Dirichlet series
| L(s) = 1 | + 9·2-s − 9·3-s + 45·4-s − 3·5-s − 81·6-s − 7·7-s + 165·8-s + 45·9-s − 27·10-s + 4·11-s − 405·12-s − 5·13-s − 63·14-s + 27·15-s + 495·16-s − 28·17-s + 405·18-s + 9·19-s − 135·20-s + 63·21-s + 36·22-s − 10·23-s − 1.48e3·24-s − 16·25-s − 45·26-s − 165·27-s − 315·28-s + ⋯ |
| L(s) = 1 | + 6.36·2-s − 5.19·3-s + 45/2·4-s − 1.34·5-s − 33.0·6-s − 2.64·7-s + 58.3·8-s + 15·9-s − 8.53·10-s + 1.20·11-s − 116.·12-s − 1.38·13-s − 16.8·14-s + 6.97·15-s + 123.·16-s − 6.79·17-s + 95.4·18-s + 2.06·19-s − 30.1·20-s + 13.7·21-s + 7.67·22-s − 2.08·23-s − 303.·24-s − 3.19·25-s − 8.82·26-s − 31.7·27-s − 59.5·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.41618\times 10^{15}\) |
| Root analytic conductor: | \(6.94590\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( ( 1 - T )^{9} \) |
| 3 | \( ( 1 + T )^{9} \) | |
| 19 | \( ( 1 - T )^{9} \) | |
| 53 | \( ( 1 + T )^{9} \) | |
| good | 5 | \( 1 + 3 T + p^{2} T^{2} + 51 T^{3} + 227 T^{4} + 43 p T^{5} + 683 T^{6} - 1284 T^{7} - 1888 T^{8} - 14576 T^{9} - 1888 p T^{10} - 1284 p^{2} T^{11} + 683 p^{3} T^{12} + 43 p^{5} T^{13} + 227 p^{5} T^{14} + 51 p^{6} T^{15} + p^{9} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 + p T + 34 T^{2} + 15 p T^{3} + 298 T^{4} + 753 T^{5} + 2365 T^{6} + 6913 T^{7} + 432 p^{2} T^{8} + 55436 T^{9} + 432 p^{3} T^{10} + 6913 p^{2} T^{11} + 2365 p^{3} T^{12} + 753 p^{4} T^{13} + 298 p^{5} T^{14} + 15 p^{7} T^{15} + 34 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 4 T + 50 T^{2} - 203 T^{3} + 1176 T^{4} - 4492 T^{5} + 17628 T^{6} - 61773 T^{7} + 205021 T^{8} - 695056 T^{9} + 205021 p T^{10} - 61773 p^{2} T^{11} + 17628 p^{3} T^{12} - 4492 p^{4} T^{13} + 1176 p^{5} T^{14} - 203 p^{6} T^{15} + 50 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 5 T + 56 T^{2} + 21 p T^{3} + 1902 T^{4} + 7937 T^{5} + 43145 T^{6} + 161365 T^{7} + 736960 T^{8} + 2391904 T^{9} + 736960 p T^{10} + 161365 p^{2} T^{11} + 43145 p^{3} T^{12} + 7937 p^{4} T^{13} + 1902 p^{5} T^{14} + 21 p^{7} T^{15} + 56 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 28 T + 463 T^{2} + 5504 T^{3} + 51856 T^{4} + 23774 p T^{5} + 157676 p T^{6} + 15365899 T^{7} + 76944328 T^{8} + 338242374 T^{9} + 76944328 p T^{10} + 15365899 p^{2} T^{11} + 157676 p^{4} T^{12} + 23774 p^{5} T^{13} + 51856 p^{5} T^{14} + 5504 p^{6} T^{15} + 463 p^{7} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 10 T + 140 T^{2} + 1061 T^{3} + 398 p T^{4} + 59854 T^{5} + 395560 T^{6} + 2254083 T^{7} + 12218033 T^{8} + 60528768 T^{9} + 12218033 p T^{10} + 2254083 p^{2} T^{11} + 395560 p^{3} T^{12} + 59854 p^{4} T^{13} + 398 p^{6} T^{14} + 1061 p^{6} T^{15} + 140 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 91 T^{2} + 306 T^{3} + 4427 T^{4} + 23594 T^{5} + 204156 T^{6} + 910810 T^{7} + 7917781 T^{8} + 28101124 T^{9} + 7917781 p T^{10} + 910810 p^{2} T^{11} + 204156 p^{3} T^{12} + 23594 p^{4} T^{13} + 4427 p^{5} T^{14} + 306 p^{6} T^{15} + 91 p^{7} T^{16} + p^{9} T^{18} \) | |
