Dirichlet series
| L(s) = 1 | − 2·2-s − 8·3-s − 3·4-s + 10·5-s + 16·6-s + 8·8-s + 22·9-s − 20·10-s + 10·11-s + 24·12-s − 8·13-s − 80·15-s + 16-s − 28·17-s − 44·18-s − 30·20-s − 20·22-s − 16·23-s − 64·24-s + 55·25-s + 16·26-s − 8·27-s + 160·30-s − 20·31-s − 10·32-s − 80·33-s + 56·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 4.61·3-s − 3/2·4-s + 4.47·5-s + 6.53·6-s + 2.82·8-s + 22/3·9-s − 6.32·10-s + 3.01·11-s + 6.92·12-s − 2.21·13-s − 20.6·15-s + 1/4·16-s − 6.79·17-s − 10.3·18-s − 6.70·20-s − 4.26·22-s − 3.33·23-s − 13.0·24-s + 11·25-s + 3.13·26-s − 1.53·27-s + 29.2·30-s − 3.59·31-s − 1.76·32-s − 13.9·33-s + 9.60·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{20} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(5^{10} \cdot 7^{20} \cdot 11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.12989\times 10^{13}\) |
| Root analytic conductor: | \(4.63893\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(10\) |
| Selberg data: | \((20,\ 5^{10} \cdot 7^{20} \cdot 11^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{10} \) |
| 7 | \( 1 \) | |
| 11 | \( ( 1 - T )^{10} \) | |
| good | 2 | \( 1 + p T + 7 T^{2} + 3 p^{2} T^{3} + 7 p^{2} T^{4} + 11 p^{2} T^{5} + 45 p T^{6} + 33 p^{2} T^{7} + 239 T^{8} + 163 p T^{9} + 517 T^{10} + 163 p^{2} T^{11} + 239 p^{2} T^{12} + 33 p^{5} T^{13} + 45 p^{5} T^{14} + 11 p^{7} T^{15} + 7 p^{8} T^{16} + 3 p^{9} T^{17} + 7 p^{8} T^{18} + p^{10} T^{19} + p^{10} T^{20} \) |
| 3 | \( 1 + 8 T + 14 p T^{2} + 56 p T^{3} + 559 T^{4} + 1612 T^{5} + 1384 p T^{6} + 9664 T^{7} + 20590 T^{8} + 40280 T^{9} + 72658 T^{10} + 40280 p T^{11} + 20590 p^{2} T^{12} + 9664 p^{3} T^{13} + 1384 p^{5} T^{14} + 1612 p^{5} T^{15} + 559 p^{6} T^{16} + 56 p^{8} T^{17} + 14 p^{9} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 + 8 T + 72 T^{2} + 360 T^{3} + 2219 T^{4} + 9004 T^{5} + 44734 T^{6} + 150044 T^{7} + 669778 T^{8} + 2057356 T^{9} + 8991730 T^{10} + 2057356 p T^{11} + 669778 p^{2} T^{12} + 150044 p^{3} T^{13} + 44734 p^{4} T^{14} + 9004 p^{5} T^{15} + 2219 p^{6} T^{16} + 360 p^{7} T^{17} + 72 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 28 T + 468 T^{2} + 5656 T^{3} + 54595 T^{4} + 440196 T^{5} + 3056868 T^{6} + 18610808 T^{7} + 100621330 T^{8} + 486651220 T^{9} + 2115417518 T^{10} + 486651220 p T^{11} + 100621330 p^{2} T^{12} + 18610808 p^{3} T^{13} + 3056868 p^{4} T^{14} + 440196 p^{5} T^{15} + 54595 p^{6} T^{16} + 5656 p^{7} T^{17} + 468 p^{8} T^{18} + 28 p^{9} T^{19} + p^{10} T^{20} \) | |
| 19 | \( 1 + 108 T^{2} + 56 T^{3} + 5717 T^{4} + 4576 T^{5} + 202136 T^{6} + 184096 T^{7} + 5360802 T^{8} + 4924760 T^{9} + 112997344 T^{10} + 4924760 p T^{11} + 5360802 p^{2} T^{12} + 184096 p^{3} T^{13} + 202136 p^{4} T^{14} + 4576 p^{5} T^{15} + 5717 p^{6} T^{16} + 56 p^{7} T^{17} + 108 p^{8} T^{18} + p^{10} T^{20} \) | |
