Dirichlet series
| L(s) = 1 | − 5·3-s − 5·5-s − 9·7-s + 5·9-s − 9·11-s + 9·13-s + 25·15-s + 13·17-s − 5·19-s + 45·21-s − 7·23-s − 3·25-s + 14·27-s − 19·29-s + 2·31-s + 45·33-s + 45·35-s + 4·37-s − 45·39-s + 10·41-s − 17·43-s − 25·45-s − 5·47-s + 45·49-s − 65·51-s − 24·53-s + 45·55-s + ⋯ |
| L(s) = 1 | − 2.88·3-s − 2.23·5-s − 3.40·7-s + 5/3·9-s − 2.71·11-s + 2.49·13-s + 6.45·15-s + 3.15·17-s − 1.14·19-s + 9.81·21-s − 1.45·23-s − 3/5·25-s + 2.69·27-s − 3.52·29-s + 0.359·31-s + 7.83·33-s + 7.60·35-s + 0.657·37-s − 7.20·39-s + 1.56·41-s − 2.59·43-s − 3.72·45-s − 0.729·47-s + 45/7·49-s − 9.10·51-s − 3.29·53-s + 6.06·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.78735\times 10^{16}\) |
| Root analytic conductor: | \(7.99651\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 2 | \( 1 \) |
| 7 | \( ( 1 + T )^{9} \) | |
| 11 | \( ( 1 + T )^{9} \) | |
| 13 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 + 5 T + 20 T^{2} + 61 T^{3} + 161 T^{4} + 365 T^{5} + 776 T^{6} + 493 p T^{7} + 913 p T^{8} + 1600 p T^{9} + 913 p^{2} T^{10} + 493 p^{3} T^{11} + 776 p^{3} T^{12} + 365 p^{4} T^{13} + 161 p^{5} T^{14} + 61 p^{6} T^{15} + 20 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + p T + 28 T^{2} + 17 p T^{3} + 303 T^{4} + 743 T^{5} + 2182 T^{6} + 4649 T^{7} + 12119 T^{8} + 23998 T^{9} + 12119 p T^{10} + 4649 p^{2} T^{11} + 2182 p^{3} T^{12} + 743 p^{4} T^{13} + 303 p^{5} T^{14} + 17 p^{7} T^{15} + 28 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 13 T + 138 T^{2} - 939 T^{3} + 5755 T^{4} - 27985 T^{5} + 133828 T^{6} - 565357 T^{7} + 2490751 T^{8} - 10032198 T^{9} + 2490751 p T^{10} - 565357 p^{2} T^{11} + 133828 p^{3} T^{12} - 27985 p^{4} T^{13} + 5755 p^{5} T^{14} - 939 p^{6} T^{15} + 138 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 5 T + 120 T^{2} + 501 T^{3} + 6879 T^{4} + 25133 T^{5} + 252736 T^{6} + 812339 T^{7} + 6555771 T^{8} + 18285248 T^{9} + 6555771 p T^{10} + 812339 p^{2} T^{11} + 252736 p^{3} T^{12} + 25133 p^{4} T^{13} + 6879 p^{5} T^{14} + 501 p^{6} T^{15} + 120 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 7 T + 135 T^{2} + 882 T^{3} + 8317 T^{4} + 49936 T^{5} + 316783 T^{6} + 1756846 T^{7} + 8844136 T^{8} + 45525138 T^{9} + 8844136 p T^{10} + 1756846 p^{2} T^{11} + 316783 p^{3} T^{12} + 49936 p^{4} T^{13} + 8317 p^{5} T^{14} + 882 p^{6} T^{15} + 135 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 19 T + 329 T^{2} + 3782 T^{3} + 39477 T^{4} + 335122 T^{5} + 2622217 T^{6} + 17766634 T^{7} + 111992816 T^{8} + 624363670 T^{9} + 111992816 p T^{10} + 17766634 p^{2} T^{11} + 2622217 p^{3} T^{12} + 335122 p^{4} T^{13} + 39477 p^{5} T^{14} + 3782 p^{6} T^{15} + 329 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 2 T + 159 T^{2} - 486 T^{3} + 12303 T^{4} - 49676 T^{5} + 627273 T^{6} - 2949818 T^{7} + 24094632 T^{8} - 112648484 T^{9} + 24094632 p T^{10} - 2949818 p^{2} T^{11} + 627273 p^{3} T^{12} - 49676 p^{4} T^{13} + 12303 p^{5} T^{14} - 486 p^{6} T^{15} + 159 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 4 T + 107 T^{2} - 1097 T^{3} + 9183 T^{4} - 79748 T^{5} + 716531 T^{6} - 4515735 T^{7} + 31612618 T^{8} - 216156032 