Dirichlet series
| L(s) = 1 | − 6·2-s − 2·3-s + 24·4-s + 3·5-s + 12·6-s − 66·8-s + 3·9-s − 18·10-s + 15·11-s − 48·12-s − 18·13-s − 6·15-s + 143·16-s − 26·17-s − 18·18-s − 19-s + 72·20-s − 90·22-s − 6·23-s + 132·24-s + 5·25-s + 108·26-s − 22·27-s + 28·29-s + 36·30-s − 10·31-s − 242·32-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 1.15·3-s + 12·4-s + 1.34·5-s + 4.89·6-s − 23.3·8-s + 9-s − 5.69·10-s + 4.52·11-s − 13.8·12-s − 4.99·13-s − 1.54·15-s + 35.7·16-s − 6.30·17-s − 4.24·18-s − 0.229·19-s + 16.0·20-s − 19.1·22-s − 1.25·23-s + 26.9·24-s + 25-s + 21.1·26-s − 4.23·27-s + 5.19·29-s + 6.57·30-s − 1.79·31-s − 42.7·32-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
| \(L(1)\) | \(\approx\) | \(0.07887126118\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07887126118\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 67 | \( 1 - 23 T + 111 T^{2} + 967 T^{3} - 6379 T^{4} - 13685 T^{5} - 6379 p T^{6} + 967 p^{2} T^{7} + 111 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 2 | \( 1 + 3 p T + 3 p^{2} T^{2} - 3 p T^{3} - 71 T^{4} - 53 p T^{5} + 39 p T^{6} + 115 p^{2} T^{7} + 459 T^{8} - 509 T^{9} - 1673 T^{10} - 509 p T^{11} + 459 p^{2} T^{12} + 115 p^{5} T^{13} + 39 p^{5} T^{14} - 53 p^{6} T^{15} - 71 p^{6} T^{16} - 3 p^{8} T^{17} + 3 p^{10} T^{18} + 3 p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 2 T + T^{2} + 2 p^{2} T^{3} + 11 p T^{4} + 4 p T^{5} + 41 p T^{6} + 232 T^{7} + 95 T^{8} + 44 p^{2} T^{9} + 1189 T^{10} + 44 p^{3} T^{11} + 95 p^{2} T^{12} + 232 p^{3} T^{13} + 41 p^{5} T^{14} + 4 p^{6} T^{15} + 11 p^{7} T^{16} + 2 p^{9} T^{17} + p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 - 3 T + 4 T^{2} - 6 p T^{3} + 14 p T^{4} - 159 T^{5} + 644 T^{6} - 1137 T^{7} + 3491 T^{8} - 9023 T^{9} + 12881 T^{10} - 9023 p T^{11} + 3491 p^{2} T^{12} - 1137 p^{3} T^{13} + 644 p^{4} T^{14} - 159 p^{5} T^{15} + 14 p^{7} T^{16} - 6 p^{8} T^{17} + 4 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 7 | \( 1 - p T^{2} + 55 T^{3} + 93 T^{4} - 440 T^{5} + 1175 T^{6} + 5577 T^{7} - 8962 T^{8} - 1562 T^{9} + 146939 T^{10} - 1562 p T^{11} - 8962 p^{2} T^{12} + 5577 p^{3} T^{13} + 1175 p^{4} T^{14} - 440 p^{5} T^{15} + 93 p^{6} T^{16} + 55 p^{7} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 - 15 T + 126 T^{2} - 757 T^{3} + 3479 T^{4} - 12112 T^{5} + 27955 T^{6} - 8431 T^{7} - 315526 T^{8} + 1986795 T^{9} - 7797811 T^{10} + 1986795 p T^{11} - 315526 p^{2} T^{12} - 8431 p^{3} T^{13} + 27955 p^{4} T^{14} - 12112 p^{5} T^{15} + 3479 p^{6} T^{16} - 757 p^{7} T^{17} + 126 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 18 T + 146 T^{2} + 53 p T^{3} + 2155 T^{4} + 5402 T^{5} + 13418 T^{6} + 33039 T^{7} + 136842 T^{8} + 73910 p T^{9} + 4602685 T^{10} + 73910 p^{2} T^{11} + 136842 p^{2} T^{12} + 33039 p^{3} T^{13} + 13418 p^{4} T^{14} + 5402 p^{5} T^{15} + 2155 p^{6} T^{16} + 53 p^{8} T^{17} + 146 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 26 T + 373 T^{2} + 3888 T^{3} + 32366 T^{4} + 226300 T^{5} + 1375228 T^{6} + 7447050 T^{7} + 36667420 T^{8} + 166910084 T^{9} + 710331073 T^{10} + 166910084 p T^{11} + 36667420 p^{2} T^{12} + 7447050 p^{3} T^{13} + 1375228 p^{4} T^{14} + 226300 p^{5} T^{15} + 32366 p^{6} T^{16} + 3888 p^{7} T^{17} + 373 p^{8} T^{18} + 26 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + T + 26 T^{2} + 216 T^{3} + 745 T^{4} + 7223 T^{5} + 26552 T^{6} + 180606 T^{7} + 43618 p T^{8} + 3043374 T^{9} + 21110165 T^{10} + 3043374 p T^{11} + 43618 p^{3} T^{12} + 180606 p^{3} T^{13} + 