Dirichlet series
| L(s) = 1 | − 10·2-s + 5·3-s + 55·4-s − 2·5-s − 50·6-s − 2·7-s − 220·8-s + 10·9-s + 20·10-s + 6·11-s + 275·12-s − 4·13-s + 20·14-s − 10·15-s + 715·16-s − 8·17-s − 100·18-s + 3·19-s − 110·20-s − 10·21-s − 60·22-s − 12·23-s − 1.10e3·24-s + 14·25-s + 40·26-s + 5·27-s − 110·28-s + ⋯ |
| L(s) = 1 | − 7.07·2-s + 2.88·3-s + 55/2·4-s − 0.894·5-s − 20.4·6-s − 0.755·7-s − 77.7·8-s + 10/3·9-s + 6.32·10-s + 1.80·11-s + 79.3·12-s − 1.10·13-s + 5.34·14-s − 2.58·15-s + 178.·16-s − 1.94·17-s − 23.5·18-s + 0.688·19-s − 24.5·20-s − 2.18·21-s − 12.7·22-s − 2.50·23-s − 224.·24-s + 14/5·25-s + 7.84·26-s + 0.962·27-s − 20.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.48140\times 10^{6}\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.005845088803\) |
| \(L(\frac12)\) | \(\approx\) | \(0.005845088803\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{10} \) |
| 3 | \( ( 1 - T + T^{2} )^{5} \) | |
| 7 | \( 1 + 2 T + 22 T^{3} + 44 T^{4} - 51 T^{5} + 44 p T^{6} + 22 p^{2} T^{7} + 2 p^{4} T^{9} + p^{5} T^{10} \) | |
| 13 | \( 1 + 4 T + 24 T^{2} + 44 T^{3} + 86 T^{4} + 15 T^{5} + 86 p T^{6} + 44 p^{2} T^{7} + 24 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 + 2 T - 2 p T^{2} - 44 T^{3} + 4 p T^{4} + 344 T^{5} + 368 T^{6} - 1484 T^{7} - 766 p T^{8} + 2784 T^{9} + 22431 T^{10} + 2784 p T^{11} - 766 p^{3} T^{12} - 1484 p^{3} T^{13} + 368 p^{4} T^{14} + 344 p^{5} T^{15} + 4 p^{7} T^{16} - 44 p^{7} T^{17} - 2 p^{9} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 11 | \( 1 - 6 T - 7 T^{2} + 140 T^{3} - 138 T^{4} - 1997 T^{5} + 5212 T^{6} + 19992 T^{7} - 102769 T^{8} - 75457 T^{9} + 1262787 T^{10} - 75457 p T^{11} - 102769 p^{2} T^{12} + 19992 p^{3} T^{13} + 5212 p^{4} T^{14} - 1997 p^{5} T^{15} - 138 p^{6} T^{16} + 140 p^{7} T^{17} - 7 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( ( 1 + 4 T + 65 T^{2} + 201 T^{3} + 109 p T^{4} + 4550 T^{5} + 109 p^{2} T^{6} + 201 p^{2} T^{7} + 65 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( 1 - 3 T - 17 T^{2} + 160 T^{3} - 459 T^{4} - 671 T^{5} + 16762 T^{6} - 63147 T^{7} + 33217 T^{8} + 901496 T^{9} - 5948112 T^{10} + 901496 p T^{11} + 33217 p^{2} T^{12} - 63147 p^{3} T^{13} + 16762 p^{4} T^{14} - 671 p^{5} T^{15} - 459 p^{6} T^{16} + 160 p^{7} T^{17} - 17 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( ( 1 + 6 T + 43 T^{2} + 15 T^{3} - 227 T^{4} - 5298 T^{5} - 227 p T^{6} + 15 p^{2} T^{7} + 43 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 29 | \( 1 - 85 T^{2} - 54 T^{3} + 3630 T^{4} + 3375 T^{5} - 96336 T^{6} - 63828 T^{7} + 1960125 T^{8} + 182925 T^{9} - 42333513 T^{10} + 182925 p T^{11} + 1960125 p^{2} T^{12} - 63828 p^{3} T^{13} - 96336 p^{4} T^{14} + 3375 p^{5} T^{15} + 3630 p^{6} T^{16} - 54 p^{7} T^{17} - 85 p^{8} T^{18} + p^{10} T^{20} \) | |
