{3} - 2100 T^{4} + 4109 T^{5} - 28067 T^{6} - 54627 T^{7} + 2335123 T^{8} - 4641566 T^{9} - 36621615 T^{10} - 4641566 p T^{11} + 2335123 p^{2} T^{12} - 54627 p^{3} T^{13} - 28067 p^{4} T^{14} + 4109 p^{5} T^{15} - 2100 p^{6} T^{16} + 446 p^{7} T^{17} + 10 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 7 T - 109 T^{2} - 756 T^{3} + 7680 T^{4} + 46315 T^{5} - 387352 T^{6} - 1698511 T^{7} + 16316674 T^{8} + 736349 p T^{9} - 612615339 T^{10} + 736349 p^{2} T^{11} + 16316674 p^{2} T^{12} - 1698511 p^{3} T^{13} - 387352 p^{4} T^{14} + 46315 p^{5} T^{15} + 7680 p^{6} T^{16} - 756 p^{7} T^{17} - 109 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 16 T + 54 T^{2} + 20 T^{3} + 1561 T^{4} + 11447 T^{5} - 277204 T^{6} + 762140 T^{7} + 3399775 T^{8} - 23542623 T^{9} + 117526931 T^{10} - 23542623 p T^{11} + 3399775 p^{2} T^{12} + 762140 p^{3} T^{13} - 277204 p^{4} T^{14} + 11447 p^{5} T^{15} + 1561 p^{6} T^{16} + 20 p^{7} T^{17} + 54 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( ( 1 - 10 T + 219 T^{2} - 1601 T^{3} + 18934 T^{4} - 101111 T^{5} + 18934 p T^{6} - 1601 p^{2} T^{7} + 219 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 - 17 T + 5 T^{2} + 1146 T^{3} + 1736 T^{4} - 96453 T^{5} + 91416 T^{6} + 3265243 T^{7} - 1007090 T^{8} - 75075363 T^{9} + 179198953 T^{10} - 75075363 p T^{11} - 1007090 p^{2} T^{12} + 3265243 p^{3} T^{13} + 91416 p^{4} T^{14} - 96453 p^{5} T^{15} + 1736 p^{6} T^{16} + 1146 p^{7} T^{17} + 5 p^{8} T^{18} - 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( ( 1 - 4 T + 154 T^{2} - 488 T^{3} + 12925 T^{4} - 37599 T^{5} + 12925 p T^{6} - 488 p^{2} T^{7} + 154 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 59 | \( ( 1 - 3 T + 34 T^{2} + 21 T^{3} + 4651 T^{4} - 9441 T^{5} + 4651 p T^{6} + 21 p^{2} T^{7} + 34 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( 1 + 6 T - 254 T^{2} - 992 T^{3} + 41373 T^{4} + 104047 T^{5} - 4668926 T^{6} - 6031242 T^{7} + 409673599 T^{8} + 170589725 T^{9} - 27891698433 T^{10} + 170589725 p T^{11} + 409673599 p^{2} T^{12} - 6031242 p^{3} T^{13} - 4668926 p^{4} T^{14} + 104047 p^{5} T^{15} + 41373 p^{6} T^{16} - 992 p^{7} T^{17} - 254 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 3 T - 150 T^{2} + 657 T^{3} + 9859 T^{4} - 69174 T^{5} - 95809 T^{6} + 5947719 T^{7} - 41172644 T^{8} - 209724327 T^{9} + 4839715151 T^{10} - 209724327 p T^{11} - 41172644 p^{2} T^{12} + 5947719 p^{3} T^{13} - 95809 p^{4} T^{14} - 69174 p^{5} T^{15} + 9859 p^{6} T^{16} + 657 p^{7} T^{17} - 150 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 34 T + 388 T^{2} + 2352 T^{3} + 40863 T^{4} + 634809 T^{5} + 4211160 T^{6} + 36296610 T^{7} + 591294201 T^{8} + 4528250233 T^{9} + 23411672563 T^{10} + 4528250233 p T^{11} + 591294201 p^{2} T^{12} + 36296610 p^{3} T^{13} + 4211160 p^{4} T^{14} + 634809 p^{5} T^{15} + 40863 p^{6} T^{16} + 2352 p^{7} T^{17} + 388 p^{8} T^{18} + 34 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 8 T - 270 T^{2} + 1818 T^{3} + 