Dirichlet series
| L(s) = 1 | − 2-s − 2·5-s + 2·7-s + 2·10-s + 5·11-s + 13·13-s − 2·14-s − 5·22-s + 32·23-s + 5·25-s − 13·26-s − 27·29-s − 8·31-s − 4·35-s − 11·37-s + 10·41-s + 34·43-s − 32·46-s − 8·47-s + 18·49-s − 5·50-s − 9·53-s − 10·55-s + 27·58-s + 21·59-s − 4·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 0.755·7-s + 0.632·10-s + 1.50·11-s + 3.60·13-s − 0.534·14-s − 1.06·22-s + 6.67·23-s + 25-s − 2.54·26-s − 5.01·29-s − 1.43·31-s − 0.676·35-s − 1.80·37-s + 1.56·41-s + 5.18·43-s − 4.71·46-s − 1.16·47-s + 18/7·49-s − 0.707·50-s − 1.23·53-s − 1.34·55-s + 3.54·58-s + 2.73·59-s − 0.512·61-s + 1.01·62-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 23^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{10} \cdot 3^{20} \cdot 23^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(155874.\) |
| Root analytic conductor: | \(1.81818\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{10} \cdot 3^{20} \cdot 23^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.217078824\) |
| \(L(\frac12)\) | \(\approx\) | \(7.217078824\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 3 | \( 1 \) | |
| 23 | \( 1 - 32 T + 496 T^{2} - 4971 T^{3} + 35982 T^{4} - 197097 T^{5} + 35982 p T^{6} - 4971 p^{2} T^{7} + 496 p^{3} T^{8} - 32 p^{4} T^{9} + p^{5} T^{10} \) | |
| good | 5 | \( 1 + 2 T - T^{2} - 34 T^{3} - 118 T^{4} - 44 T^{5} + 28 p^{2} T^{6} + 4 p^{4} T^{7} + 2534 T^{8} - 8862 T^{9} - 33771 T^{10} - 8862 p T^{11} + 2534 p^{2} T^{12} + 4 p^{7} T^{13} + 28 p^{6} T^{14} - 44 p^{5} T^{15} - 118 p^{6} T^{16} - 34 p^{7} T^{17} - p^{8} T^{18} + 2 p^{9} T^{19} + p^{10} T^{20} \) |
| 7 | \( 1 - 2 T - 2 p T^{2} + 64 T^{3} + 2 p T^{4} - 520 T^{5} + 997 T^{6} + 1074 T^{7} - 636 p T^{8} + 3872 T^{9} + 4929 T^{10} + 3872 p T^{11} - 636 p^{3} T^{12} + 1074 p^{3} T^{13} + 997 p^{4} T^{14} - 520 p^{5} T^{15} + 2 p^{7} T^{16} + 64 p^{7} T^{17} - 2 p^{9} T^{18} - 2 p^{9} T^{19} + p^{10} T^{20} \) | |
| 11 | \( 1 - 5 T + 14 T^{2} - 48 T^{3} + 284 T^{4} - 1508 T^{5} + 6022 T^{6} - 19374 T^{7} + 63683 T^{8} - 245485 T^{9} + 973336 T^{10} - 245485 p T^{11} + 63683 p^{2} T^{12} - 19374 p^{3} T^{13} + 6022 p^{4} T^{14} - 1508 p^{5} T^{15} + 284 p^{6} T^{16} - 48 p^{7} T^{17} + 14 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20} \) | |
| 13 | \( 1 - p T + 112 T^{2} - 737 T^{3} + 4297 T^{4} - 22212 T^{5} + 104349 T^{6} - 448349 T^{7} + 1815610 T^{8} - 7003501 T^{9} + 25874507 T^{10} - 7003501 p T^{11} + 1815610 p^{2} T^{12} - 448349 p^{3} T^{13} + 104349 p^{4} T^{14} - 22212 p^{5} T^{15} + 4297 p^{6} T^{16} - 737 p^{7} T^{17} + 112 p^{8} T^{18} - p^{10} T^{19} + p^{10} T^{20} \) | |
