Dirichlet series
| L(s) = 1 | + 4·3-s + 4·5-s + 9·7-s + 9-s − 9·11-s + 9·13-s + 16·15-s − 5·17-s + 17·19-s + 36·21-s − 8·23-s + 3·25-s − 20·27-s − 3·29-s + 9·31-s − 36·33-s + 36·35-s + 7·37-s + 36·39-s + 14·41-s + 21·43-s + 4·45-s + 47-s + 45·49-s − 20·51-s − 8·53-s − 36·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 3.40·7-s + 1/3·9-s − 2.71·11-s + 2.49·13-s + 4.13·15-s − 1.21·17-s + 3.90·19-s + 7.85·21-s − 1.66·23-s + 3/5·25-s − 3.84·27-s − 0.557·29-s + 1.61·31-s − 6.26·33-s + 6.08·35-s + 1.15·37-s + 5.76·39-s + 2.18·41-s + 3.20·43-s + 0.596·45-s + 0.145·47-s + 45/7·49-s − 2.80·51-s − 1.09·53-s − 4.85·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.49092\times 10^{13}\) |
| Root analytic conductor: | \(5.65438\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{18} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(288.4553031\) |
| \(L(\frac12)\) | \(\approx\) | \(288.4553031\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 4 T + 5 p T^{2} - 4 p^{2} T^{3} + 29 p T^{4} - 161 T^{5} + 299 T^{6} - 473 T^{7} + 815 T^{8} - 49 p^{3} T^{9} + 815 p T^{10} - 473 p^{2} T^{11} + 299 p^{3} T^{12} - 161 p^{4} T^{13} + 29 p^{6} T^{14} - 4 p^{8} T^{15} + 5 p^{8} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - 4 T + 13 T^{2} - 24 T^{3} + 49 T^{4} - 63 T^{5} + 131 T^{6} - 541 T^{7} + 1767 T^{8} - 4913 T^{9} + 1767 p T^{10} - 541 p^{2} T^{11} + 131 p^{3} T^{12} - 63 p^{4} T^{13} + 49 p^{5} T^{14} - 24 p^{6} T^{15} + 13 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 5 T + 112 T^{2} + 435 T^{3} + 5849 T^{4} + 18779 T^{5} + 11354 p T^{6} + 525275 T^{7} + 4474639 T^{8} + 10441272 T^{9} + 4474639 p T^{10} + 525275 p^{2} T^{11} + 11354 p^{4} T^{12} + 18779 p^{4} T^{13} + 5849 p^{5} T^{14} + 435 p^{6} T^{15} + 112 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 17 T + 232 T^{2} - 113 p T^{3} + 17621 T^{4} - 118587 T^{5} + 739064 T^{6} - 4008751 T^{7} + 20396271 T^{8} - 91757468 T^{9} + 20396271 p T^{10} - 4008751 p^{2} T^{11} + 739064 p^{3} T^{12} - 118587 p^{4} T^{13} + 17621 p^{5} T^{14} - 113 p^{7} T^{15} + 232 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 8 T + 142 T^{2} + 900 T^{3} + 8348 T^{4} + 43931 T^{5} + 277858 T^{6} + 1305284 T^{7} + 6658435 T^{8} + 30989946 T^{9} + 6658435 p T^{10} + 1305284 p^{2} T^{11} + 277858 p^{3} T^{12} + 43931 p^{4} T^{13} + 8348 p^{5} T^{14} + 900 p^{6} T^{15} + 142 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 3 T + 133 T^{2} + 432 T^{3} + 8825 T^{4} + 31164 T^{5} + 408405 T^{6} + 1502576 T^{7} + 14859460 T^{8} + 51506786 T^{9} + 14859460 p T^{10} + 1502576 p^{2} T^{11} + 408405 p^{3} T^{12} + 31164 p^{4} T^{13} + 8825 p^{5} T^{14} + 432 p^{6} T^{15} + 133 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 9 T + 189 T^{2} - 1422 T^{3} + 17963 T^{4} - 115249 T^{5} + 1088445 T^{6} - 6024758 T^{7} + 46368142 T^{8} - 220727252 T^{9} + 46368142 p T^{10} - 6024758 p^{2} T^{11} + 1088445 p^{3} T^{12} - 115249 p^{4} T^{13} + 17963 p^{5} T^{14} - 1422 p^{6} T^{15} + 189 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 7 T + 147 T^{2} - 663 T^{3} + 8476 