Dirichlet series
| L(s) = 1 | + 9·2-s + 2·3-s + 45·4-s + 4·5-s + 18·6-s + 9·7-s + 165·8-s − 5·9-s + 36·10-s + 8·11-s + 90·12-s + 2·13-s + 81·14-s + 8·15-s + 495·16-s + 6·17-s − 45·18-s + 2·19-s + 180·20-s + 18·21-s + 72·22-s + 330·24-s − 3·25-s + 18·26-s − 14·27-s + 405·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 6.36·2-s + 1.15·3-s + 45/2·4-s + 1.78·5-s + 7.34·6-s + 3.40·7-s + 58.3·8-s − 5/3·9-s + 11.3·10-s + 2.41·11-s + 25.9·12-s + 0.554·13-s + 21.6·14-s + 2.06·15-s + 123.·16-s + 1.45·17-s − 10.6·18-s + 0.458·19-s + 40.2·20-s + 3.92·21-s + 15.3·22-s + 67.3·24-s − 3/5·25-s + 3.53·26-s − 2.69·27-s + 76.5·28-s + 1.85·29-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{9} \cdot 79^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 7^{9} \cdot 79^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 7^{9} \cdot 79^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.26806\times 10^{8}\) |
| Root analytic conductor: | \(2.97177\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 7^{9} \cdot 79^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9232.906384\) |
| \(L(\frac12)\) | \(\approx\) | \(9232.906384\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{9} \) |
| 7 | \( ( 1 - T )^{9} \) | |
| 79 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 - 2 T + p^{2} T^{2} - 14 T^{3} + 17 p T^{4} - 76 T^{5} + 229 T^{6} - 100 p T^{7} + 10 p^{4} T^{8} - 4 p^{5} T^{9} + 10 p^{5} T^{10} - 100 p^{3} T^{11} + 229 p^{3} T^{12} - 76 p^{4} T^{13} + 17 p^{6} T^{14} - 14 p^{6} T^{15} + p^{9} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - 4 T + 19 T^{2} - 8 p T^{3} + 26 p T^{4} - 256 T^{5} + 902 T^{6} - 1608 T^{7} + 184 p^{2} T^{8} - 1368 p T^{9} + 184 p^{3} T^{10} - 1608 p^{2} T^{11} + 902 p^{3} T^{12} - 256 p^{4} T^{13} + 26 p^{6} T^{14} - 8 p^{7} T^{15} + 19 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 8 T + 7 p T^{2} - 472 T^{3} + 2883 T^{4} - 13880 T^{5} + 65929 T^{6} - 262456 T^{7} + 1023098 T^{8} - 3435744 T^{9} + 1023098 p T^{10} - 262456 p^{2} T^{11} + 65929 p^{3} T^{12} - 13880 p^{4} T^{13} + 2883 p^{5} T^{14} - 472 p^{6} T^{15} + 7 p^{8} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 2 T + 45 T^{2} - 74 T^{3} + 1385 T^{4} - 178 p T^{5} + 29571 T^{6} - 42490 T^{7} + 37846 p T^{8} - 672392 T^{9} + 37846 p^{2} T^{10} - 42490 p^{2} T^{11} + 29571 p^{3} T^{12} - 178 p^{5} T^{13} + 1385 p^{5} T^{14} - 74 p^{6} T^{15} + 45 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 6 T + 57 T^{2} - 202 T^{3} + 1337 T^{4} - 3226 T^{5} + 26019 T^{6} - 70678 T^{7} + 585202 T^{8} - 1521408 T^{9} + 585202 p T^{10} - 70678 p^{2} T^{11} + 26019 p^{3} T^{12} - 3226 p^{4} T^{13} + 1337 p^{5} T^{14} - 202 p^{6} T^{15} + 57 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 2 T + 51 T^{2} - 116 T^{3} + 1116 T^{4} - 4208 T^{5} + 24340 T^{6} - 101100 T^{7} + 612678 T^{8} - 1867996 T^{9} + 612678 p T^{10} - 101100 p^{2} T^{11} + 24340 p^{3} T^{12} - 4208 p^{4} T^{13} + 1116 p^{5} T^{14} - 116 p^{6} T^{15} + 51 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 55 T^{2} + 340 T^{3} + 1948 T^{4} + 14824 T^{5} + 102020 T^{6} + 448172 T^{7} + 2703014 T^{8} + 13952176 T^{9} + 2703014 p T^{10} + 448172 p^{2} T^{11} + 102020 p^{3} T^{12} + 14824 p^{4} T^{13} + 1948 p^{5} T^{14} + 340 p^{6} T^{15} + 55 p^{7} T^{16} + p^{9} T^{18} \) | |
