Dirichlet series
| L(s) = 1 | − 6·2-s + 9·3-s + 13·4-s + 9·5-s − 54·6-s − 12·7-s − 9·8-s + 45·9-s − 54·10-s − 10·11-s + 117·12-s + 9·13-s + 72·14-s + 81·15-s − 2·16-s − 17·17-s − 270·18-s + 6·19-s + 117·20-s − 108·21-s + 60·22-s − 23·23-s − 81·24-s + 45·25-s − 54·26-s + 165·27-s − 156·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 5.19·3-s + 13/2·4-s + 4.02·5-s − 22.0·6-s − 4.53·7-s − 3.18·8-s + 15·9-s − 17.0·10-s − 3.01·11-s + 33.7·12-s + 2.49·13-s + 19.2·14-s + 20.9·15-s − 1/2·16-s − 4.12·17-s − 63.6·18-s + 1.37·19-s + 26.1·20-s − 23.5·21-s + 12.7·22-s − 4.79·23-s − 16.5·24-s + 9·25-s − 10.5·26-s + 31.7·27-s − 29.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.42252\times 10^{15}\) |
| Root analytic conductor: | \(6.94763\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 3^{9} \cdot 5^{9} \cdot 13^{9} \cdot 31^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | 3 | \( ( 1 - T )^{9} \) |
| 5 | \( ( 1 - T )^{9} \) | |
| 13 | \( ( 1 - T )^{9} \) | |
| 31 | \( ( 1 - T )^{9} \) | |
| good | 2 | \( 1 + 3 p T + 23 T^{2} + 69 T^{3} + 171 T^{4} + 367 T^{5} + 175 p^{2} T^{6} + 1209 T^{7} + 1917 T^{8} + 1407 p T^{9} + 1917 p T^{10} + 1209 p^{2} T^{11} + 175 p^{5} T^{12} + 367 p^{4} T^{13} + 171 p^{5} T^{14} + 69 p^{6} T^{15} + 23 p^{7} T^{16} + 3 p^{9} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + 12 T + 96 T^{2} + 569 T^{3} + 2836 T^{4} + 12102 T^{5} + 45856 T^{6} + 22130 p T^{7} + 474445 T^{8} + 187807 p T^{9} + 474445 p T^{10} + 22130 p^{3} T^{11} + 45856 p^{3} T^{12} + 12102 p^{4} T^{13} + 2836 p^{5} T^{14} + 569 p^{6} T^{15} + 96 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 10 T + 82 T^{2} + 422 T^{3} + 183 p T^{4} + 7684 T^{5} + 31505 T^{6} + 116662 T^{7} + 458426 T^{8} + 1530726 T^{9} + 458426 p T^{10} + 116662 p^{2} T^{11} + 31505 p^{3} T^{12} + 7684 p^{4} T^{13} + 183 p^{6} T^{14} + 422 p^{6} T^{15} + 82 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + p T + 198 T^{2} + 1579 T^{3} + 10164 T^{4} + 50783 T^{5} + 211606 T^{6} + 700783 T^{7} + 2157490 T^{8} + 7148926 T^{9} + 2157490 p T^{10} + 700783 p^{2} T^{11} + 211606 p^{3} T^{12} + 50783 p^{4} T^{13} + 10164 p^{5} T^{14} + 1579 p^{6} T^{15} + 198 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 6 T + 103 T^{2} - 564 T^{3} + 5257 T^{4} - 25844 T^{5} + 180543 T^{6} - 781626 T^{7} + 4579137 T^{8} - 17217784 T^{9} + 4579137 p T^{10} - 781626 p^{2} T^{11} + 180543 p^{3} T^{12} - 25844 p^{4} T^{13} + 5257 p^{5} T^{14} - 564 p^{6} T^{15} + 103 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + p T + 350 T^{2} + 3884 T^{3} + 35446 T^{4} + 275528 T^{5} + 1892253 T^{6} + 11617327 T^{7} + 64576901 T^{8} + 324785234 T^{9} + 64576901 p T^{10} + 11617327 p^{2} T^{11} + 1892253 p^{3} T^{12} + 275528 p^{4} T^{13} + 35446 p^{5} T^{14} + 3884 p^{6} T^{15} + 350 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 4 T + 45 T^{2} + 18 T^{3} + 1410 T^{4} + 3563 T^{5} + 51490 T^{6} + 137119 T^{7} + 1579520 T^{8} + 6166205 T^{9} + 1579520 p T^{10} + 137119 p^{2} T^{11} + 51490 p^{3} T^{12} + 3563 p^{4} T^{13} + 1410 p^{5} T^{14} + 18 p^{6} T^{15} + 45 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 11 T + 210 T^{2} - 