T^{4} - 29797 T^{5} + 380684 T^{6} - 1550701 T^{7} + 18320960 T^{8} - 73203984 T^{9} + 18320960 p T^{10} - 1550701 p^{2} T^{11} + 380684 p^{3} T^{12} - 29797 p^{4} T^{13} + 8476 p^{5} T^{14} - 663 p^{6} T^{15} + 147 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 14 T + 215 T^{2} - 1941 T^{3} + 17113 T^{4} - 117718 T^{5} + 752369 T^{6} - 4616331 T^{7} + 25703150 T^{8} - 172452312 T^{9} + 25703150 p T^{10} - 4616331 p^{2} T^{11} + 752369 p^{3} T^{12} - 117718 p^{4} T^{13} + 17113 p^{5} T^{14} - 1941 p^{6} T^{15} + 215 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 21 T + 395 T^{2} - 4694 T^{3} + 49931 T^{4} - 403182 T^{5} + 3019603 T^{6} - 18398443 T^{7} + 117340899 T^{8} - 694152558 T^{9} + 117340899 p T^{10} - 18398443 p^{2} T^{11} + 3019603 p^{3} T^{12} - 403182 p^{4} T^{13} + 49931 p^{5} T^{14} - 4694 p^{6} T^{15} + 395 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - T + 97 T^{2} + 200 T^{3} + 5791 T^{4} + 21672 T^{5} + 416053 T^{6} + 1080856 T^{7} + 22279878 T^{8} + 55018994 T^{9} + 22279878 p T^{10} + 1080856 p^{2} T^{11} + 416053 p^{3} T^{12} + 21672 p^{4} T^{13} + 5791 p^{5} T^{14} + 200 p^{6} T^{15} + 97 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 8 T + 287 T^{2} + 2286 T^{3} + 41111 T^{4} + 313016 T^{5} + 3907455 T^{6} + 27566774 T^{7} + 272219877 T^{8} + 1718117526 T^{9} + 272219877 p T^{10} + 27566774 p^{2} T^{11} + 3907455 p^{3} T^{12} + 313016 p^{4} T^{13} + 41111 p^{5} T^{14} + 2286 p^{6} T^{15} + 287 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 4 T + 267 T^{2} + 620 T^{3} + 35132 T^{4} + 34249 T^{5} + 3125936 T^{6} - 33172 T^{7} + 218144930 T^{8} - 76875498 T^{9} + 218144930 p T^{10} - 33172 p^{2} T^{11} + 3125936 p^{3} T^{12} + 34249 p^{4} T^{13} + 35132 p^{5} T^{14} + 620 p^{6} T^{15} + 267 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 30 T + 716 T^{2} - 11943 T^{3} + 175473 T^{4} - 2139867 T^{5} + 23818644 T^{6} - 232195520 T^{7} + 2101070175 T^{8} - 16987691396 T^{9} + 2101070175 p T^{10} - 232195520 p^{2} T^{11} + 23818644 p^{3} T^{12} - 2139867 p^{4} T^{13} + 175473 p^{5} T^{14} - 11943 p^{6} T^{15} + 716 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 15 T + 274 T^{2} - 3475 T^{3} + 50073 T^{4} - 537804 T^{5} + 5945176 T^{6} - 55267900 T^{7} + 531241911 T^{8} - 4317339305 T^{9} + 531241911 p T^{10} - 55267900 p^{2} T^{11} + 5945176 p^{3} T^{12} - 537804 p^{4} T^{13} + 50073 p^{5} T^{14} - 3475 p^{6} T^{15} + 274 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + T + 308 T^{2} + 263 T^{3} + 52152 T^{4} + 15517 T^{5} + 6102418 T^{6} - 606011 T^{7} + 542916157 T^{8} - 122916004 T^{9} + 542916157 p T^{10} - 606011 p^{2} T^{11} + 6102418 p^{3} T^{12} + 15517 p^{4} T^{13} + 52152 p^{5} T^{14} + 263 p^{6} T^{15} + 308 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 18 T + 507 T^{2} - 6843 T^{3} + 113603 T^{4} - 1227540 T^{5} + 15477139 T^{6} - 140656093 T^{7} + 1485962414 T^{8} - 11757700036 T^{9} + 1485962414 p T^{10} - 140656093 p^{2} T^{11} + 15477139 p^{3} T^{12} - 1227540 p^{4} T^{13} + 113603 p^{5} T^{14} - 6843 p^{6} T^{15} + 507 