T^{9} + 31612618 p T^{10} - 4515735 p^{2} T^{11} + 716531 p^{3} T^{12} - 79748 p^{4} T^{13} + 9183 p^{5} T^{14} - 1097 p^{6} T^{15} + 107 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 10 T + 305 T^{2} - 2593 T^{3} + 42979 T^{4} - 313468 T^{5} + 3711175 T^{6} - 23176807 T^{7} + 216910916 T^{8} - 1147525828 T^{9} + 216910916 p T^{10} - 23176807 p^{2} T^{11} + 3711175 p^{3} T^{12} - 313468 p^{4} T^{13} + 42979 p^{5} T^{14} - 2593 p^{6} T^{15} + 305 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 17 T + 233 T^{2} + 2784 T^{3} + 27999 T^{4} + 258892 T^{5} + 2171739 T^{6} + 16696805 T^{7} + 121449257 T^{8} + 821565760 T^{9} + 121449257 p T^{10} + 16696805 p^{2} T^{11} + 2171739 p^{3} T^{12} + 258892 p^{4} T^{13} + 27999 p^{5} T^{14} + 2784 p^{6} T^{15} + 233 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 5 T + 171 T^{2} + 1432 T^{3} + 18895 T^{4} + 166252 T^{5} + 1568597 T^{6} + 12737544 T^{7} + 95627888 T^{8} + 707398878 T^{9} + 95627888 p T^{10} + 12737544 p^{2} T^{11} + 1568597 p^{3} T^{12} + 166252 p^{4} T^{13} + 18895 p^{5} T^{14} + 1432 p^{6} T^{15} + 171 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 24 T + 635 T^{2} + 9974 T^{3} + 153135 T^{4} + 1794104 T^{5} + 20023523 T^{6} + 185341542 T^{7} + 1623719005 T^{8} + 12157983174 T^{9} + 1623719005 p T^{10} + 185341542 p^{2} T^{11} + 20023523 p^{3} T^{12} + 1794104 p^{4} T^{13} + 153135 p^{5} T^{14} + 9974 p^{6} T^{15} + 635 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 19 T + 272 T^{2} + 3015 T^{3} + 30368 T^{4} + 277598 T^{5} + 2498362 T^{6} + 21379857 T^{7} + 181132049 T^{8} + 1446807518 T^{9} + 181132049 p T^{10} + 21379857 p^{2} T^{11} + 2498362 p^{3} T^{12} + 277598 p^{4} T^{13} + 30368 p^{5} T^{14} + 3015 p^{6} T^{15} + 272 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 184 T^{2} - 107 T^{3} + 23133 T^{4} - 35349 T^{5} + 2105984 T^{6} - 2837584 T^{7} + 160711743 T^{8} - 220063166 T^{9} + 160711743 p T^{10} - 2837584 p^{2} T^{11} + 2105984 p^{3} T^{12} - 35349 p^{4} T^{13} + 23133 p^{5} T^{14} - 107 p^{6} T^{15} + 184 p^{7} T^{16} + p^{9} T^{18} \) | |
| 67 | \( 1 - 2 T + 232 T^{2} + 218 T^{3} + 32861 T^{4} + 80466 T^{5} + 3212210 T^{6} + 13028828 T^{7} + 253369485 T^{8} + 1062328520 T^{9} + 253369485 p T^{10} + 13028828 p^{2} T^{11} + 3212210 p^{3} T^{12} + 80466 p^{4} T^{13} + 32861 p^{5} T^{14} + 218 p^{6} T^{15} + 232 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 2 T + 442 T^{2} + 1474 T^{3} + 92256 T^{4} + 390440 T^{5} + 12280054 T^{6} + 55691390 T^{7} + 1168753487 T^{8} + 4925309932 T^{9} + 1168753487 p T^{10} + 55691390 p^{2} T^{11} + 12280054 p^{3} T^{12} + 390440 p^{4} T^{13} + 92256 p^{5} T^{14} + 1474 p^{6} T^{15} + 442 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 34 T + 917 T^{2} - 17149 T^{3} + 282119 T^{4} - 3831704 T^{5} + 47461111 T^{6} - 512294339 T^{7} + 5117529820 T^{8} - 45367142732 T^{9} + 5117529820 p T^{10} - 512294339 p^{2} T^{11} + 47461111 p^{3} T^{12} - 3831704 p^{4} T^{13} + 282119 p^{5} T^{14} - 17149 p^{6} T^{15} + 917 p^{7} T^{16} - 34 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 5 T + 523 T^{2} - 2086 T^{3} + 130785 