Dirichlet series
| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 4·9-s + 2·11-s − 4·12-s − 4·13-s + 2·16-s + 5·17-s − 8·18-s + 6·19-s − 4·22-s + 10·23-s + 2·25-s + 8·26-s + 8·27-s − 16·29-s − 5·32-s − 4·33-s − 10·34-s + 8·36-s − 8·37-s − 12·38-s + 8·39-s − 36·41-s + 16·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 4/3·9-s + 0.603·11-s − 1.15·12-s − 1.10·13-s + 1/2·16-s + 1.21·17-s − 1.88·18-s + 1.37·19-s − 0.852·22-s + 2.08·23-s + 2/5·25-s + 1.56·26-s + 1.53·27-s − 2.97·29-s − 0.883·32-s − 0.696·33-s − 1.71·34-s + 4/3·36-s − 1.31·37-s − 1.94·38-s + 1.28·39-s − 5.62·41-s + 2.43·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{20} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{20} \cdot 17^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.69520\times 10^{8}\) |
| Root analytic conductor: | \(2.57905\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{20} \cdot 17^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.03131094526\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03131094526\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 17 | \( ( 1 - T + T^{2} )^{5} \) | |
| good | 2 | \( 1 + p T + p T^{2} - 3 p T^{4} - 11 T^{5} - p T^{6} + p^{3} T^{7} + 5 p T^{8} - 5 p T^{9} - 39 T^{10} - 5 p^{2} T^{11} + 5 p^{3} T^{12} + p^{6} T^{13} - p^{5} T^{14} - 11 p^{5} T^{15} - 3 p^{7} T^{16} + p^{9} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 2 T - 16 T^{3} - 10 p T^{4} - 16 T^{5} + 140 T^{6} + 104 p T^{7} + 256 T^{8} - 244 p T^{9} - 625 p T^{10} - 244 p^{2} T^{11} + 256 p^{2} T^{12} + 104 p^{4} T^{13} + 140 p^{4} T^{14} - 16 p^{5} T^{15} - 10 p^{7} T^{16} - 16 p^{7} T^{17} + 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 5 | \( 1 - 2 T^{2} - 36 T^{3} - 32 T^{4} + 38 T^{5} + 612 T^{6} + 218 p T^{7} + 26 p^{2} T^{8} - 5508 T^{9} - 16701 T^{10} - 5508 p T^{11} + 26 p^{4} T^{12} + 218 p^{4} T^{13} + 612 p^{4} T^{14} + 38 p^{5} T^{15} - 32 p^{6} T^{16} - 36 p^{7} T^{17} - 2 p^{8} T^{18} + p^{10} T^{20} \) | |
| 11 | \( 1 - 2 T - 7 T^{2} + 74 T^{3} - 229 T^{4} + 108 T^{5} + 4338 T^{6} - 15228 T^{7} + 24133 T^{8} + 137098 T^{9} - 741573 T^{10} + 137098 p T^{11} + 24133 p^{2} T^{12} - 15228 p^{3} T^{13} + 4338 p^{4} T^{14} + 108 p^{5} T^{15} - 229 p^{6} T^{16} + 74 p^{7} T^{17} - 7 p^{8} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( ( 1 + 2 T + 25 T^{2} + 48 T^{3} + 482 T^{4} + 1116 T^{5} + 482 p T^{6} + 48 p^{2} T^{7} + 25 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 19 | \( 1 - 6 T - 47 T^{2} + 302 T^{3} + 1515 T^{4} - 8444 T^{5} - 39422 T^{6} + 141676 T^{7} + 945381 T^{8} - 1144098 T^{9} - 18964573 T^{10} - 1144098 p T^{11} + 945381 p^{2} T^{12} + 141676 p^{3} T^{13} - 39422 p^{4} T^{14} - 8444 p^{5} T^{15} + 1515 p^{6} T^{16} + 302 p^{7} T^{17} - 47 p^{8} T^{18} - 6 p^{9} T^{19} + p^{10} T^{20} \) | |
