Dirichlet series
| L(s) = 1 | − 9·2-s − 9·3-s + 45·4-s − 5-s + 81·6-s + 4·7-s − 165·8-s + 45·9-s + 9·10-s + 12·11-s − 405·12-s − 13-s − 36·14-s + 9·15-s + 495·16-s + 12·17-s − 405·18-s − 9·19-s − 45·20-s − 36·21-s − 108·22-s + 11·23-s + 1.48e3·24-s − 17·25-s + 9·26-s − 165·27-s + 180·28-s + ⋯ |
| L(s) = 1 | − 6.36·2-s − 5.19·3-s + 45/2·4-s − 0.447·5-s + 33.0·6-s + 1.51·7-s − 58.3·8-s + 15·9-s + 2.84·10-s + 3.61·11-s − 116.·12-s − 0.277·13-s − 9.62·14-s + 2.32·15-s + 123.·16-s + 2.91·17-s − 95.4·18-s − 2.06·19-s − 10.0·20-s − 7.85·21-s − 23.0·22-s + 2.29·23-s + 303.·24-s − 3.39·25-s + 1.76·26-s − 31.7·27-s + 34.0·28-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: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 3^{9} \cdot 19^{9} \cdot 53^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3883269734\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3883269734\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + T + 18 T^{2} + 18 T^{3} + 141 T^{4} + 197 T^{5} + 29 p^{2} T^{6} + 1824 T^{7} + 683 p T^{8} + 11648 T^{9} + 683 p^{2} T^{10} + 1824 p^{2} T^{11} + 29 p^{5} T^{12} + 197 p^{4} T^{13} + 141 p^{5} T^{14} + 18 p^{6} T^{15} + 18 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 4 T + 38 T^{2} - 153 T^{3} + 804 T^{4} - 2780 T^{5} + 11178 T^{6} - 32911 T^{7} + 107771 T^{8} - 273808 T^{9} + 107771 p T^{10} - 32911 p^{2} T^{11} + 11178 p^{3} T^{12} - 2780 p^{4} T^{13} + 804 p^{5} T^{14} - 153 p^{6} T^{15} + 38 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 12 T + 130 T^{2} - 949 T^{3} + 6296 T^{4} - 33826 T^{5} + 167392 T^{6} - 710051 T^{7} + 2794025 T^{8} - 9610148 T^{9} + 2794025 p T^{10} - 710051 p^{2} T^{11} + 167392 p^{3} T^{12} - 33826 p^{4} T^{13} + 6296 p^{5} T^{14} - 949 p^{6} T^{15} + 130 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + T + 55 T^{2} + 85 T^{3} + 1550 T^{4} + 2920 T^{5} + 31630 T^{6} + 63651 T^{7} + 39256 p T^{8} + 984590 T^{9} + 39256 p^{2} T^{10} + 63651 p^{2} T^{11} + 31630 p^{3} T^{12} + 2920 p^{4} T^{13} + 1550 p^{5} T^{14} + 85 p^{6} T^{15} + 55 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 12 T + 154 T^{2} - 1245 T^{3} + 9878 T^{4} - 61316 T^{5} + 21696 p T^{6} - 1860731 T^{7} + 9079727 T^{8} - 38054384 T^{9} + 9079727 p T^{10} - 1860731 p^{2} T^{11} + 21696 p^{4} T^{12} - 61316 p^{4} T^{13} + 9878 p^{5} T^{14} - 1245 p^{6} T^{15} + 154 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 11 T + 186 T^{2} - 1480 T^{3} + 14895 T^{4} - 95265 T^{5} + 718295 T^{6} - 166596 p T^{7} + 23398985 T^{8} - 105303104 T^{9} + 23398985 p T^{10} - 166596 p^{3} T^{11} + 718295 p^{3} T^{12} - 95265 p^{4} T^{13} + 14895 p^{5} T^{14} - 1480 p^{6} T^{15} + 186 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 7 T + 152 T^{2} - 1024 T^{3} + 11598 T^{4} - 70476 T^{5} + 587392 T^{6} - 3107984 T^{7} + 22026585 T^{8} - 101811162 T^{9} + 22026585 p T^{10} - 3107984 p^{2} T^{11} + 587392 p^{3} T^{12} - 70476 p^{4} T^{13} + 11598 p^{5} T^{14} - 1024 p^{6} T^{15} + 152 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 12 T + 254 T^{2} + 80 p T^{3} + 29455 T^{4} + 236744 T^{5} + 2036883 T^{6} + 13602480 T^{7} + 92589857 T^{8} + 513923480 T^{9} + 92589857 p T^{10} + 13602480 p^{2} T^{11} + 2036883 p^{3} T^{12} + 236744 p^{4} T^{13} + 29455 p^{5} T^{14} + 80 p^{7} T^{15} + 254 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 9 T + 241 T^{2} + 1763 T^{3} + 27470 T^{4} + 171864 T^{5} + 1989422 T^{6} + 10764093 T^{7} + 100997566 T^{8} + 470700878 T^{9} + 100997566 p T^{10} + 10764093 p^{2} T^{11} + 1989422 p^{3} T^{12} + 171864 p^{4} T^{13} + 27470 p^{5} T^{14} + 1763 p^{6} T^{15} + 241 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 4 T + 234 T^{2} + 833 T^{3} + 28007 T^{4} + 87198 T^{5} + 2177373 T^{6} + 5926383 T^{7} + 120832265 T^{8} + 285048700 T^{9} + 120832265 p T^{10} + 5926383 p^{2} T^{11} + 2177373 p^{3} T^{12} + 87198 p^{4} T^{13} + 28007 p^{5} T^{14} + 833 p^{6} T^{15} + 234 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 23 T + 488 T^{2} - 6590 T^{3} + 82757 T^{4} - 813497 T^{5} + 7608593 T^{6} - 59917394 T^{7} + 457786639 