Dirichlet series
| L(s) = 1 | − 9·2-s + 9·3-s + 45·4-s + 5-s − 81·6-s − 165·8-s + 45·9-s − 9·10-s + 6·11-s + 405·12-s + 2·13-s + 9·15-s + 495·16-s + 9·17-s − 405·18-s − 5·19-s + 45·20-s − 54·22-s + 15·23-s − 1.48e3·24-s − 22·25-s − 18·26-s + 165·27-s + 11·29-s − 81·30-s − 5·31-s − 1.28e3·32-s + ⋯ |
| L(s) = 1 | − 6.36·2-s + 5.19·3-s + 45/2·4-s + 0.447·5-s − 33.0·6-s − 58.3·8-s + 15·9-s − 2.84·10-s + 1.80·11-s + 116.·12-s + 0.554·13-s + 2.32·15-s + 123.·16-s + 2.18·17-s − 95.4·18-s − 1.14·19-s + 10.0·20-s − 11.5·22-s + 3.12·23-s − 303.·24-s − 4.39·25-s − 3.53·26-s + 31.7·27-s + 2.04·29-s − 14.7·30-s − 0.898·31-s − 227.·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.36635\times 10^{15}\) |
| Root analytic conductor: | \(6.93209\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 3^{9} \cdot 17^{9} \cdot 59^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(65.79084310\) |
| \(L(\frac12)\) | \(\approx\) | \(65.79084310\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{9} \) |
| 3 | \( ( 1 - T )^{9} \) | |
| 17 | \( ( 1 - T )^{9} \) | |
| 59 | \( ( 1 - T )^{9} \) | |
| good | 5 | \( 1 - T + 23 T^{2} - 4 p T^{3} + 259 T^{4} - 206 T^{5} + 2049 T^{6} - 1572 T^{7} + 12884 T^{8} - 9178 T^{9} + 12884 p T^{10} - 1572 p^{2} T^{11} + 2049 p^{3} T^{12} - 206 p^{4} T^{13} + 259 p^{5} T^{14} - 4 p^{7} T^{15} + 23 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + 32 T^{2} + 529 T^{4} - 93 T^{5} + 6017 T^{6} - 1999 T^{7} + 52139 T^{8} - 18868 T^{9} + 52139 p T^{10} - 1999 p^{2} T^{11} + 6017 p^{3} T^{12} - 93 p^{4} T^{13} + 529 p^{5} T^{14} + 32 p^{7} T^{16} + p^{9} T^{18} \) | |
| 11 | \( 1 - 6 T + 85 T^{2} - 417 T^{3} + 3275 T^{4} - 13496 T^{5} + 76345 T^{6} - 267103 T^{7} + 1197882 T^{8} - 3544564 T^{9} + 1197882 p T^{10} - 267103 p^{2} T^{11} + 76345 p^{3} T^{12} - 13496 p^{4} T^{13} + 3275 p^{5} T^{14} - 417 p^{6} T^{15} + 85 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 2 T + 68 T^{2} - 12 p T^{3} + 2398 T^{4} - 5588 T^{5} + 56694 T^{6} - 124472 T^{7} + 979455 T^{8} - 1917444 T^{9} + 979455 p T^{10} - 124472 p^{2} T^{11} + 56694 p^{3} T^{12} - 5588 p^{4} T^{13} + 2398 p^{5} T^{14} - 12 p^{7} T^{15} + 68 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 5 T + 116 T^{2} + 31 p T^{3} + 6716 T^{4} + 32481 T^{5} + 250913 T^{6} + 1100870 T^{7} + 6586328 T^{8} + 25160586 T^{9} + 6586328 p T^{10} + 1100870 p^{2} T^{11} + 250913 p^{3} T^{12} + 32481 p^{4} T^{13} + 6716 p^{5} T^{14} + 31 p^{7} T^{15} + 116 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 15 T + 254 T^{2} - 2548 T^{3} + 25515 T^{4} - 192934 T^{5} + 1421195 T^{6} - 8540045 T^{7} + 49687745 T^{8} - 242325216 T^{9} + 49687745 p T^{10} - 8540045 p^{2} T^{11} + 1421195 p^{3} T^{12} - 192934 p^{4} T^{13} + 25515 p^{5} T^{14} - 2548 p^{6} T^{15} + 254 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 11 T + 239 T^{2} - 1998 T^{3} + 25121 T^{4} - 170556 T^{5} + 54327 p T^{6} - 8915038 T^{7} + 65813996 T^{8} - 312186338 T^{9} + 65813996 p T^{10} - 8915038 p^{2} T^{11} + 54327 p^{4} T^{12} - 170556 p^{4} T^{13} + 25121 p^{5} T^{14} - 1998 p^{6} T^{15} + 239 