| 23 | \( 1 + 16 T + 192 T^{2} + 1824 T^{3} + 15179 T^{4} + 107764 T^{5} + 703184 T^{6} + 4161500 T^{7} + 23102264 T^{8} + 119359600 T^{9} + 590861948 T^{10} + 119359600 p T^{11} + 23102264 p^{2} T^{12} + 4161500 p^{3} T^{13} + 703184 p^{4} T^{14} + 107764 p^{5} T^{15} + 15179 p^{6} T^{16} + 1824 p^{7} T^{17} + 192 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 29 | \( 1 + 70 T^{2} - 112 T^{3} + 2625 T^{4} - 3816 T^{5} + 116656 T^{6} - 7032 T^{7} + 3773414 T^{8} - 2246928 T^{9} + 93067620 T^{10} - 2246928 p T^{11} + 3773414 p^{2} T^{12} - 7032 p^{3} T^{13} + 116656 p^{4} T^{14} - 3816 p^{5} T^{15} + 2625 p^{6} T^{16} - 112 p^{7} T^{17} + 70 p^{8} T^{18} + p^{10} T^{20} \) | |
| 31 | \( 1 + 20 T + 292 T^{2} + 3172 T^{3} + 31073 T^{4} + 267028 T^{5} + 2109618 T^{6} + 15070796 T^{7} + 100597982 T^{8} + 621652600 T^{9} + 3601225638 T^{10} + 621652600 p T^{11} + 100597982 p^{2} T^{12} + 15070796 p^{3} T^{13} + 2109618 p^{4} T^{14} + 267028 p^{5} T^{15} + 31073 p^{6} T^{16} + 3172 p^{7} T^{17} + 292 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 36 T + 714 T^{2} + 9652 T^{3} + 97711 T^{4} + 762180 T^{5} + 4561860 T^{6} + 19330924 T^{7} + 37589440 T^{8} - 209546648 T^{9} - 2331036104 T^{10} - 209546648 p T^{11} + 37589440 p^{2} T^{12} + 19330924 p^{3} T^{13} + 4561860 p^{4} T^{14} + 762180 p^{5} T^{15} + 97711 p^{6} T^{16} + 9652 p^{7} T^{17} + 714 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 + 36 T + 834 T^{2} + 13852 T^{3} + 186805 T^{4} + 2103916 T^{5} + 504406 p T^{6} + 179822220 T^{7} + 1417820634 T^{8} + 10204357424 T^{9} + 67996785762 T^{10} + 10204357424 p T^{11} + 1417820634 p^{2} T^{12} + 179822220 p^{3} T^{13} + 504406 p^{5} T^{14} + 2103916 p^{5} T^{15} + 186805 p^{6} T^{16} + 13852 p^{7} T^{17} + 834 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 + 4 T + 226 T^{2} + 1324 T^{3} + 28523 T^{4} + 178124 T^{5} + 2513324 T^{6} + 15080996 T^{7} + 162564076 T^{8} + 894355664 T^{9} + 7990847176 T^{10} + 894355664 p T^{11} + 162564076 p^{2} T^{12} + 15080996 p^{3} T^{13} + 2513324 p^{4} T^{14} + 178124 p^{5} T^{15} + 28523 p^{6} T^{16} + 1324 p^{7} T^{17} + 226 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( 1 + 12 T + 304 T^{2} + 2880 T^{3} + 44295 T^{4} + 353648 T^{5} + 4199722 T^{6} + 29164256 T^{7} + 290369194 T^{8} + 1778727616 T^{9} + 15433155834 T^{10} + 1778727616 p T^{11} + 290369194 p^{2} T^{12} + 29164256 p^{3} T^{13} + 4199722 p^{4} T^{14} + 353648 p^{5} T^{15} + 44295 p^{6} T^{16} + 2880 p^{7} T^{17} + 304 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 53 | \( 1 + 16 T + 282 T^{2} + 2276 T^{3} + 18563 T^{4} + 55424 T^{5} + 204824 T^{6} + 810856 T^{7} + 42357180 T^{8} + 574099988 