| 31 | \( 1 - 5 T + 98 T^{2} - 296 T^{3} + 4910 T^{4} - 2740 T^{5} + 140640 T^{6} + 312096 T^{7} + 3503411 T^{8} + 17203362 T^{9} + 3503411 p T^{10} + 312096 p^{2} T^{11} + 140640 p^{3} T^{12} - 2740 p^{4} T^{13} + 4910 p^{5} T^{14} - 296 p^{6} T^{15} + 98 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 25 T + 334 T^{2} + 3231 T^{3} + 26434 T^{4} + 198031 T^{5} + 1451457 T^{6} + 10527915 T^{7} + 72728030 T^{8} + 462404356 T^{9} + 72728030 p T^{10} + 10527915 p^{2} T^{11} + 1451457 p^{3} T^{12} + 198031 p^{4} T^{13} + 26434 p^{5} T^{14} + 3231 p^{6} T^{15} + 334 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 7 T + 229 T^{2} + 19 p T^{3} + 20930 T^{4} + 18604 T^{5} + 1241203 T^{6} - 747759 T^{7} + 61086109 T^{8} - 57378350 T^{9} + 61086109 p T^{10} - 747759 p^{2} T^{11} + 1241203 p^{3} T^{12} + 18604 p^{4} T^{13} + 20930 p^{5} T^{14} + 19 p^{7} T^{15} + 229 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 16 T + 370 T^{2} + 4261 T^{3} + 57936 T^{4} + 527928 T^{5} + 5341518 T^{6} + 40173363 T^{7} + 328239423 T^{8} + 2070822992 T^{9} + 328239423 p T^{10} + 40173363 p^{2} T^{11} + 5341518 p^{3} T^{12} + 527928 p^{4} T^{13} + 57936 p^{5} T^{14} + 4261 p^{6} T^{15} + 370 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + 9 T + 217 T^{2} + 1377 T^{3} + 17386 T^{4} + 55778 T^{5} + 463249 T^{6} - 2301431 T^{7} - 12983767 T^{8} - 264529406 T^{9} - 12983767 p T^{10} - 2301431 p^{2} T^{11} + 463249 p^{3} T^{12} + 55778 p^{4} T^{13} + 17386 p^{5} T^{14} + 1377 p^{6} T^{15} + 217 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 3 T + 95 T^{2} + 36 T^{3} + 10809 T^{4} + 21035 T^{5} + 734662 T^{6} + 259951 T^{7} + 52896539 T^{8} + 1857482 p T^{9} + 52896539 p T^{10} + 259951 p^{2} T^{11} + 734662 p^{3} T^{12} + 21035 p^{4} T^{13} + 10809 p^{5} T^{14} + 36 p^{6} T^{15} + 95 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 16 T + 479 T^{2} + 6584 T^{3} + 107417 T^{4} + 1230672 T^{5} + 14591754 T^{6} + 137872698 T^{7} + 1308299009 T^{8} + 10207642764 T^{9} + 1308299009 p T^{10} + 137872698 p^{2} T^{11} + 14591754 p^{3} T^{12} + 1230672 p^{4} T^{13} + 107417 p^{5} T^{14} + 6584 p^{6} T^{15} + 479 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 13 T + 293 T^{2} + 1957 T^{3} + 33772 T^{4} + 184886 T^{5} + 3544848 T^{6} + 18535043 T^{7} + 296098764 T^{8} + 1325387994 T^{9} + 296098764 p T^{10} + 18535043 p^{2} T^{11} + 3544848 p^{3} T^{12} + 184886 p^{4} T^{13} + 33772 p^{5} T^{14} + 1957 p^{6} T^{15} + 293 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 4 T + 130 T^{2} + 583 T^{3} + 15287 T^{4} + 96846 T^{5} + 1598307 T^{6} + 10813961 T^{7} + 124006133 T^{8} + 711461340 T^{9} + 124006133 p T^{10} + 10813961 p^{2} T^{11} + 1598307 p^{3} T^{12} + 96846 p^{4} T^{13} + 15287 p^{5} T^{14} + 583 p^{6} T^{15} + 130 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 29 T + 635 T^{2} + 9791 T^{3} + 136916 T^{4} + 1627014 T^{5} + 18475288 T^{6} + 186639953 T^{7} + 1810039792 T^{8} + 15782839098 T^{9} + 1810039792 p T^{10} + 186639953 p^{2} T^{11} + 18475288 p^{3} T^{12} + 1627014 p^{4} T^{13} + 136916 p^{5} T^{14} + 9791 p^{6} T^{15} + 635 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 13 T + 517 T^{2} - 6255 T^{3} + 128837 