Dirichlet series
| L(s) = 1 | − 5·2-s − 9·3-s + 9·4-s − 4·5-s + 45·6-s + 5·7-s − 8·8-s + 45·9-s + 20·10-s + 11-s − 81·12-s + 13-s − 25·14-s + 36·15-s + 15·16-s − 2·17-s − 225·18-s − 9·19-s − 36·20-s − 45·21-s − 5·22-s − 4·23-s + 72·24-s − 14·25-s − 5·26-s − 165·27-s + 45·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s − 5.19·3-s + 9/2·4-s − 1.78·5-s + 18.3·6-s + 1.88·7-s − 2.82·8-s + 15·9-s + 6.32·10-s + 0.301·11-s − 23.3·12-s + 0.277·13-s − 6.68·14-s + 9.29·15-s + 15/4·16-s − 0.485·17-s − 53.0·18-s − 2.06·19-s − 8.04·20-s − 9.81·21-s − 1.06·22-s − 0.834·23-s + 14.6·24-s − 2.79·25-s − 0.980·26-s − 31.7·27-s + 8.50·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 29^{18}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(5.46700\times 10^{11}\) |
| Root analytic conductor: | \(4.48845\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 3^{9} \cdot 29^{18} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 3 | \( ( 1 + T )^{9} \) |
| 29 | \( 1 \) | |
| good | 2 | \( 1 + 5 T + p^{4} T^{2} + 43 T^{3} + 3 p^{5} T^{4} + 187 T^{5} + 335 T^{6} + 137 p^{2} T^{7} + 105 p^{3} T^{8} + 1225 T^{9} + 105 p^{4} T^{10} + 137 p^{4} T^{11} + 335 p^{3} T^{12} + 187 p^{4} T^{13} + 3 p^{10} T^{14} + 43 p^{6} T^{15} + p^{11} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 4 T + 6 p T^{2} + 92 T^{3} + 433 T^{4} + 1113 T^{5} + 4008 T^{6} + 8849 T^{7} + 26633 T^{8} + 51301 T^{9} + 26633 p T^{10} + 8849 p^{2} T^{11} + 4008 p^{3} T^{12} + 1113 p^{4} T^{13} + 433 p^{5} T^{14} + 92 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 - 5 T + 47 T^{2} - 172 T^{3} + 137 p T^{4} - 2893 T^{5} + 12304 T^{6} - 4576 p T^{7} + 113343 T^{8} - 258348 T^{9} + 113343 p T^{10} - 4576 p^{3} T^{11} + 12304 p^{3} T^{12} - 2893 p^{4} T^{13} + 137 p^{6} T^{14} - 172 p^{6} T^{15} + 47 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - T + 49 T^{2} + 1099 T^{4} + 1345 T^{5} + 15750 T^{6} + 39934 T^{7} + 180715 T^{8} + 581228 T^{9} + 180715 p T^{10} + 39934 p^{2} T^{11} + 15750 p^{3} T^{12} + 1345 p^{4} T^{13} + 1099 p^{5} T^{14} + 49 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - T + 30 T^{2} - 149 T^{3} + 716 T^{4} - 2996 T^{5} + 20065 T^{6} - 53796 T^{7} + 270263 T^{8} - 1030329 T^{9} + 270263 p T^{10} - 53796 p^{2} T^{11} + 20065 p^{3} T^{12} - 2996 p^{4} T^{13} + 716 p^{5} T^{14} - 149 p^{6} T^{15} + 30 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 2 T + 90 T^{2} + 154 T^{3} + 4290 T^{4} + 6733 T^{5} + 135098 T^{6} + 189810 T^{7} + 3077115 T^{8} + 3827129 T^{9} + 3077115 p T^{10} + 189810 p^{2} T^{11} + 135098 p^{3} T^{12} + 6733 p^{4} T^{13} + 4290 p^{5} T^{14} + 154 p^{6} T^{15} + 90 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 9 T + 92 T^{2} + 594 T^{3} + 4533 T^{4} + 24609 T^{5} + 146900 T^{6} + 687176 T^{7} + 3585054 T^{8} + 14915472 T^{9} + 3585054 p T^{10} + 687176 p^{2} T^{11} + 146900 p^{3} T^{12} + 24609 p^{4} T^{13} + 4533 p^{5} T^{14} + 594 p^{6} T^{15} + 92 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 4 T + 88 T^{2} + 262 T^{3} + 3461 T^{4} + 8620 T^{5} + 92680 T^{6} + 254144 T^{7} + 2259442 T^{8} + 6737116 T^{9} + 2259442 p T^{10} + 254144 p^{2} T^{11} + 92680 p^{3} T^{12} + 8620 p^{4} T^{13} + 3461 p^{5} T^{14} + 262 p^{6} T^{15} + 88 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 8 T + 223 T^{2} + 