Dirichlet series
| L(s) = 1 | + 6·2-s + 24·4-s − 3·5-s + 66·8-s − 18·10-s − 15·11-s − 18·13-s + 143·16-s + 26·17-s − 19-s − 72·20-s − 90·22-s + 6·23-s + 5·25-s − 108·26-s − 28·29-s − 10·31-s + 242·32-s + 156·34-s − 22·37-s − 6·38-s − 198·40-s − 9·41-s + 21·43-s − 360·44-s + 36·46-s + 22·47-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 12·4-s − 1.34·5-s + 23.3·8-s − 5.69·10-s − 4.52·11-s − 4.99·13-s + 35.7·16-s + 6.30·17-s − 0.229·19-s − 16.0·20-s − 19.1·22-s + 1.25·23-s + 25-s − 21.1·26-s − 5.19·29-s − 1.79·31-s + 42.7·32-s + 26.7·34-s − 3.61·37-s − 0.973·38-s − 31.3·40-s − 1.40·41-s + 3.20·43-s − 54.2·44-s + 5.30·46-s + 3.20·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 67^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(3^{20} \cdot 67^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.69798\times 10^{6}\) |
| Root analytic conductor: | \(2.19430\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 3^{20} \cdot 67^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.882001395\) |
| \(L(\frac12)\) | \(\approx\) | \(2.882001395\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 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 | \( 1 - 3 p T + 3 p^{2} T^{2} + 3 p T^{3} - 71 T^{4} + 53 p T^{5} + 39 p T^{6} - 115 p^{2} T^{7} + 459 T^{8} + 509 T^{9} - 1673 T^{10} + 509 p T^{11} + 459 p^{2} T^{12} - 115 p^{5} T^{13} + 39 p^{5} T^{14} + 53 p^{6} T^{15} - 71 p^{6} T^{16} + 3 p^{8} T^{17} + 3 p^{10} T^{18} - 3 p^{10} T^{19} + p^{10} T^{20} \) |
| 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
−3.82805765384972525148359326130, −3.59947838356394479538081397869, −3.58949016611154597947017899553, −3.58887360803093043247024618245, −3.52570416991534183306482559626, −3.33775799935851921365722701537, −3.30565607537856393568988092299, −3.20684973109054256193427264571, −3.08907308448644199698628209039, −2.93893343373837981583439905691, −2.93050252170428469120488699276, −2.74013911355282468530920458822, −2.70003766971037348830941518443, −2.36463525996506841411781129009, −2.32323364236448530636620553960, −2.18751565949888116367777673191, −2.18179639537375902277389449260, −2.14098014581204963466086970247, −1.86435107126152840142272667530, −1.75405296622268896890758592323, −1.64690327924133919259547578105, −0.929493408372804059500641295079, −0.71407386859022166056039411978, −0.66795626669810595608214244917, −0.07706590972374041151095123412, 0.07706590972374041151095123412, 0.66795626669810595608214244917, 0.71407386859022166056039411978, 0.929493408372804059500641295079, 1.64690327924133919259547578105, 1.75405296622268896890758592323, 1.86435107126152840142272667530, 2.14098014581204963466086970247, 2.18179639537375902277389449260, 2.18751565949888116367777673191, 2.32323364236448530636620553960, 2.36463525996506841411781129009, 2.70003766971037348830941518443, 2.74013911355282468530920458822, 2.93050252170428469120488699276, 2.93893343373837981583439905691, 3.08907308448644199698628209039, 3.20684973109054256193427264571, 3.30565607537856393568988092299, 3.33775799935851921365722701537, 3.52570416991534183306482559626, 3.58887360803093043247024618245, 3.58949016611154597947017899553, 3.59947838356394479538081397869, 3.82805765384972525148359326130
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.