| 31 | \( 1 + 10 T - 29 T^{2} - 192 T^{3} + 3510 T^{4} + 6771 T^{5} - 90246 T^{6} + 119556 T^{7} + 1279461 T^{8} - 4544141 T^{9} - 16306697 T^{10} - 4544141 p T^{11} + 1279461 p^{2} T^{12} + 119556 p^{3} T^{13} - 90246 p^{4} T^{14} + 6771 p^{5} T^{15} + 3510 p^{6} T^{16} - 192 p^{7} T^{17} - 29 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( ( 1 + T + 147 T^{2} + 165 T^{3} + 9840 T^{4} + 8907 T^{5} + 9840 p T^{6} + 165 p^{2} T^{7} + 147 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 41 | \( 1 + 4 T - 157 T^{2} - 366 T^{3} + 14718 T^{4} + 17565 T^{5} - 1035036 T^{6} - 695484 T^{7} + 57489417 T^{8} + 14323171 T^{9} - 2587515041 T^{10} + 14323171 p T^{11} + 57489417 p^{2} T^{12} - 695484 p^{3} T^{13} - 1035036 p^{4} T^{14} + 17565 p^{5} T^{15} + 14718 p^{6} T^{16} - 366 p^{7} T^{17} - 157 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 3 T - 104 T^{2} + 223 T^{3} + 6210 T^{4} - 11084 T^{5} - 116252 T^{6} + 216573 T^{7} - 4797452 T^{8} - 4910440 T^{9} + 487444146 T^{10} - 4910440 p T^{11} - 4797452 p^{2} T^{12} + 216573 p^{3} T^{13} - 116252 p^{4} T^{14} - 11084 p^{5} T^{15} + 6210 p^{6} T^{16} + 223 p^{7} T^{17} - 104 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 15 T + 44 T^{2} - 685 T^{3} - 8256 T^{4} - 47792 T^{5} - 110249 T^{6} + 1247169 T^{7} + 15610607 T^{8} + 42731021 T^{9} - 121164885 T^{10} + 42731021 p T^{11} + 15610607 p^{2} T^{12} + 1247169 p^{3} T^{13} - 110249 p^{4} T^{14} - 47792 p^{5} T^{15} - 8256 p^{6} T^{16} - 685 p^{7} T^{17} + 44 p^{8} T^{18} + 15 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 17 T + 14 T^{2} - 497 T^{3} + 6152 T^{4} + 22370 T^{5} - 660061 T^{6} - 2005769 T^{7} + 23181553 T^{8} - 11517771 T^{9} - 1562966193 T^{10} - 11517771 p T^{11} + 23181553 p^{2} T^{12} - 2005769 p^{3} T^{13} - 660061 p^{4} T^{14} + 22370 p^{5} T^{15} + 6152 p^{6} T^{16} - 497 p^{7} T^{17} + 14 p^{8} T^{18} + 17 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( ( 1 + 2 T + 128 T^{2} + 912 T^{3} + 7118 T^{4} + 94960 T^{5} + 7118 p T^{6} + 912 p^{2} T^{7} + 128 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 61 | \( 1 - 11 T - 176 T^{2} + 1301 T^{3} + 28355 T^{4} - 111154 T^{5} - 3026620 T^{6} + 5812490 T^{7} + 246886997 T^{8} - 120782473 T^{9} - 16707910924 T^{10} - 120782473 p T^{11} + 246886997 p^{2} T^{12} + 5812490 p^{3} T^{13} - 3026620 p^{4} T^{14} - 111154 p^{5} T^{15} + 28355 p^{6} T^{16} + 1301 p^{7} T^{17} - 176 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + T - 197 T^{2} + 60 T^{3} + 21063 T^{4} - 22194 T^{5} - 1152135 T^{6} + 2653485 T^{7} + 27902568 T^{8} - 81311426 T^{9} + 308648563 T^{10} - 81311426 p T^{11} + 27902568 p^{2} T^{12} + 2653485 p^{3} T^{13} - 1152135 p^{4} T^{14} - 22194 p^{5} T^{15} + 21063 p^{6} T^{16} + 60 p^{7} T^{17} - 197 p^{8} T^{18} + p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 18 T - 40 T^{2} + 1108 T^{3} + 16884 T^{4} - 76762 T^{5} - 2223296 T^{6} + 6435546 T^{7} + 147203672 T^{8} - 9125804 T^{9} - 12149042091 T^{10} - 9125804 p T^{11} + 147203672 p^{2} T^{12} + 6435546 p^{3} T^{13} - 2223296 p^{4} T^{14} - 76762 p^{5} T^{15} + 16884 p^{6} T^{16} + 1108 p^{7} T^{17} - 40 p^{8} T^{18} - 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 12 T + 43 T^{2} - 472 T^{3} - 87 p T^{4} + 132407 T^{5} - 768992 T^{6} + 7360740 T^{7} + 