T^{9} + 5251868688 T^{10} + 574099988 p T^{11} + 42357180 p^{2} T^{12} + 810856 p^{3} T^{13} + 204824 p^{4} T^{14} + 55424 p^{5} T^{15} + 18563 p^{6} T^{16} + 2276 p^{7} T^{17} + 282 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 + 32 T + 654 T^{2} + 9728 T^{3} + 125149 T^{4} + 1405740 T^{5} + 14631590 T^{6} + 139635092 T^{7} + 1260449918 T^{8} + 10552974808 T^{9} + 83917029890 T^{10} + 10552974808 p T^{11} + 1260449918 p^{2} T^{12} + 139635092 p^{3} T^{13} + 14631590 p^{4} T^{14} + 1405740 p^{5} T^{15} + 125149 p^{6} T^{16} + 9728 p^{7} T^{17} + 654 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 - 16 T + 462 T^{2} - 4996 T^{3} + 86089 T^{4} - 719704 T^{5} + 9777068 T^{6} - 68360664 T^{7} + 812927518 T^{8} - 4997202660 T^{9} + 54330774270 T^{10} - 4997202660 p T^{11} + 812927518 p^{2} T^{12} - 68360664 p^{3} T^{13} + 9777068 p^{4} T^{14} - 719704 p^{5} T^{15} + 86089 p^{6} T^{16} - 4996 p^{7} T^{17} + 462 p^{8} T^{18} - 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 20 T + 458 T^{2} + 6624 T^{3} + 87771 T^{4} + 978688 T^{5} + 9693440 T^{6} + 87295464 T^{7} + 741374676 T^{8} + 5909725268 T^{9} + 49383725888 T^{10} + 5909725268 p T^{11} + 741374676 p^{2} T^{12} + 87295464 p^{3} T^{13} + 9693440 p^{4} T^{14} + 978688 p^{5} T^{15} + 87771 p^{6} T^{16} + 6624 p^{7} T^{17} + 458 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 - 12 T + 394 T^{2} - 3508 T^{3} + 70597 T^{4} - 493912 T^{5} + 7911736 T^{6} - 43998920 T^{7} + 653518698 T^{8} - 3083701432 T^{9} + 47311953804 T^{10} - 3083701432 p T^{11} + 653518698 p^{2} T^{12} - 43998920 p^{3} T^{13} + 7911736 p^{4} T^{14} - 493912 p^{5} T^{15} + 70597 p^{6} T^{16} - 3508 p^{7} T^{17} + 394 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 + 20 T + 546 T^{2} + 7916 T^{3} + 131903 T^{4} + 1525892 T^{5} + 19516288 T^{6} + 190410104 T^{7} + 2049626582 T^{8} + 17557091768 T^{9} + 167116501146 T^{10} + 17557091768 p T^{11} + 2049626582 p^{2} T^{12} + 190410104 p^{3} T^{13} + 19516288 p^{4} T^{14} + 1525892 p^{5} T^{15} + 131903 p^{6} T^{16} + 7916 p^{7} T^{17} + 546 p^{8} T^{18} + 20 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 - 12 T + 318 T^{2} - 3796 T^{3} + 57157 T^{4} - 603908 T^{5} + 7348416 T^{6} - 67061828 T^{7} + 724306158 T^{8} - 6082275872 T^{9} + 60444368288 T^{10} - 6082275872 p T^{11} + 724306158 p^{2} T^{12} - 67061828 p^{3} T^{13} + 7348416 p^{4} T^{14} - 603908 p^{5} T^{15} + 57157 p^{6} T^{16} - 3796 p^{7} T^{17} + 318 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 + 8 T + 380 T^{2} + 1804 T^{3} + 71837 T^{4} + 228148 T^{5} + 10097756 T^{6} + 24074052 T^{7} + 1122117730 T^{8} + 2012814020 T^{9} + 101247578472 T^{10} + 2012814020 p T^{11} + 1122117730 p^{2} T^{12} + 24074052 p^{3} T^{13} + 10097756 