| 17 | \( 1 + 38 T^{2} - 66 T^{3} + 641 T^{4} - 1749 T^{5} + 6879 T^{6} - 44561 T^{7} + 32129 T^{8} - 973544 T^{9} + 572177 T^{10} - 973544 p T^{11} + 32129 p^{2} T^{12} - 44561 p^{3} T^{13} + 6879 p^{4} T^{14} - 1749 p^{5} T^{15} + 641 p^{6} T^{16} - 66 p^{7} T^{17} + 38 p^{8} T^{18} + p^{10} T^{20} \) | |
| 19 | \( 1 - p T^{2} + 176 T^{3} + 614 T^{4} - 4686 T^{5} + 12248 T^{6} + 137500 T^{7} - 458146 T^{8} - 654258 T^{9} + 15808321 T^{10} - 654258 p T^{11} - 458146 p^{2} T^{12} + 137500 p^{3} T^{13} + 12248 p^{4} T^{14} - 4686 p^{5} T^{15} + 614 p^{6} T^{16} + 176 p^{7} T^{17} - p^{9} T^{18} + p^{10} T^{20} \) | |
| 29 | \( 1 + 27 T + 293 T^{2} + 1430 T^{3} + 1051 T^{4} - 9903 T^{5} + 188648 T^{6} + 2818838 T^{7} + 14843777 T^{8} + 21931694 T^{9} - 76635483 T^{10} + 21931694 p T^{11} + 14843777 p^{2} T^{12} + 2818838 p^{3} T^{13} + 188648 p^{4} T^{14} - 9903 p^{5} T^{15} + 1051 p^{6} T^{16} + 1430 p^{7} T^{17} + 293 p^{8} T^{18} + 27 p^{9} T^{19} + p^{10} T^{20} \) | |
| 31 | \( 1 + 8 T + 77 T^{2} + 544 T^{3} + 4660 T^{4} + 34408 T^{5} + 238318 T^{6} + 1455522 T^{7} + 9039162 T^{8} + 54519920 T^{9} + 329077869 T^{10} + 54519920 p T^{11} + 9039162 p^{2} T^{12} + 1455522 p^{3} T^{13} + 238318 p^{4} T^{14} + 34408 p^{5} T^{15} + 4660 p^{6} T^{16} + 544 p^{7} T^{17} + 77 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20} \) | |
| 37 | \( 1 + 11 T + 29 T^{2} + 33 T^{3} - 1832 T^{4} - 19360 T^{5} - 62126 T^{6} - 574167 T^{7} + 401382 T^{8} + 35935163 T^{9} + 206672157 T^{10} + 35935163 p T^{11} + 401382 p^{2} T^{12} - 574167 p^{3} T^{13} - 62126 p^{4} T^{14} - 19360 p^{5} T^{15} - 1832 p^{6} T^{16} + 33 p^{7} T^{17} + 29 p^{8} T^{18} + 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 41 | \( 1 - 10 T - 18 T^{2} + 1261 T^{3} - 9287 T^{4} - 8903 T^{5} + 605317 T^{6} - 101826 p T^{7} + 4849935 T^{8} + 127696826 T^{9} - 1265704111 T^{10} + 127696826 p T^{11} + 4849935 p^{2} T^{12} - 101826 p^{4} T^{13} + 605317 p^{4} T^{14} - 8903 p^{5} T^{15} - 9287 p^{6} T^{16} + 1261 p^{7} T^{17} - 18 p^{8} T^{18} - 10 p^{9} T^{19} + p^{10} T^{20} \) | |
| 43 | \( 1 - 34 T + 552 T^{2} - 6317 T^{3} + 57733 T^{4} - 406425 T^{5} + 2101299 T^{6} - 6135040 T^{7} - 25498593 T^{8} + 515102632 T^{9} - 4155248415 T^{10} + 515102632 p T^{11} - 25498593 p^{2} T^{12} - 6135040 p^{3} T^{13} + 2101299 p^{4} T^{14} - 406425 p^{5} T^{15} + 57733 p^{6} T^{16} - 6317 p^{7} T^{17} + 552 p^{8} T^{18} - 34 p^{9} T^{19} + p^{10} T^{20} \) | |
| 47 | \( ( 1 + 4 T + 160 T^{2} + 780 T^{3} + 259 p T^{4} + 54791 T^{5} + 259 p^{2} T^{6} + 780 p^{2} T^{7} + 160 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) | |