T^{4} - 1416535 T^{5} + 20305237 T^{6} - 198575152 T^{7} + 2225833152 T^{8} - 18848897366 T^{9} + 2225833152 p T^{10} - 198575152 p^{2} T^{11} + 20305237 p^{3} T^{12} - 1416535 p^{4} T^{13} + 128837 p^{5} T^{14} - 6255 p^{6} T^{15} + 517 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 35 T + 856 T^{2} + 15883 T^{3} + 250682 T^{4} + 3470149 T^{5} + 43270557 T^{6} + 489084637 T^{7} + 5059367602 T^{8} + 48046886080 T^{9} + 5059367602 p T^{10} + 489084637 p^{2} T^{11} + 43270557 p^{3} T^{12} + 3470149 p^{4} T^{13} + 250682 p^{5} T^{14} + 15883 p^{6} T^{15} + 856 p^{7} T^{16} + 35 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 19 T + 518 T^{2} + 7530 T^{3} + 123908 T^{4} + 1512318 T^{5} + 19662938 T^{6} + 209238302 T^{7} + 2312049979 T^{8} + 21500675150 T^{9} + 2312049979 p T^{10} + 209238302 p^{2} T^{11} + 19662938 p^{3} T^{12} + 1512318 p^{4} T^{13} + 123908 p^{5} T^{14} + 7530 p^{6} T^{15} + 518 p^{7} T^{16} + 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 12 T + 502 T^{2} + 3297 T^{3} + 97924 T^{4} + 221390 T^{5} + 11188394 T^{6} - 18864561 T^{7} + 1046986971 T^{8} - 3929804244 T^{9} + 1046986971 p T^{10} - 18864561 p^{2} T^{11} + 11188394 p^{3} T^{12} + 221390 p^{4} T^{13} + 97924 p^{5} T^{14} + 3297 p^{6} T^{15} + 502 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.56836996036454159809479061878, −3.55422138404802778128491822218, −3.42490942792825554921993438559, −3.39047359564176896379084504767, −3.18785707470660497751574789458, −3.15567154722777175630730593326, −2.83171110759526534127373046526, −2.70390803357321514316604026314, −2.69326641427338254616445625241, −2.64324077324142926096735952585, −2.56482202906282570881553622897, −2.48137724712774803097346858956, −2.28166807277994895212400932906, −2.25613832459353263939742342146, −2.23259694483595218780632133659, −2.12719828227820341572428166625, −1.64039392811028791958701503413, −1.56201830513640398649758287821, −1.52770950815214070196058270210, −1.51460493072694165509068053057, −1.44787124870630696238800433801, −1.44309448470157258780314808659, −1.38176620675941921431646070705, −1.17037574507301169004035442803, −1.07260975283014983490707644190, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.07260975283014983490707644190, 1.17037574507301169004035442803, 1.38176620675941921431646070705, 1.44309448470157258780314808659, 1.44787124870630696238800433801, 1.51460493072694165509068053057, 1.52770950815214070196058270210, 1.56201830513640398649758287821, 1.64039392811028791958701503413, 2.12719828227820341572428166625, 2.23259694483595218780632133659, 2.25613832459353263939742342146, 2.28166807277994895212400932906, 2.48137724712774803097346858956, 2.56482202906282570881553622897, 2.64324077324142926096735952585, 2.69326641427338254616445625241, 2.70390803357321514316604026314, 2.83171110759526534127373046526, 3.15567154722777175630730593326, 3.18785707470660497751574789458, 3.39047359564176896379084504767, 3.42490942792825554921993438559, 3.55422138404802778128491822218, 3.56836996036454159809479061878
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.