20692501 T^{8} - 799259342 T^{9} + 4955096646 T^{10} - 799259342 p T^{11} + 20692501 p^{2} T^{12} + 7360740 p^{3} T^{13} - 768992 p^{4} T^{14} + 132407 p^{5} T^{15} - 87 p^{7} T^{16} - 472 p^{7} T^{17} + 43 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 4 T - 308 T^{2} - 1152 T^{3} + 53712 T^{4} + 168192 T^{5} - 6769152 T^{6} - 13333176 T^{7} + 690029892 T^{8} + 445430536 T^{9} - 59061818879 T^{10} + 445430536 p T^{11} + 690029892 p^{2} T^{12} - 13333176 p^{3} T^{13} - 6769152 p^{4} T^{14} + 168192 p^{5} T^{15} + 53712 p^{6} T^{16} - 1152 p^{7} T^{17} - 308 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 + 355 T^{2} - 27 T^{3} + 54355 T^{4} - 3996 T^{5} + 54355 p T^{6} - 27 p^{2} T^{7} + 355 p^{3} T^{8} + p^{5} T^{10} )^{2} \) | |
| 89 | \( ( 1 + 7 T - 19 T^{2} - 1278 T^{3} + 5897 T^{4} + 114872 T^{5} + 5897 p T^{6} - 1278 p^{2} T^{7} - 19 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 97 | \( 1 + 6 T - 206 T^{2} - 2336 T^{3} + 22716 T^{4} + 420385 T^{5} + 876934 T^{6} - 45385950 T^{7} - 466860704 T^{8} + 1843252112 T^{9} + 68681302440 T^{10} + 1843252112 p T^{11} - 466860704 p^{2} T^{12} - 45385950 p^{3} T^{13} + 876934 p^{4} T^{14} + 420385 p^{5} T^{15} + 22716 p^{6} T^{16} - 2336 p^{7} T^{17} - 206 p^{8} T^{18} + 6 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.71278363059800471022160630826, −3.68344569906954652173139568837, −3.58811405118349492471083770161, −3.51734602380226913905034998266, −3.37090577516217940796669472164, −3.25834089034242479170059033165, −3.04114643256215599596645519179, −2.94136081144713257299257382002, −2.90965862046940005784072720396, −2.89035846000635803807751267572, −2.88434261874344011050096735341, −2.26223663438874164996318449779, −2.20233622014064226469727346574, −2.13767372488385520035785765756, −2.07672024970241327805232997466, −2.02452400719138416987039130327, −1.99307053318053842454897535633, −1.93479114242809629275524679903, −1.58552163656874047479931164612, −1.13469009728378678193571582098, −1.07907436797511927983477601073, −1.02575454851706550483919886889, −0.955500380296911027281977551228, −0.14849969111678716171626430907, −0.11533513123380136627020962463, 0.11533513123380136627020962463, 0.14849969111678716171626430907, 0.955500380296911027281977551228, 1.02575454851706550483919886889, 1.07907436797511927983477601073, 1.13469009728378678193571582098, 1.58552163656874047479931164612, 1.93479114242809629275524679903, 1.99307053318053842454897535633, 2.02452400719138416987039130327, 2.07672024970241327805232997466, 2.13767372488385520035785765756, 2.20233622014064226469727346574, 2.26223663438874164996318449779, 2.88434261874344011050096735341, 2.89035846000635803807751267572, 2.90965862046940005784072720396, 2.94136081144713257299257382002, 3.04114643256215599596645519179, 3.25834089034242479170059033165, 3.37090577516217940796669472164, 3.51734602380226913905034998266, 3.58811405118349492471083770161, 3.68344569906954652173139568837, 3.71278363059800471022160630826
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.