1452 T^{3} + 22492 T^{4} + 124237 T^{5} + 1395045 T^{6} + 6653364 T^{7} + 59826447 T^{8} + 245448446 T^{9} + 59826447 p T^{10} + 6653364 p^{2} T^{11} + 1395045 p^{3} T^{12} + 124237 p^{4} T^{13} + 22492 p^{5} T^{14} + 1452 p^{6} T^{15} + 223 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 27 T + 544 T^{2} + 7852 T^{3} + 96462 T^{4} + 991317 T^{5} + 9025832 T^{6} + 71977894 T^{7} + 517158242 T^{8} + 3306525525 T^{9} + 517158242 p T^{10} + 71977894 p^{2} T^{11} + 9025832 p^{3} T^{12} + 991317 p^{4} T^{13} + 96462 p^{5} T^{14} + 7852 p^{6} T^{15} + 544 p^{7} T^{16} + 27 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 12 T + 226 T^{2} + 2136 T^{3} + 24552 T^{4} + 199425 T^{5} + 1767092 T^{6} + 12614876 T^{7} + 94224175 T^{8} + 594125521 T^{9} + 94224175 p T^{10} + 12614876 p^{2} T^{11} + 1767092 p^{3} T^{12} + 199425 p^{4} T^{13} + 24552 p^{5} T^{14} + 2136 p^{6} T^{15} + 226 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 16 T + 349 T^{2} + 4222 T^{3} + 53464 T^{4} + 515381 T^{5} + 4875605 T^{6} + 38702340 T^{7} + 298086737 T^{8} + 1984494250 T^{9} + 298086737 p T^{10} + 38702340 p^{2} T^{11} + 4875605 p^{3} T^{12} + 515381 p^{4} T^{13} + 53464 p^{5} T^{14} + 4222 p^{6} T^{15} + 349 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 8 T + 380 T^{2} - 2565 T^{3} + 65031 T^{4} - 372743 T^{5} + 6647480 T^{6} - 32363927 T^{7} + 451278644 T^{8} - 1849968562 T^{9} + 451278644 p T^{10} - 32363927 p^{2} T^{11} + 6647480 p^{3} T^{12} - 372743 p^{4} T^{13} + 65031 p^{5} T^{14} - 2565 p^{6} T^{15} + 380 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 8 T + 273 T^{2} + 2044 T^{3} + 36211 T^{4} + 260714 T^{5} + 3153661 T^{6} + 21870087 T^{7} + 207103767 T^{8} + 1338205431 T^{9} + 207103767 p T^{10} + 21870087 p^{2} T^{11} + 3153661 p^{3} T^{12} + 260714 p^{4} T^{13} + 36211 p^{5} T^{14} + 2044 p^{6} T^{15} + 273 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 16 T + 393 T^{2} + 3630 T^{3} + 49303 T^{4} + 216050 T^{5} + 1990154 T^{6} - 10149150 T^{7} - 52334721 T^{8} - 1597546588 T^{9} - 52334721 p T^{10} - 10149150 p^{2} T^{11} + 1990154 p^{3} T^{12} + 216050 p^{4} T^{13} + 49303 p^{5} T^{14} + 3630 p^{6} T^{15} + 393 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 21 T + 593 T^{2} + 9039 T^{3} + 147416 T^{4} + 1763827 T^{5} + 20995514 T^{6} + 203924756 T^{7} + 1916422141 T^{8} + 15273534255 T^{9} + 1916422141 p T^{10} + 203924756 p^{2} T^{11} + 20995514 p^{3} T^{12} + 1763827 p^{4} T^{13} + 147416 p^{5} T^{14} + 9039 p^{6} T^{15} + 593 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 3 T + 379 T^{2} - 1384 T^{3} + 72420 T^{4} - 274006 T^{5} + 9106607 T^{6} - 32870880 T^{7} + 821083453 T^{8} - 2656323014 T^{9} + 821083453 p T^{10} - 32870880 p^{2} T^{11} + 9106607 p^{3} T^{12} - 274006 p^{4} T^{13} + 72420 p^{5} T^{14} - 1384 p^{6} T^{15} + 379 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 33 T + 817 T^{2} + 13790 T^{3} + 199530 T^{4} + 2418510 T^{5} + 27125871 T^{6} + 276226520 T^{7} + 2646386681 T^{8} + 23124430206 T^{9} + 2646386681 p T^{10} + 276226520 p^{2} T^{11} + 27125871 p^{3} T^{12} + 2418510 p^{4} T^{13} + 199530 p^{5} T^{14} + 13790 p^{6} T^{15} + 817 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 3 T + 369 T^{2} + 774 T^{3} + 68199 T^{4} + 91024 T^{5} + 8591261 T^{6} + 8183155 