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 5 T + 197 T^{2} + 1028 T^{3} + 19331 T^{4} + 96082 T^{5} + 1210371 T^{6} + 5438504 T^{7} + 52499892 T^{8} + 204464842 T^{9} + 52499892 p T^{10} + 5438504 p^{2} T^{11} + 1210371 p^{3} T^{12} + 96082 p^{4} T^{13} + 19331 p^{5} T^{14} + 1028 p^{6} T^{15} + 197 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 9 T + 221 T^{2} - 1608 T^{3} + 23645 T^{4} - 145620 T^{5} + 1637363 T^{6} - 8751644 T^{7} + 81530090 T^{8} - 378508470 T^{9} + 81530090 p T^{10} - 8751644 p^{2} T^{11} + 1637363 p^{3} T^{12} - 145620 p^{4} T^{13} + 23645 p^{5} T^{14} - 1608 p^{6} T^{15} + 221 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - T + 192 T^{2} + 120 T^{3} + 18151 T^{4} + 28740 T^{5} + 29727 p T^{6} + 2080389 T^{7} + 64488045 T^{8} + 96163704 T^{9} + 64488045 p T^{10} + 2080389 p^{2} T^{11} + 29727 p^{4} T^{12} + 28740 p^{4} T^{13} + 18151 p^{5} T^{14} + 120 p^{6} T^{15} + 192 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 4 T + 161 T^{2} - 471 T^{3} + 14737 T^{4} - 41884 T^{5} + 1010927 T^{6} - 2722481 T^{7} + 53128430 T^{8} - 129157984 T^{9} + 53128430 p T^{10} - 2722481 p^{2} T^{11} + 1010927 p^{3} T^{12} - 41884 p^{4} T^{13} + 14737 p^{5} T^{14} - 471 p^{6} T^{15} + 161 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 14 T + 281 T^{2} - 3151 T^{3} + 40005 T^{4} - 375022 T^{5} + 3680619 T^{6} - 29291021 T^{7} + 237980750 T^{8} - 1620320752 T^{9} + 237980750 p T^{10} - 29291021 p^{2} T^{11} + 3680619 p^{3} T^{12} - 375022 p^{4} T^{13} + 40005 p^{5} T^{14} - 3151 p^{6} T^{15} + 281 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 4 T + 282 T^{2} - 1180 T^{3} + 40111 T^{4} - 155071 T^{5} + 3789769 T^{6} - 12985727 T^{7} + 262088125 T^{8} - 793219050 T^{9} + 262088125 p T^{10} - 12985727 p^{2} T^{11} + 3789769 p^{3} T^{12} - 155071 p^{4} T^{13} + 40111 p^{5} T^{14} - 1180 p^{6} T^{15} + 282 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 10 T + 221 T^{2} - 1949 T^{3} + 32485 T^{4} - 245698 T^{5} + 3260791 T^{6} - 22873143 T^{7} + 255049270 T^{8} - 1544108912 T^{9} + 255049270 p T^{10} - 22873143 p^{2} T^{11} + 3260791 p^{3} T^{12} - 245698 p^{4} T^{13} + 32485 p^{5} T^{14} - 1949 p^{6} T^{15} + 221 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + T + 420 T^{2} + 349 T^{3} + 82094 T^{4} + 55794 T^{5} + 10081652 T^{6} + 5615495 T^{7} + 886788121 T^{8} + 421298402 T^{9} + 886788121 p T^{10} + 5615495 p^{2} T^{11} + 10081652 p^{3} T^{12} + 55794 p^{4} T^{13} + 82094 p^{5} T^{14} + 349 p^{6} T^{15} + 420 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 14 T + 445 T^{2} - 6025 T^{3} + 99225 T^{4} - 1182622 T^{5} + 14346989 T^{6} - 2008345 p T^{7} + 1441846360 T^{8} - 11926719920 T^{9} + 1441846360 p T^{10} - 2008345 p^{3} T^{11} + 14346989 p^{3} T^{12} - 1182622 p^{4} T^{13} + 99225 p^{5} T^{14} - 6025 p^{6} T^{15} + 445 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + T + 304 T^{2} + 862 T^{3} + 50121 T^{4} + 204334 T^{5} + 5877735 T^{6} + 26890993 T^{7} + 538795915 T^{8} + 2350149524 T^{9} + 538795915 p T^{10} + 26890993 p^{2} T^{11} + 5877735 p^{3} T^{12} + 204334 p^{4} T^{13} + 50121 p^{5} T^{14} + 862 