26552 p^{4} T^{14} + 7223 p^{5} T^{15} + 745 p^{6} T^{16} + 216 p^{7} T^{17} + 26 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 + 6 T + 2 p T^{2} + 292 T^{3} + 2014 T^{4} + 12892 T^{5} + 71697 T^{6} + 391682 T^{7} + 2068724 T^{8} + 10751196 T^{9} + 57015969 T^{10} + 10751196 p T^{11} + 2068724 p^{2} T^{12} + 391682 p^{3} T^{13} + 71697 p^{4} T^{14} + 12892 p^{5} T^{15} + 2014 p^{6} T^{16} + 292 p^{7} T^{17} + 2 p^{9} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 - 14 T + 186 T^{2} - 1499 T^{3} + 11494 T^{4} - 63593 T^{5} + 11494 p T^{6} - 1499 p^{2} T^{7} + 186 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 + 10 T + 102 T^{2} + 1249 T^{3} + 10329 T^{4} + 81687 T^{5} + 630673 T^{6} + 4474506 T^{7} + 28121671 T^{8} + 171796966 T^{9} + 1032212413 T^{10} + 171796966 p T^{11} + 28121671 p^{2} T^{12} + 4474506 p^{3} T^{13} + 630673 p^{4} T^{14} + 81687 p^{5} T^{15} + 10329 p^{6} T^{16} + 1249 p^{7} T^{17} + 102 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 + 11 T + 130 T^{2} + 1177 T^{3} + 9389 T^{4} + 56001 T^{5} + 9389 p T^{6} + 1177 p^{2} T^{7} + 130 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 - 9 T + 40 T^{2} + 306 T^{3} + 160 T^{4} - 11313 T^{5} + 147826 T^{6} + 497850 T^{7} - 658742 T^{8} - 2759394 T^{9} + 350382319 T^{10} - 2759394 p T^{11} - 658742 p^{2} T^{12} + 497850 p^{3} T^{13} + 147826 p^{4} T^{14} - 11313 p^{5} T^{15} + 160 p^{6} T^{16} + 306 p^{7} T^{17} + 40 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 21 T + 288 T^{2} - 2769 T^{3} + 21840 T^{4} - 140055 T^{5} + 652423 T^{6} - 1065890 T^{7} - 17392825 T^{8} + 230128679 T^{9} - 1793081113 T^{10} + 230128679 p T^{11} - 17392825 p^{2} T^{12} - 1065890 p^{3} T^{13} + 652423 p^{4} T^{14} - 140055 p^{5} T^{15} + 21840 p^{6} T^{16} - 2769 p^{7} T^{17} + 288 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 22 T + 250 T^{2} + 1386 T^{3} - 1982 T^{4} - 113916 T^{5} - 1065993 T^{6} - 4765420 T^{7} + 5013254 T^{8} + 268925668 T^{9} + 2504576689 T^{10} + 268925668 p T^{11} + 5013254 p^{2} T^{12} - 4765420 p^{3} T^{13} - 1065993 p^{4} T^{14} - 113916 p^{5} T^{15} - 1982 p^{6} T^{16} + 1386 p^{7} T^{17} + 250 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 - 20 T + 116 T^{2} + 489 T^{3} - 12903 T^{4} + 90166 T^{5} - 42792 T^{6} - 3807843 T^{7} + 23205826 T^{8} - 707784 T^{9} - 545699835 T^{10} - 707784 p T^{11} + 23205826 p^{2} T^{12} - 3807843 p^{3} T^{13} - 42792 p^{4} T^{14} + 90166 p^{5} T^{15} - 12903 p^{6} T^{16} + 489 p^{7} T^{17} + 116 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 17 T + 241 T^{2} + 2599 T^{3} + 22814 T^{4} + 177187 T^{5} + 1323866 T^{6} + 8330851 T^{7} + 63828025 T^{8} + 477832256 T^{9} + 3640127217 T^{10} + 477832256 p T^{11} + 63828025 p^{2} T^{12} + 8330851 p^{3} T^{13} + 1323866 p^{4} T^{14} + 177187 p^{5} T^{15} + 22814 p^{6} T^{16} + 2599 p^{7} T^{17} + 241 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 13 T + 174 T^{2} - 1337 T^{3} + 19571 T^{4} - 174296 T^{5} + 1981255 T^{6} - 12996507 T^{7} + 132793130 T^{8} - 952257907 T^{9} + 9819441367 T^{10} - 952257907 p T^{11} + 132793130 p^{2} T^{12} - 12996507 p^{3} T^{13} + 1981255 p^{4} T^{14} - 174296 p^{5} T^{15} + 19571 p^{6} T^{16} - 1337 p^{7} T^{17} + 174 p^{8} T^{18} - 13 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - T - 81 T^{2} + 603 T^{3} - 9581 T^{4} - 22881 T^{5} + 1272030 T^{6} - 6773213 T^{7} + 338422 p T^{8} + 397226390 T^{9} - 8711303457 T^{10} + 397226390 p T^{11} + 338422 p^{3} T^{12} - 6773213 p^{3} T^{13} + 