44292 T^{4} - 214561 T^{5} - 5666239 T^{6} + 13542807 T^{7} + 620129991 T^{8} - 388513982 T^{9} - 54751185079 T^{10} - 388513982 p T^{11} + 620129991 p^{2} T^{12} + 13542807 p^{3} T^{13} - 5666239 p^{4} T^{14} - 214561 p^{5} T^{15} + 44292 p^{6} T^{16} + 1818 p^{7} T^{17} - 270 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 28 T + 186 T^{2} + 30 T^{3} + 19592 T^{4} - 224101 T^{5} - 1853131 T^{6} + 16994897 T^{7} + 143855677 T^{8} + 392032870 T^{9} - 27099579299 T^{10} + 392032870 p T^{11} + 143855677 p^{2} T^{12} + 16994897 p^{3} T^{13} - 1853131 p^{4} T^{14} - 224101 p^{5} T^{15} + 19592 p^{6} T^{16} + 30 p^{7} T^{17} + 186 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( ( 1 + 42 T + 1118 T^{2} + 20069 T^{3} + 277354 T^{4} + 2932285 T^{5} + 277354 p T^{6} + 20069 p^{2} T^{7} + 1118 p^{3} T^{8} + 42 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 + 20 T - 44 T^{2} - 3226 T^{3} - 5804 T^{4} + 234797 T^{5} - 280491 T^{6} - 18929403 T^{7} + 72426183 T^{8} + 1043577628 T^{9} - 2763958057 T^{10} + 1043577628 p T^{11} + 72426183 p^{2} T^{12} - 18929403 p^{3} T^{13} - 280491 p^{4} T^{14} + 234797 p^{5} T^{15} - 5804 p^{6} T^{16} - 3226 p^{7} T^{17} - 44 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−6.06375906657695665541462855225, −5.92986916436990438978361244108, −5.76466962035263440497215796025, −5.71062011387641909357999275819, −5.58787978256886913866914554812, −5.57539516799930994422659901410, −5.54702644610128287032589570767, −5.51098457778676529920569326575, −5.48484728028709492369061459611, −4.84998166752050669690295095316, −4.67101565087284020947347403334, −4.39797302216410795662837401628, −4.38304448587423974830019764898, −4.17212179153597783017201839980, −4.00650152661581402275430941861, −3.94622924273193361353095244981, −3.94315650197534710671974393400, −3.63053812417645499552815285497, −3.14542774796868383053637302967, −2.78643885898268681904839512760, −2.71601557415435324682696258754, −2.69123970027674781180484686574, −2.10559577302920571385192015993, −1.77841415788511703592157090336, −0.58456001798576908724228650320, 0.58456001798576908724228650320, 1.77841415788511703592157090336, 2.10559577302920571385192015993, 2.69123970027674781180484686574, 2.71601557415435324682696258754, 2.78643885898268681904839512760, 3.14542774796868383053637302967, 3.63053812417645499552815285497, 3.94315650197534710671974393400, 3.94622924273193361353095244981, 4.00650152661581402275430941861, 4.17212179153597783017201839980, 4.38304448587423974830019764898, 4.39797302216410795662837401628, 4.67101565087284020947347403334, 4.84998166752050669690295095316, 5.48484728028709492369061459611, 5.51098457778676529920569326575, 5.54702644610128287032589570767, 5.57539516799930994422659901410, 5.58787978256886913866914554812, 5.71062011387641909357999275819, 5.76466962035263440497215796025, 5.92986916436990438978361244108, 6.06375906657695665541462855225
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.