| 29 | \( 1 - 10 T + 189 T^{2} - 1486 T^{3} + 15860 T^{4} - 106476 T^{5} + 833444 T^{6} - 4948130 T^{7} + 31545366 T^{8} - 166126804 T^{9} + 31545366 p T^{10} - 4948130 p^{2} T^{11} + 833444 p^{3} T^{12} - 106476 p^{4} T^{13} + 15860 p^{5} T^{14} - 1486 p^{6} T^{15} + 189 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 6 T + 215 T^{2} + 1296 T^{3} + 22189 T^{4} + 125164 T^{5} + 1434405 T^{6} + 7176960 T^{7} + 63311446 T^{8} + 270535092 T^{9} + 63311446 p T^{10} + 7176960 p^{2} T^{11} + 1434405 p^{3} T^{12} + 125164 p^{4} T^{13} + 22189 p^{5} T^{14} + 1296 p^{6} T^{15} + 215 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 10 T + 103 T^{2} - 486 T^{3} + 3939 T^{4} - 12326 T^{5} + 160947 T^{6} - 741396 T^{7} + 240018 p T^{8} - 36829636 T^{9} + 240018 p^{2} T^{10} - 741396 p^{2} T^{11} + 160947 p^{3} T^{12} - 12326 p^{4} T^{13} + 3939 p^{5} T^{14} - 486 p^{6} T^{15} + 103 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 10 T + 267 T^{2} - 2416 T^{3} + 33318 T^{4} - 276040 T^{5} + 2604794 T^{6} - 19674144 T^{7} + 143827228 T^{8} - 961713084 T^{9} + 143827228 p T^{10} - 19674144 p^{2} T^{11} + 2604794 p^{3} T^{12} - 276040 p^{4} T^{13} + 33318 p^{5} T^{14} - 2416 p^{6} T^{15} + 267 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 22 T + 519 T^{2} - 178 p T^{3} + 105296 T^{4} - 1160308 T^{5} + 11591960 T^{6} - 100189578 T^{7} + 779979122 T^{8} - 5392150892 T^{9} + 779979122 p T^{10} - 100189578 p^{2} T^{11} + 11591960 p^{3} T^{12} - 1160308 p^{4} T^{13} + 105296 p^{5} T^{14} - 178 p^{7} T^{15} + 519 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 14 T + 371 T^{2} + 4216 T^{3} + 63888 T^{4} + 595992 T^{5} + 6615144 T^{6} + 51318920 T^{7} + 453136346 T^{8} + 2929536692 T^{9} + 453136346 p T^{10} + 51318920 p^{2} T^{11} + 6615144 p^{3} T^{12} + 595992 p^{4} T^{13} + 63888 p^{5} T^{14} + 4216 p^{6} T^{15} + 371 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 12 T + 207 T^{2} - 1050 T^{3} + 10699 T^{4} + 3024 T^{5} + 467203 T^{6} - 1421212 T^{7} + 64319666 T^{8} - 290437660 T^{9} + 64319666 p T^{10} - 1421212 p^{2} T^{11} + 467203 p^{3} T^{12} + 3024 p^{4} T^{13} + 10699 p^{5} T^{14} - 1050 p^{6} T^{15} + 207 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 2 T + 169 T^{2} + 60 T^{3} + 15127 T^{4} - 9682 T^{5} + 1073605 T^{6} - 790726 T^{7} + 67016030 T^{8} - 51548016 T^{9} + 67016030 p T^{10} - 790726 p^{2} T^{11} + 1073605 p^{3} T^{12} - 9682 p^{4} T^{13} + 15127 p^{5} T^{14} + 60 p^{6} T^{15} + 169 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 6 T + 363 T^{2} - 2290 T^{3} + 61558 T^{4} - 392996 T^{5} + 6626306 T^{6} - 41032702 T^{7} + 519469720 T^{8} - 2954733740 T^{9} + 519469720 p T^{10} - 41032702 p^{2} T^{11} + 6626306 p^{3} T^{12} - 392996 p^{4} T^{13} + 61558 p^{5} T^{14} - 2290 p^{6} T^{15} + 363 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 24 T + 393 T^{2} - 3604 T^{3} + 26703 T^{4} - 81056 T^{5} + 58953 T^{6} + 5878436 T^{7} - 25864550 T^{8} + 370923472 T^{9} - 25864550 p T^{10} + 5878436 p^{2} T^{11} + 58953 p^{3} T^{12} - 81056 p^{4} T^{13} + 26703 p^{5} T^{14} - 3604 p^{6} T^{15} + 393 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 20 T + 493 T^{2} - 7436 T^{3} + 116799 T^{4} - 1401556 T^{5} + 16948349 T^{6} - 170158500 T^{7} + 