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 13 T + 347 T^{2} - 3078 T^{3} + 56981 T^{4} - 453136 T^{5} + 7041845 T^{6} - 49917247 T^{7} + 659623475 T^{8} - 4209775716 T^{9} + 659623475 p T^{10} - 49917247 p^{2} T^{11} + 7041845 p^{3} T^{12} - 453136 p^{4} T^{13} + 56981 p^{5} T^{14} - 3078 p^{6} T^{15} + 347 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 2 T + 364 T^{2} - 1367 T^{3} + 69441 T^{4} - 357405 T^{5} + 9123390 T^{6} - 55363134 T^{7} + 925368647 T^{8} - 5606818522 T^{9} + 925368647 p T^{10} - 55363134 p^{2} T^{11} + 9123390 p^{3} T^{12} - 357405 p^{4} T^{13} + 69441 p^{5} T^{14} - 1367 p^{6} T^{15} + 364 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 459 T^{2} - 602 T^{3} + 107035 T^{4} - 232167 T^{5} + 16743355 T^{6} - 43127025 T^{7} + 1933853371 T^{8} - 4847269619 T^{9} + 1933853371 p T^{10} - 43127025 p^{2} T^{11} + 16743355 p^{3} T^{12} - 232167 p^{4} T^{13} + 107035 p^{5} T^{14} - 602 p^{6} T^{15} + 459 p^{7} T^{16} + p^{9} T^{18} \) | |
| 97 | \( 1 - 29 T + 869 T^{2} - 15245 T^{3} + 269006 T^{4} - 3475931 T^{5} + 46249846 T^{6} - 489985103 T^{7} + 5557828502 T^{8} - 52338872712 T^{9} + 5557828502 p T^{10} - 489985103 p^{2} T^{11} + 46249846 p^{3} T^{12} - 3475931 p^{4} T^{13} + 269006 p^{5} T^{14} - 15245 p^{6} T^{15} + 869 p^{7} T^{16} - 29 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.99295163989557297627759944518, −2.92839404230870183346388614718, −2.91822234153038156981024919036, −2.87804845451985656366453343100, −2.75021634315379066655213619419, −2.74576932846564588632036607666, −2.67575800109061707538133475959, −2.27149474331131056345072238116, −2.19673559904405652086617766504, −2.16807812457011578594282158652, −2.11270213275765651641371274531, −2.03536330982297771478332068973, −2.00064130933092290739206469438, −1.94765473223938689715950985619, −1.75324870552607220471431575002, −1.68684936772159536036445978402, −1.50206247134581292400787287364, −1.18066889434452593838810602032, −1.10993196539471686031337163582, −0.881845823439036896017926428357, −0.881321676699097153529019547944, −0.790852294047762591475101495691, −0.71868963667970795682473811754, −0.50230893851552229705817560105, −0.33843276115774597643132156995, 0.33843276115774597643132156995, 0.50230893851552229705817560105, 0.71868963667970795682473811754, 0.790852294047762591475101495691, 0.881321676699097153529019547944, 0.881845823439036896017926428357, 1.10993196539471686031337163582, 1.18066889434452593838810602032, 1.50206247134581292400787287364, 1.68684936772159536036445978402, 1.75324870552607220471431575002, 1.94765473223938689715950985619, 2.00064130933092290739206469438, 2.03536330982297771478332068973, 2.11270213275765651641371274531, 2.16807812457011578594282158652, 2.19673559904405652086617766504, 2.27149474331131056345072238116, 2.67575800109061707538133475959, 2.74576932846564588632036607666, 2.75021634315379066655213619419, 2.87804845451985656366453343100, 2.91822234153038156981024919036, 2.92839404230870183346388614718, 2.99295163989557297627759944518
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.