| 23 | \( 1 - 10 T - 7 T^{2} + 482 T^{3} - 1321 T^{4} - 340 p T^{5} + 46686 T^{6} - 4484 p T^{7} + 288013 T^{8} + 2301282 T^{9} - 27273737 T^{10} + 2301282 p T^{11} + 288013 p^{2} T^{12} - 4484 p^{4} T^{13} + 46686 p^{4} T^{14} - 340 p^{6} T^{15} - 1321 p^{6} T^{16} + 482 p^{7} T^{17} - 7 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( ( 1 + 8 T + 73 T^{2} + 16 p T^{3} + 3362 T^{4} + 16048 T^{5} + 3362 p T^{6} + 16 p^{3} T^{7} + 73 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 31 | \( 1 - 122 T^{2} + 188 T^{3} + 8420 T^{4} - 17312 T^{5} - 379004 T^{6} + 727986 T^{7} + 13152610 T^{8} - 11124172 T^{9} - 411431899 T^{10} - 11124172 p T^{11} + 13152610 p^{2} T^{12} + 727986 p^{3} T^{13} - 379004 p^{4} T^{14} - 17312 p^{5} T^{15} + 8420 p^{6} T^{16} + 188 p^{7} T^{17} - 122 p^{8} T^{18} + p^{10} T^{20} \) | |
| 37 | \( 1 + 8 T - 17 T^{2} - 856 T^{3} - 5185 T^{4} - 1408 T^{5} + 290514 T^{6} + 2278208 T^{7} + 5950285 T^{8} - 64405144 T^{9} - 651604271 T^{10} - 64405144 p T^{11} + 5950285 p^{2} T^{12} + 2278208 p^{3} T^{13} + 290514 p^{4} T^{14} - 1408 p^{5} T^{15} - 5185 p^{6} T^{16} - 856 p^{7} T^{17} - 17 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( ( 1 + 18 T + 284 T^{2} + 3016 T^{3} + 26390 T^{4} + 186634 T^{5} + 26390 p T^{6} + 3016 p^{2} T^{7} + 284 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 184 T^{2} - 1160 T^{3} + 14648 T^{4} - 71228 T^{5} + 14648 p T^{6} - 1160 p^{2} T^{7} + 184 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 47 | \( 1 + 10 T - 87 T^{2} - 258 T^{3} + 11711 T^{4} + 140 T^{5} - 576698 T^{6} + 35860 p T^{7} + 18948893 T^{8} - 50459410 T^{9} - 574319601 T^{10} - 50459410 p T^{11} + 18948893 p^{2} T^{12} + 35860 p^{4} T^{13} - 576698 p^{4} T^{14} + 140 p^{5} T^{15} + 11711 p^{6} T^{16} - 258 p^{7} T^{17} - 87 p^{8} T^{18} + 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 4 T - 216 T^{2} - 616 T^{3} + 27592 T^{4} + 53174 T^{5} - 2557048 T^{6} - 2536764 T^{7} + 188155648 T^{8} + 54851304 T^{9} - 11074176673 T^{10} + 54851304 p T^{11} + 188155648 p^{2} T^{12} - 2536764 p^{3} T^{13} - 2557048 p^{4} T^{14} + 53174 p^{5} T^{15} + 27592 p^{6} T^{16} - 616 p^{7} T^{17} - 216 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 8 T - 151 T^{2} + 776 T^{3} + 15335 T^{4} - 28464 T^{5} - 1227498 T^{6} + 392592 T^{7} + 79109629 T^{8} + 6761320 T^{9} - 4754678001 T^{10} + 6761320 p T^{11} + 79109629 p^{2} T^{12} + 392592 p^{3} T^{13} - 1227498 p^{4} T^{14} - 28464 p^{5} T^{15} + 15335 p^{6} T^{16} + 776 p^{7} T^{17} - 151 p^{8} T^{18} - 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 22 T + 36 T^{2} + 960 T^{3} + 20726 T^{4} - 229206 T^{5} - 1431408 T^{6} + 7610292 T^{7} + 185235172 T^{8} - 317522464 T^{9} - 11655910785 T^{10} - 317522464 p T^{11} + 185235172 p^{2} T^{12} + 7610292 p^{3} T^{13} - 1431408 p^{4} T^{14} - 229206 p^{5} T^{15} + 20726 p^{6} T^{16} + 960 p^{7} T^{17} + 36 p^{8} T^{18} - 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 16 T - 128 T^{2} - 1824 T^{3} + 30720 T^{4} + 222452 T^{5} - 3589184 T^{6} - 12433392 T^{7} + 354427968 T^{8} + 359391712 T^{9} - 26398984099 T^{10} + 359391712 p T^{11} + 354427968 p^{2} T^{12} - 12433392 p^{3} T^{13} - 3589184 p^{4} T^{14} + 222452 p^{5} T^{15} + 30720 p^{6} T^{16} - 1824 p^{7} T^{17} - 128 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( ( 1 + 2 T + 119 T^{2} - 304 T^{3} + 7614 T^{4} - 49636 T^{5} + 7614 p T^{6} - 304 p^{2} T^{7} + 119 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 73 | \( 1 - 10 T - 88 T^{2} - 464 T^{3} + 17954 T^{4} + 65190 T^{5} - 10816 T^{6} - 15626056 