| 53 | \( 1 + 9 T + 50 T^{2} + 897 T^{3} + 2189 T^{4} + 4016 T^{5} + 178011 T^{6} - 2691595 T^{7} - 12753174 T^{8} - 56421387 T^{9} - 1653818693 T^{10} - 56421387 p T^{11} - 12753174 p^{2} T^{12} - 2691595 p^{3} T^{13} + 178011 p^{4} T^{14} + 4016 p^{5} T^{15} + 2189 p^{6} T^{16} + 897 p^{7} T^{17} + 50 p^{8} T^{18} + 9 p^{9} T^{19} + p^{10} T^{20} \) | |
| 59 | \( 1 - 21 T + 140 T^{2} + 1104 T^{3} - 28727 T^{4} + 166749 T^{5} + 1114192 T^{6} - 23861000 T^{7} + 112118264 T^{8} + 781962002 T^{9} - 12853732783 T^{10} + 781962002 p T^{11} + 112118264 p^{2} T^{12} - 23861000 p^{3} T^{13} + 1114192 p^{4} T^{14} + 166749 p^{5} T^{15} - 28727 p^{6} T^{16} + 1104 p^{7} T^{17} + 140 p^{8} T^{18} - 21 p^{9} T^{19} + p^{10} T^{20} \) | |
| 61 | \( 1 + 4 T - 56 T^{2} - 314 T^{3} - 4792 T^{4} + 1438 T^{5} + 420417 T^{6} + 840714 T^{7} + 6916578 T^{8} - 21284604 T^{9} - 1602435691 T^{10} - 21284604 p T^{11} + 6916578 p^{2} T^{12} + 840714 p^{3} T^{13} + 420417 p^{4} T^{14} + 1438 p^{5} T^{15} - 4792 p^{6} T^{16} - 314 p^{7} T^{17} - 56 p^{8} T^{18} + 4 p^{9} T^{19} + p^{10} T^{20} \) | |
| 67 | \( 1 + 32 T + 561 T^{2} + 7624 T^{3} + 94885 T^{4} + 1100561 T^{5} + 11593935 T^{6} + 110958000 T^{7} + 1007238837 T^{8} + 8910214435 T^{9} + 75409396017 T^{10} + 8910214435 p T^{11} + 1007238837 p^{2} T^{12} + 110958000 p^{3} T^{13} + 11593935 p^{4} T^{14} + 1100561 p^{5} T^{15} + 94885 p^{6} T^{16} + 7624 p^{7} T^{17} + 561 p^{8} T^{18} + 32 p^{9} T^{19} + p^{10} T^{20} \) | |
| 71 | \( 1 + 22 T + 127 T^{2} - 1210 T^{3} - 10557 T^{4} + 166232 T^{5} + 1883713 T^{6} - 11310112 T^{7} - 232400467 T^{8} - 135995222 T^{9} + 11981403895 T^{10} - 135995222 p T^{11} - 232400467 p^{2} T^{12} - 11310112 p^{3} T^{13} + 1883713 p^{4} T^{14} + 166232 p^{5} T^{15} - 10557 p^{6} T^{16} - 1210 p^{7} T^{17} + 127 p^{8} T^{18} + 22 p^{9} T^{19} + p^{10} T^{20} \) | |
| 73 | \( 1 - 43 T + 940 T^{2} - 14753 T^{3} + 178075 T^{4} - 1583506 T^{5} + 9182057 T^{6} - 2553979 T^{7} - 807802326 T^{8} + 12695771399 T^{9} - 126625496097 T^{10} + 12695771399 p T^{11} - 807802326 p^{2} T^{12} - 2553979 p^{3} T^{13} + 9182057 p^{4} T^{14} - 1583506 p^{5} T^{15} + 178075 p^{6} T^{16} - 14753 p^{7} T^{17} + 940 p^{8} T^{18} - 43 p^{9} T^{19} + p^{10} T^{20} \) | |
| 79 | \( 1 + 16 T - 65 T^{2} - 1732 T^{3} + 1040 T^{4} + 21226 T^{5} - 941544 T^{6} - 4082648 T^{7} + 53721368 T^{8} + 440846464 T^{9} + 1534736521 T^{10} + 440846464 p T^{11} + 53721368 p^{2} T^{12} - 4082648 p^{3} T^{13} - 941544 p^{4} T^{14} + 21226 p^{5} T^{15} + 1040 p^{6} T^{16} - 1732 p^{7} T^{17} - 65 p^{8} T^{18} + 16 p^{9} T^{19} + p^{10} T^{20} \) | |
| 83 | \( 1 - 3 T - 162 T^{2} + 262 T^{3} + 19986 T^{4} - 56283 T^{5} - 1274730 T^{6} + 3818798 T^{7} + 49483114 T^{8} - 125012970 T^{9} - 843229773 T^{10} - 125012970 p T^{11} + 49483114 p^{2} T^{12} + 3818798 p^{3} T^{13} - 1274730 p^{4} T^{14} - 56283 p^{5} T^{15} + 19986 p^{6} T^{16} + 262 p^{7} T^{17} - 162 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20} \) | |