p^{6} T^{15} + 304 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 4 T + 375 T^{2} - 2354 T^{3} + 77515 T^{4} - 513570 T^{5} + 11244401 T^{6} - 69689698 T^{7} + 1178333420 T^{8} - 6609437660 T^{9} + 1178333420 p T^{10} - 69689698 p^{2} T^{11} + 11244401 p^{3} T^{12} - 513570 p^{4} T^{13} + 77515 p^{5} T^{14} - 2354 p^{6} T^{15} + 375 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 22 T + 609 T^{2} - 9620 T^{3} + 163590 T^{4} - 2078533 T^{5} + 27322287 T^{6} - 290781145 T^{7} + 3160030789 T^{8} - 28527857400 T^{9} + 3160030789 p T^{10} - 290781145 p^{2} T^{11} + 27322287 p^{3} T^{12} - 2078533 p^{4} T^{13} + 163590 p^{5} T^{14} - 9620 p^{6} T^{15} + 609 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 22 T + 683 T^{2} - 11268 T^{3} + 200176 T^{4} - 2721425 T^{5} + 35682015 T^{6} - 415670113 T^{7} + 4404947441 T^{8} - 44002465932 T^{9} + 4404947441 p T^{10} - 415670113 p^{2} T^{11} + 35682015 p^{3} T^{12} - 2721425 p^{4} T^{13} + 200176 p^{5} T^{14} - 11268 p^{6} T^{15} + 683 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 15 T + 617 T^{2} + 6686 T^{3} + 160594 T^{4} + 1273112 T^{5} + 24401087 T^{6} + 144569669 T^{7} + 2686663609 T^{8} + 13642367100 T^{9} + 2686663609 p T^{10} + 144569669 p^{2} T^{11} + 24401087 p^{3} T^{12} + 1273112 p^{4} T^{13} + 160594 p^{5} T^{14} + 6686 p^{6} T^{15} + 617 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.77312709221866923193693817661, −2.72472786851801657652489953587, −2.72445592841772857550680264735, −2.57992129071941453638064274830, −2.53351528635618893664108480892, −2.43184786807255772901072729275, −2.41593143244427406698702891629, −1.97268285896971531200014845711, −1.95275856808568802719210280750, −1.85534371369605523467758374725, −1.80987351627867543701288623078, −1.79058232275963886165760179103, −1.78665444305740388187410510702, −1.73348860278761330565881106755, −1.71449420693554404211869624461, −1.62829163196057802641827715252, −1.11184534513317322919471253254, −1.05074650655486732572232897167, −0.987792757555782904020879986209, −0.948113362683230798741488401762, −0.76340938214273792396636765566, −0.74044174141145370992249839382, −0.54079908865446791115528818691, −0.50075025475228711510103397993, −0.38573575486112775988325009865, 0.38573575486112775988325009865, 0.50075025475228711510103397993, 0.54079908865446791115528818691, 0.74044174141145370992249839382, 0.76340938214273792396636765566, 0.948113362683230798741488401762, 0.987792757555782904020879986209, 1.05074650655486732572232897167, 1.11184534513317322919471253254, 1.62829163196057802641827715252, 1.71449420693554404211869624461, 1.73348860278761330565881106755, 1.78665444305740388187410510702, 1.79058232275963886165760179103, 1.80987351627867543701288623078, 1.85534371369605523467758374725, 1.95275856808568802719210280750, 1.97268285896971531200014845711, 2.41593143244427406698702891629, 2.43184786807255772901072729275, 2.53351528635618893664108480892, 2.57992129071941453638064274830, 2.72445592841772857550680264735, 2.72472786851801657652489953587, 2.77312709221866923193693817661
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.