T^{7} - 38472344 T^{8} + 111514864 T^{9} + 12217591683 T^{10} + 111514864 p T^{11} - 38472344 p^{2} T^{12} - 15626056 p^{3} T^{13} - 10816 p^{4} T^{14} + 65190 p^{5} T^{15} + 17954 p^{6} T^{16} - 464 p^{7} T^{17} - 88 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 18 T - 111 T^{2} - 2458 T^{3} + 27415 T^{4} + 330796 T^{5} - 3242722 T^{6} - 22364388 T^{7} + 344455757 T^{8} + 659142630 T^{9} - 30517436849 T^{10} + 659142630 p T^{11} + 344455757 p^{2} T^{12} - 22364388 p^{3} T^{13} - 3242722 p^{4} T^{14} + 330796 p^{5} T^{15} + 27415 p^{6} T^{16} - 2458 p^{7} T^{17} - 111 p^{8} T^{18} + 18 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( ( 1 - 12 T + 351 T^{2} - 3032 T^{3} + 51082 T^{4} - 339960 T^{5} + 51082 p T^{6} - 3032 p^{2} T^{7} + 351 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 89 | \( 1 - 20 T + 55 T^{2} + 236 T^{3} + 9347 T^{4} - 9048 T^{5} + 115358 T^{6} - 18729272 T^{7} + 70022805 T^{8} + 321343508 T^{9} + 2223302725 T^{10} + 321343508 p T^{11} + 70022805 p^{2} T^{12} - 18729272 p^{3} T^{13} + 115358 p^{4} T^{14} - 9048 p^{5} T^{15} + 9347 p^{6} T^{16} + 236 p^{7} T^{17} + 55 p^{8} T^{18} - 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( ( 1 + 12 T + 246 T^{2} + 1890 T^{3} + 26704 T^{4} + 140626 T^{5} + 26704 p T^{6} + 1890 p^{2} T^{7} + 246 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
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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
−3.61290622328630246613613785999, −3.60143441767355678167968091353, −3.57319769113942052999650135666, −3.55725142384839940136331053143, −3.48829124190202128634937440661, −3.22498440221638579024495872032, −2.91928572722945563006361825655, −2.83897262040444831133543342953, −2.78590686399840408898511631156, −2.71520711042459873059246037554, −2.56943159171694245143573290403, −2.52572408928614389652683360278, −2.33060190955236257756060936564, −2.12407676530089895449020053449, −1.86208296288184310282254609845, −1.67159705497359184285802303203, −1.58129077884038498623860087882, −1.54682602394832086575259599223, −1.37083950948169089901850399822, −1.23976728102592441070394986796, −1.02494604996650200951131583561, −0.964061150274693267439179648030, −0.915346391482747927679348436060, −0.26641869411353927192761001453, −0.04529328960981488577251694267, 0.04529328960981488577251694267, 0.26641869411353927192761001453, 0.915346391482747927679348436060, 0.964061150274693267439179648030, 1.02494604996650200951131583561, 1.23976728102592441070394986796, 1.37083950948169089901850399822, 1.54682602394832086575259599223, 1.58129077884038498623860087882, 1.67159705497359184285802303203, 1.86208296288184310282254609845, 2.12407676530089895449020053449, 2.33060190955236257756060936564, 2.52572408928614389652683360278, 2.56943159171694245143573290403, 2.71520711042459873059246037554, 2.78590686399840408898511631156, 2.83897262040444831133543342953, 2.91928572722945563006361825655, 3.22498440221638579024495872032, 3.48829124190202128634937440661, 3.55725142384839940136331053143, 3.57319769113942052999650135666, 3.60143441767355678167968091353, 3.61290622328630246613613785999
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.