p^{4} T^{14} + 228148 p^{5} T^{15} + 71837 p^{6} T^{16} + 1804 p^{7} T^{17} + 380 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 + 68 T + 2796 T^{2} + 81868 T^{3} + 1901341 T^{4} + 36456656 T^{5} + 597306552 T^{6} + 8502225840 T^{7} + 106778029058 T^{8} + 1191339409968 T^{9} + 11884728172512 T^{10} + 1191339409968 p T^{11} + 106778029058 p^{2} T^{12} + 8502225840 p^{3} T^{13} + 597306552 p^{4} T^{14} + 36456656 p^{5} T^{15} + 1901341 p^{6} T^{16} + 81868 p^{7} T^{17} + 2796 p^{8} T^{18} + 68 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 + 36 T + 1082 T^{2} + 23200 T^{3} + 433609 T^{4} + 6890436 T^{5} + 98751540 T^{6} + 1272689436 T^{7} + 15116728182 T^{8} + 165655878652 T^{9} + 1691629475884 T^{10} + 165655878652 p T^{11} + 15116728182 p^{2} T^{12} + 1272689436 p^{3} T^{13} + 98751540 p^{4} T^{14} + 6890436 p^{5} T^{15} + 433609 p^{6} T^{16} + 23200 p^{7} T^{17} + 1082 p^{8} T^{18} + 36 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−3.59785872443314821734095731207, −3.54019919363138770918860078988, −3.45292229658917716693884288732, −3.42139623771665067661089331097, −3.17970355297055825941005151822, −2.87581181722159069568988826244, −2.85471518311817364837493881201, −2.80346546252412498256373569184, −2.77905498371038008724927367435, −2.75895378494430435875103371306, −2.45276633963271785572898596312, −2.37312701435264007339565680378, −2.23819314135425932973906297726, −1.95199968659934401787967555799, −1.94967499257513988262629216262, −1.92710821144806957905472571567, −1.84901479548460317571770613999, −1.83055435499033680280819180411, −1.73359644586573820089458140571, −1.66100589221573998788756513421, −1.39240809169510890835729973918, −1.38058107589483084845028209938, −1.23034445140009722951913589733, −1.13963713169875181307397835449, −1.01171065632545581910488420368, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.01171065632545581910488420368, 1.13963713169875181307397835449, 1.23034445140009722951913589733, 1.38058107589483084845028209938, 1.39240809169510890835729973918, 1.66100589221573998788756513421, 1.73359644586573820089458140571, 1.83055435499033680280819180411, 1.84901479548460317571770613999, 1.92710821144806957905472571567, 1.94967499257513988262629216262, 1.95199968659934401787967555799, 2.23819314135425932973906297726, 2.37312701435264007339565680378, 2.45276633963271785572898596312, 2.75895378494430435875103371306, 2.77905498371038008724927367435, 2.80346546252412498256373569184, 2.85471518311817364837493881201, 2.87581181722159069568988826244, 3.17970355297055825941005151822, 3.42139623771665067661089331097, 3.45292229658917716693884288732, 3.54019919363138770918860078988, 3.59785872443314821734095731207
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.