1690879834 T^{8} - 14322013904 T^{9} + 1690879834 p T^{10} - 170158500 p^{2} T^{11} + 16948349 p^{3} T^{12} - 1401556 p^{4} T^{13} + 116799 p^{5} T^{14} - 7436 p^{6} T^{15} + 493 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 12 T + 305 T^{2} + 2020 T^{3} + 37780 T^{4} + 223128 T^{5} + 4081444 T^{6} + 24174236 T^{7} + 348126462 T^{8} + 1855553336 T^{9} + 348126462 p T^{10} + 24174236 p^{2} T^{11} + 4081444 p^{3} T^{12} + 223128 p^{4} T^{13} + 37780 p^{5} T^{14} + 2020 p^{6} T^{15} + 305 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 22 T + 779 T^{2} + 12604 T^{3} + 256484 T^{4} + 3283216 T^{5} + 48766460 T^{6} + 510248548 T^{7} + 5998805182 T^{8} + 51790088692 T^{9} + 5998805182 p T^{10} + 510248548 p^{2} T^{11} + 48766460 p^{3} T^{12} + 3283216 p^{4} T^{13} + 256484 p^{5} T^{14} + 12604 p^{6} T^{15} + 779 p^{7} T^{16} + 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 20 T + 497 T^{2} - 7020 T^{3} + 103668 T^{4} - 1235496 T^{5} + 13907236 T^{6} - 149006612 T^{7} + 1414219950 T^{8} - 14238821896 T^{9} + 1414219950 p T^{10} - 149006612 p^{2} T^{11} + 13907236 p^{3} T^{12} - 1235496 p^{4} T^{13} + 103668 p^{5} T^{14} - 7020 p^{6} T^{15} + 497 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 6 T + 633 T^{2} + 2960 T^{3} + 184548 T^{4} + 643720 T^{5} + 33365044 T^{6} + 86470768 T^{7} + 4274837582 T^{8} + 9002906852 T^{9} + 4274837582 p T^{10} + 86470768 p^{2} T^{11} + 33365044 p^{3} T^{12} + 643720 p^{4} T^{13} + 184548 p^{5} T^{14} + 2960 p^{6} T^{15} + 633 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.76194019345979290072263226843, −3.75167633633478920127177629537, −3.67701681192123669105777237441, −3.64928051801754849046610650571, −3.59703535650634340248251237242, −3.52092540490701759206249317802, −3.28030485413308116331440749747, −3.00269960507725297676786744434, −2.83609067384239253320140957669, −2.71189864934618821040542944534, −2.69377906661848549324473330955, −2.63402515507457283138031099267, −2.40472635708357624334459034410, −2.39092602075790743585314849093, −2.38151127712076401153359836851, −2.33535286394862542795414317591, −1.93057623726446159232837324345, −1.84383826167973415421689135603, −1.69339623408531843507847032757, −1.46279649457522244716145777514, −1.41563438339045254120927568620, −1.29925587636223275804576431778, −1.08422103535501466732075168282, −1.03877920654495532803840386592, −0.76452042604513579335813713169, 0.76452042604513579335813713169, 1.03877920654495532803840386592, 1.08422103535501466732075168282, 1.29925587636223275804576431778, 1.41563438339045254120927568620, 1.46279649457522244716145777514, 1.69339623408531843507847032757, 1.84383826167973415421689135603, 1.93057623726446159232837324345, 2.33535286394862542795414317591, 2.38151127712076401153359836851, 2.39092602075790743585314849093, 2.40472635708357624334459034410, 2.63402515507457283138031099267, 2.69377906661848549324473330955, 2.71189864934618821040542944534, 2.83609067384239253320140957669, 3.00269960507725297676786744434, 3.28030485413308116331440749747, 3.52092540490701759206249317802, 3.59703535650634340248251237242, 3.64928051801754849046610650571, 3.67701681192123669105777237441, 3.75167633633478920127177629537, 3.76194019345979290072263226843
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.