1272030 p^{4} T^{14} - 22881 p^{5} T^{15} - 9581 p^{6} T^{16} + 603 p^{7} T^{17} - 81 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 18 T + 207 T^{2} + 1796 T^{3} + 19065 T^{4} + 229904 T^{5} + 2431707 T^{6} + 18414708 T^{7} + 149245213 T^{8} + 1425344942 T^{9} + 13293131103 T^{10} + 1425344942 p T^{11} + 149245213 p^{2} T^{12} + 18414708 p^{3} T^{13} + 2431707 p^{4} T^{14} + 229904 p^{5} T^{15} + 19065 p^{6} T^{16} + 1796 p^{7} T^{17} + 207 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 5 T - 76 T^{2} + 2590 T^{3} - 1270 T^{4} - 218797 T^{5} + 2276594 T^{6} + 15204858 T^{7} - 184964002 T^{8} + 26446076 T^{9} + 19410983107 T^{10} + 26446076 p T^{11} - 184964002 p^{2} T^{12} + 15204858 p^{3} T^{13} + 2276594 p^{4} T^{14} - 218797 p^{5} T^{15} - 1270 p^{6} T^{16} + 2590 p^{7} T^{17} - 76 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 12 T - 115 T^{2} - 1540 T^{3} + 6465 T^{4} + 30181 T^{5} - 945413 T^{6} - 42680 p T^{7} + 7145785 T^{8} + 397493039 T^{9} + 7663760257 T^{10} + 397493039 p T^{11} + 7145785 p^{2} T^{12} - 42680 p^{4} T^{13} - 945413 p^{4} T^{14} + 30181 p^{5} T^{15} + 6465 p^{6} T^{16} - 1540 p^{7} T^{17} - 115 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 51 T + 1126 T^{2} - 15003 T^{3} + 164417 T^{4} - 1953568 T^{5} + 22220139 T^{6} - 202706651 T^{7} + 1729561976 T^{8} - 17858590849 T^{9} + 185096728543 T^{10} - 17858590849 p T^{11} + 1729561976 p^{2} T^{12} - 202706651 p^{3} T^{13} + 22220139 p^{4} T^{14} - 1953568 p^{5} T^{15} + 164417 p^{6} T^{16} - 15003 p^{7} T^{17} + 1126 p^{8} T^{18} - 51 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 5 T + 396 T^{2} + 1653 T^{3} + 69681 T^{4} + 225603 T^{5} + 69681 p T^{6} + 1653 p^{2} T^{7} + 396 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−6.36482481701108108846189516670, −6.34075392809389608307121277984, −6.33339869963557778593805852611, −6.07500849640953219260349586335, −5.96497745018869722875971425292, −5.87441071277092371701818614499, −5.41294864253551256224147896572, −5.03773897762151392801797217235, −4.98974771778447620779401953629, −4.87753131370298889005593320325, −4.80551532546529382123211624942, −4.72610928698281295875661100000, −4.18178578843873943528256330715, −4.04353129471180137993356740970, −4.00649036787726745642882726746, −3.89270894046844294842534682873, −3.71232404412472618660734369885, −2.75257143067634281787472220995, −2.51637970927187035342813907583, −2.48231968057767050624373350024, −2.27074473713084819478402558799, −2.13604216291331806054805858341, −1.81691943013905374783199452849, −1.75870967665180397768350887951, −1.49269717254852779529034636948, 1.49269717254852779529034636948, 1.75870967665180397768350887951, 1.81691943013905374783199452849, 2.13604216291331806054805858341, 2.27074473713084819478402558799, 2.48231968057767050624373350024, 2.51637970927187035342813907583, 2.75257143067634281787472220995, 3.71232404412472618660734369885, 3.89270894046844294842534682873, 4.00649036787726745642882726746, 4.04353129471180137993356740970, 4.18178578843873943528256330715, 4.72610928698281295875661100000, 4.80551532546529382123211624942, 4.87753131370298889005593320325, 4.98974771778447620779401953629, 5.03773897762151392801797217235, 5.41294864253551256224147896572, 5.87441071277092371701818614499, 5.96497745018869722875971425292, 6.07500849640953219260349586335, 6.33339869963557778593805852611, 6.34075392809389608307121277984, 6.36482481701108108846189516670
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.