T^{7} + 814407150 T^{8} + 656168425 T^{9} + 814407150 p T^{10} + 8183155 p^{2} T^{11} + 8591261 p^{3} T^{12} + 91024 p^{4} T^{13} + 68199 p^{5} T^{14} + 774 p^{6} T^{15} + 369 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 3 T + 439 T^{2} + 408 T^{3} + 88556 T^{4} - 87082 T^{5} + 11391461 T^{6} - 27467106 T^{7} + 1100412555 T^{8} - 3084319398 T^{9} + 1100412555 p T^{10} - 27467106 p^{2} T^{11} + 11391461 p^{3} T^{12} - 87082 p^{4} T^{13} + 88556 p^{5} T^{14} + 408 p^{6} T^{15} + 439 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 13 T + 543 T^{2} - 7006 T^{3} + 147206 T^{4} - 1731416 T^{5} + 25372343 T^{6} - 259949004 T^{7} + 2994947111 T^{8} - 26073244458 T^{9} + 2994947111 p T^{10} - 259949004 p^{2} T^{11} + 25372343 p^{3} T^{12} - 1731416 p^{4} T^{13} + 147206 p^{5} T^{14} - 7006 p^{6} T^{15} + 543 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 6 T + 228 T^{2} + 913 T^{3} + 24716 T^{4} - 30276 T^{5} + 2218155 T^{6} - 9025240 T^{7} + 212571166 T^{8} - 639205597 T^{9} + 212571166 p T^{10} - 9025240 p^{2} T^{11} + 2218155 p^{3} T^{12} - 30276 p^{4} T^{13} + 24716 p^{5} T^{14} + 913 p^{6} T^{15} + 228 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 4 T + 518 T^{2} + 2545 T^{3} + 130982 T^{4} + 678282 T^{5} + 21974185 T^{6} + 107665712 T^{7} + 2747966160 T^{8} + 12054120693 T^{9} + 2747966160 p T^{10} + 107665712 p^{2} T^{11} + 21974185 p^{3} T^{12} + 678282 p^{4} T^{13} + 130982 p^{5} T^{14} + 2545 p^{6} T^{15} + 518 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.81991437265013158049913916639, −3.76613563215253767129588921614, −3.76585720490627090889591404919, −3.65413949556455987749308763057, −3.62621806303080916307682918794, −3.29603854325230641985739974403, −3.28612249788794028265077392208, −3.18113135874819428182427088810, −3.08532735565992705697302543922, −3.06557022268368583177071956968, −2.75683451687524491558477579123, −2.51416836061186733291986915043, −2.29231254260122489743550136062, −2.19304005538454843855552151192, −1.95780500793912690913660712556, −1.86912535166975849954487176050, −1.84333927328006301035292600534, −1.83894462438426462464386859131, −1.64080213974236598639818711791, −1.39612627759359059550349974421, −1.33779143665464148399065480815, −1.26717115835034167213497768007, −1.21667908133787997145854605263, −1.00941985767264951657191395090, −0.932572860602934845927688802050, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.932572860602934845927688802050, 1.00941985767264951657191395090, 1.21667908133787997145854605263, 1.26717115835034167213497768007, 1.33779143665464148399065480815, 1.39612627759359059550349974421, 1.64080213974236598639818711791, 1.83894462438426462464386859131, 1.84333927328006301035292600534, 1.86912535166975849954487176050, 1.95780500793912690913660712556, 2.19304005538454843855552151192, 2.29231254260122489743550136062, 2.51416836061186733291986915043, 2.75683451687524491558477579123, 3.06557022268368583177071956968, 3.08532735565992705697302543922, 3.18113135874819428182427088810, 3.28612249788794028265077392208, 3.29603854325230641985739974403, 3.62621806303080916307682918794, 3.65413949556455987749308763057, 3.76585720490627090889591404919, 3.76613563215253767129588921614, 3.81991437265013158049913916639
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.