| 89 | \( 1 - 11 T - 34 T^{2} + 2882 T^{3} - 33274 T^{4} + 42845 T^{5} + 3955742 T^{6} - 54651762 T^{7} + 192802710 T^{8} + 3282916736 T^{9} - 54976219101 T^{10} + 3282916736 p T^{11} + 192802710 p^{2} T^{12} - 54651762 p^{3} T^{13} + 3955742 p^{4} T^{14} + 42845 p^{5} T^{15} - 33274 p^{6} T^{16} + 2882 p^{7} T^{17} - 34 p^{8} T^{18} - 11 p^{9} T^{19} + p^{10} T^{20} \) | |
| 97 | \( 1 - 39 T + 676 T^{2} - 5839 T^{3} - 16403 T^{4} + 1329124 T^{5} - 20724837 T^{6} + 16999 p^{2} T^{7} + 104971296 T^{8} - 19946567527 T^{9} + 276660180643 T^{10} - 19946567527 p T^{11} + 104971296 p^{2} T^{12} + 16999 p^{5} T^{13} - 20724837 p^{4} T^{14} + 1329124 p^{5} T^{15} - 16403 p^{6} T^{16} - 5839 p^{7} T^{17} + 676 p^{8} T^{18} - 39 p^{9} T^{19} + p^{10} T^{20} \) | |
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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
−4.19495842919904058708291621265, −4.05856895741895308224798494162, −3.99907641662399050410067006606, −3.89686901634859544997865870779, −3.74926837812827055996931266916, −3.56986006795664743328949383669, −3.48148844000271675878438681917, −3.42373810581689663602888372028, −3.37465477958789807756286491374, −3.14422051027192550446337481931, −3.13412567513716610528838814406, −3.06081924946057397469322390482, −2.69403813274574026979379339045, −2.42862906184095370909443622852, −2.36127175084848787855739697153, −2.27309091049374158579589047337, −1.95234106407029569459481523450, −1.90621843097277047063820267114, −1.66168538140030909842434082430, −1.44963493488086127291137236478, −1.25081170583059260511368707525, −1.00987733979762569761962712388, −0.986994147402523752343717990887, −0.67365840120219943113683167781, −0.66403404568143706696033044759, 0.66403404568143706696033044759, 0.67365840120219943113683167781, 0.986994147402523752343717990887, 1.00987733979762569761962712388, 1.25081170583059260511368707525, 1.44963493488086127291137236478, 1.66168538140030909842434082430, 1.90621843097277047063820267114, 1.95234106407029569459481523450, 2.27309091049374158579589047337, 2.36127175084848787855739697153, 2.42862906184095370909443622852, 2.69403813274574026979379339045, 3.06081924946057397469322390482, 3.13412567513716610528838814406, 3.14422051027192550446337481931, 3.37465477958789807756286491374, 3.42373810581689663602888372028, 3.48148844000271675878438681917, 3.56986006795664743328949383669, 3.74926837812827055996931266916, 3.89686901634859544997865870779, 3.99907641662399050410067006606, 4.05856895741895308224798494162, 4.19495842919904058708291621265
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.