1959 T^{3} + 22319 T^{4} - 4800 p T^{5} + 1546392 T^{6} - 10628146 T^{7} + 76648501 T^{8} - 458056038 T^{9} + 76648501 p T^{10} - 10628146 p^{2} T^{11} + 1546392 p^{3} T^{12} - 4800 p^{5} T^{13} + 22319 p^{5} T^{14} - 1959 p^{6} T^{15} + 210 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 8 T + 224 T^{2} + 1906 T^{3} + 27401 T^{4} + 217103 T^{5} + 2187165 T^{6} + 15627101 T^{7} + 123581568 T^{8} + 765435461 T^{9} + 123581568 p T^{10} + 15627101 p^{2} T^{11} + 2187165 p^{3} T^{12} + 217103 p^{4} T^{13} + 27401 p^{5} T^{14} + 1906 p^{6} T^{15} + 224 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 15 T + 268 T^{2} + 2695 T^{3} + 29711 T^{4} + 219751 T^{5} + 1843185 T^{6} + 11089462 T^{7} + 83173948 T^{8} + 466195437 T^{9} + 83173948 p T^{10} + 11089462 p^{2} T^{11} + 1843185 p^{3} T^{12} + 219751 p^{4} T^{13} + 29711 p^{5} T^{14} + 2695 p^{6} T^{15} + 268 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 33 T + 731 T^{2} + 11689 T^{3} + 155668 T^{4} + 1750831 T^{5} + 17423194 T^{6} + 153511633 T^{7} + 1224155805 T^{8} + 8791568684 T^{9} + 1224155805 p T^{10} + 153511633 p^{2} T^{11} + 17423194 p^{3} T^{12} + 1750831 p^{4} T^{13} + 155668 p^{5} T^{14} + 11689 p^{6} T^{15} + 731 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 38 T + 882 T^{2} + 15467 T^{3} + 222161 T^{4} + 2721796 T^{5} + 29162504 T^{6} + 277344973 T^{7} + 2362366855 T^{8} + 18113761198 T^{9} + 2362366855 p T^{10} + 277344973 p^{2} T^{11} + 29162504 p^{3} T^{12} + 2721796 p^{4} T^{13} + 222161 p^{5} T^{14} + 15467 p^{6} T^{15} + 882 p^{7} T^{16} + 38 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 25 T + 598 T^{2} + 8836 T^{3} + 125382 T^{4} + 1372240 T^{5} + 14739200 T^{6} + 132023743 T^{7} + 1178625939 T^{8} + 9043301631 T^{9} + 1178625939 p T^{10} + 132023743 p^{2} T^{11} + 14739200 p^{3} T^{12} + 1372240 p^{4} T^{13} + 125382 p^{5} T^{14} + 8836 p^{6} T^{15} + 598 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 10 T + 213 T^{2} + 1636 T^{3} + 24454 T^{4} + 163520 T^{5} + 2052270 T^{6} + 12263416 T^{7} + 147500303 T^{8} + 819376538 T^{9} + 147500303 p T^{10} + 12263416 p^{2} T^{11} + 2052270 p^{3} T^{12} + 163520 p^{4} T^{13} + 24454 p^{5} T^{14} + 1636 p^{6} T^{15} + 213 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 19 T + 364 T^{2} + 3968 T^{3} + 47546 T^{4} + 407038 T^{5} + 4235261 T^{6} + 32861357 T^{7} + 4757589 p T^{8} + 2313027443 T^{9} + 4757589 p^{2} T^{10} + 32861357 p^{2} T^{11} + 4235261 p^{3} T^{12} + 407038 p^{4} T^{13} + 47546 p^{5} T^{14} + 3968 p^{6} T^{15} + 364 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 43 T + 1250 T^{2} + 26537 T^{3} + 464310 T^{4} + 6823092 T^{5} + 87406863 T^{6} + 981563961 T^{7} + 9819184763 T^{8} + 87402239446 T^{9} + 9819184763 p T^{10} + 981563961 p^{2} T^{11} + 87406863 p^{3} T^{12} + 6823092 p^{4} T^{13} + 464310 p^{5} T^{14} + 26537 p^{6} T^{15} + 1250 p^{7} T^{16} + 43 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - T + 201 T^{2} + 624 T^{3} + 27268 T^{4} + 76833 T^{5} + 3362478 T^{6} + 7658689 T^{7} + 282006259 T^{8} + 821117972 T^{9} + 282006259 p T^{10} + 7658689 p^{2} T^{11} + 3362478 p^{3} T^{12} + 76833 p^{4} T^{13} + 27268 