T^{4} - 432318 T^{5} + 20668229 T^{6} - 57814181 T^{7} + 2269679833 T^{8} - 5404767844 T^{9} + 2269679833 p T^{10} - 57814181 p^{2} T^{11} + 20668229 p^{3} T^{12} - 432318 p^{4} T^{13} + 130785 p^{5} T^{14} - 2086 p^{6} T^{15} + 523 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 24 T + 668 T^{2} + 9323 T^{3} + 141569 T^{4} + 1289391 T^{5} + 13743858 T^{6} + 84075958 T^{7} + 829150087 T^{8} + 4609755868 T^{9} + 829150087 p T^{10} + 84075958 p^{2} T^{11} + 13743858 p^{3} T^{12} + 1289391 p^{4} T^{13} + 141569 p^{5} T^{14} + 9323 p^{6} T^{15} + 668 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 11 T + 440 T^{2} + 4713 T^{3} + 92131 T^{4} + 904647 T^{5} + 12523078 T^{6} + 110132047 T^{7} + 1302015983 T^{8} + 10536868290 T^{9} + 1302015983 p T^{10} + 110132047 p^{2} T^{11} + 12523078 p^{3} T^{12} + 904647 p^{4} T^{13} + 92131 p^{5} T^{14} + 4713 p^{6} T^{15} + 440 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 187 T^{2} - 1603 T^{3} + 28071 T^{4} - 201640 T^{5} + 5440447 T^{6} - 25034565 T^{7} + 521044510 T^{8} - 4053338400 T^{9} + 521044510 p T^{10} - 25034565 p^{2} T^{11} + 5440447 p^{3} T^{12} - 201640 p^{4} T^{13} + 28071 p^{5} T^{14} - 1603 p^{6} T^{15} + 187 p^{7} T^{16} + p^{9} T^{18} \) | |
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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
−3.26735352790871534191158819853, −3.26449867368768780535341296307, −3.23101821478083412859434920271, −3.17344207400602782724478120743, −3.06519450075259942600700178556, −2.97162959512240260661442094171, −2.91062557399143668562379059807, −2.65054770659988963899292305218, −2.43257926750476706629533171938, −2.42276717121744625715963843461, −2.35338955824438668775045981045, −2.23895908340095495325154537212, −2.18498534910628714541173138007, −2.06148906938658328450940433553, −2.05169330503775681848210551701, −1.84994902470689886245016274434, −1.69684116898056896303961352166, −1.54783459887139965148739125286, −1.26210625505133149774252222335, −1.19877135401816043941041774133, −1.19281610043289661356718411880, −1.09334766305120558374520663526, −0.839142224914833621474890261342, −0.824261623891900948178565869615, −0.798407547366719683670602259706, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.798407547366719683670602259706, 0.824261623891900948178565869615, 0.839142224914833621474890261342, 1.09334766305120558374520663526, 1.19281610043289661356718411880, 1.19877135401816043941041774133, 1.26210625505133149774252222335, 1.54783459887139965148739125286, 1.69684116898056896303961352166, 1.84994902470689886245016274434, 2.05169330503775681848210551701, 2.06148906938658328450940433553, 2.18498534910628714541173138007, 2.23895908340095495325154537212, 2.35338955824438668775045981045, 2.42276717121744625715963843461, 2.43257926750476706629533171938, 2.65054770659988963899292305218, 2.91062557399143668562379059807, 2.97162959512240260661442094171, 3.06519450075259942600700178556, 3.17344207400602782724478120743, 3.23101821478083412859434920271, 3.26449867368768780535341296307, 3.26735352790871534191158819853
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.