p^{5} T^{14} + 624 p^{6} T^{15} + 201 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 9 T + 459 T^{2} + 4009 T^{3} + 98119 T^{4} + 855507 T^{5} + 13286490 T^{6} + 115060659 T^{7} + 1319858202 T^{8} + 10753106806 T^{9} + 1319858202 p T^{10} + 115060659 p^{2} T^{11} + 13286490 p^{3} T^{12} + 855507 p^{4} T^{13} + 98119 p^{5} T^{14} + 4009 p^{6} T^{15} + 459 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 12 T + 403 T^{2} + 4312 T^{3} + 88615 T^{4} + 830672 T^{5} + 12887502 T^{6} + 107980426 T^{7} + 1390571815 T^{8} + 10298966981 T^{9} + 1390571815 p T^{10} + 107980426 p^{2} T^{11} + 12887502 p^{3} T^{12} + 830672 p^{4} T^{13} + 88615 p^{5} T^{14} + 4312 p^{6} T^{15} + 403 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 5 T + 333 T^{2} + 1789 T^{3} + 62420 T^{4} + 266030 T^{5} + 8358386 T^{6} + 27361400 T^{7} + 861715531 T^{8} + 2548208682 T^{9} + 861715531 p T^{10} + 27361400 p^{2} T^{11} + 8358386 p^{3} T^{12} + 266030 p^{4} T^{13} + 62420 p^{5} T^{14} + 1789 p^{6} T^{15} + 333 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 4 T + 501 T^{2} + 1151 T^{3} + 129781 T^{4} + 218247 T^{5} + 23001178 T^{6} + 31050897 T^{7} + 2957902923 T^{8} + 3288593497 T^{9} + 2957902923 p T^{10} + 31050897 p^{2} T^{11} + 23001178 p^{3} T^{12} + 218247 p^{4} T^{13} + 129781 p^{5} T^{14} + 1151 p^{6} T^{15} + 501 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.26700506261509548794460627814, −3.07220138324799967827674191869, −3.05852445757666497038566976214, −3.02472894705310445833661851681, −3.00085759467330370623096853054, −2.84456181263803202935315624851, −2.75571835206311490349287477001, −2.69592326476705424278154443959, −2.65255194839425757352489722142, −2.61184603933533587132663069402, −2.59649165986445663089896628778, −2.34521350937610748393618828898, −2.13140974015822140859778528882, −2.07937277850727288801110376985, −2.03520735588460010939195204273, −1.85235401735717265750587940367, −1.82619612445172632827720123030, −1.44829725828415336920708363136, −1.43980562959346340881924119380, −1.40459705815022101443054199686, −1.36569949871766833189691658110, −1.36340613316965033432206716624, −1.31799395118324248028392764967, −1.16508811092243715164513613645, −1.10823242347320029807002105304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.10823242347320029807002105304, 1.16508811092243715164513613645, 1.31799395118324248028392764967, 1.36340613316965033432206716624, 1.36569949871766833189691658110, 1.40459705815022101443054199686, 1.43980562959346340881924119380, 1.44829725828415336920708363136, 1.82619612445172632827720123030, 1.85235401735717265750587940367, 2.03520735588460010939195204273, 2.07937277850727288801110376985, 2.13140974015822140859778528882, 2.34521350937610748393618828898, 2.59649165986445663089896628778, 2.61184603933533587132663069402, 2.65255194839425757352489722142, 2.69592326476705424278154443959, 2.75571835206311490349287477001, 2.84456181263803202935315624851, 3.00085759467330370623096853054, 3.02472894705310445833661851681, 3.05852445757666497038566976214, 3.07220138324799967827674191869, 3.26700506261509548794460627814
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.