T^{8} - 3035426000 T^{9} + 457786639 p T^{10} - 59917394 p^{2} T^{11} + 7608593 p^{3} T^{12} - 813497 p^{4} T^{13} + 82757 p^{5} T^{14} - 6590 p^{6} T^{15} + 488 p^{7} T^{16} - 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 35 T + 818 T^{2} - 13847 T^{3} + 193227 T^{4} - 2263832 T^{5} + 23203575 T^{6} - 209101393 T^{7} + 1688272269 T^{8} - 12186761370 T^{9} + 1688272269 p T^{10} - 209101393 p^{2} T^{11} + 23203575 p^{3} T^{12} - 2263832 p^{4} T^{13} + 193227 p^{5} T^{14} - 13847 p^{6} T^{15} + 818 p^{7} T^{16} - 35 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 14 T + 355 T^{2} - 3767 T^{3} + 55803 T^{4} - 469223 T^{5} + 5270813 T^{6} - 37409639 T^{7} + 364776252 T^{8} - 2355300378 T^{9} + 364776252 p T^{10} - 37409639 p^{2} T^{11} + 5270813 p^{3} T^{12} - 469223 p^{4} T^{13} + 55803 p^{5} T^{14} - 3767 p^{6} T^{15} + 355 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 14 T + 418 T^{2} - 4419 T^{3} + 77909 T^{4} - 685740 T^{5} + 9158979 T^{6} - 69482597 T^{7} + 761575889 T^{8} - 4986790796 T^{9} + 761575889 p T^{10} - 69482597 p^{2} T^{11} + 9158979 p^{3} T^{12} - 685740 p^{4} T^{13} + 77909 p^{5} T^{14} - 4419 p^{6} T^{15} + 418 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 10 T + 316 T^{2} + 3381 T^{3} + 52944 T^{4} + 539910 T^{5} + 5980494 T^{6} + 56470739 T^{7} + 502506209 T^{8} + 4352382816 T^{9} + 502506209 p T^{10} + 56470739 p^{2} T^{11} + 5980494 p^{3} T^{12} + 539910 p^{4} T^{13} + 52944 p^{5} T^{14} + 3381 p^{6} T^{15} + 316 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 4 T + 347 T^{2} - 888 T^{3} + 61392 T^{4} - 117856 T^{5} + 7414408 T^{6} - 11385864 T^{7} + 669936618 T^{8} - 875285560 T^{9} + 669936618 p T^{10} - 11385864 p^{2} T^{11} + 7414408 p^{3} T^{12} - 117856 p^{4} T^{13} + 61392 p^{5} T^{14} - 888 p^{6} T^{15} + 347 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 5 T + 387 T^{2} + 1657 T^{3} + 78808 T^{4} + 296890 T^{5} + 10683452 T^{6} + 35592151 T^{7} + 1046082720 T^{8} + 3033081026 T^{9} + 1046082720 p T^{10} + 35592151 p^{2} T^{11} + 10683452 p^{3} T^{12} + 296890 p^{4} T^{13} + 78808 p^{5} T^{14} + 1657 p^{6} T^{15} + 387 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 14 T + 370 T^{2} + 3233 T^{3} + 58794 T^{4} + 447314 T^{5} + 7154740 T^{6} + 50178255 T^{7} + 682819339 T^{8} + 4325003040 T^{9} + 682819339 p T^{10} + 50178255 p^{2} T^{11} + 7154740 p^{3} T^{12} + 447314 p^{4} T^{13} + 58794 p^{5} T^{14} + 3233 p^{6} T^{15} + 370 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 37 T + 975 T^{2} - 19221 T^{3} + 321998 T^{4} - 4669300 T^{5} + 60411650 T^{6} - 699529371 T^{7} + 7354919590 T^{8} - 70078464318 T^{9} + 7354919590 p T^{10} - 699529371 p^{2} T^{11} + 60411650 p^{3} T^{12} - 4669300 p^{4} T^{13} + 321998 p^{5} T^{14} - 19221 p^{6} T^{15} + 975 p^{7} T^{16} - 37 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 20 T + 625 T^{2} + 10919 T^{3} + 196203 T^{4} + 2803097 T^{5} + 38033347 T^{6} + 446055007 T^{7} + 4909103832 T^{8} + 47869278570 T^{9} + 4909103832 p T^{10} + 446055007 p^{2} T^{11} + 38033347 p^{3} T^{12} + 2803097 p^{4} T^{13} + 196203 p^{5} T^{14} + 10919 p^{6} T^{15} + 625 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 2 T + 504 T^{2} + 161 T^{3} + 129388 T^{4} - 33272 T^{5} + 22693242 T^{6} - 9390905 T^{7} + 2906526433 T^{8} - 1285370468 T^{9} + 2906526433 p T^{10} - 9390905 p^{2} T^{11} + 22693242 p^{3} T^{12} - 33272 p^{4} T^{13} + 129388 p^{5} T^{14} + 161 p^{6} T^{15} + 504 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.69743345209173527885264808550, −2.69601434949686808590367355545, −2.62055618991723166657816606241, −2.54857098065108725937764034718, −2.38694264051638749270700240973, −2.32758649515763948012257597169, −2.32508499707073854781336406735, −1.81739584912946824172333081621, −1.76419693301777717539836953657, −1.72312463618480209611960700353, −1.66530426499429613251523338973, −1.61688660085569852581291242326, −1.56438450508867976930619362418, −1.51576833862843940560399500814, −1.42201657527127267504581576758, −1.34645643144581499011267084480, −1.07198025537385256956213218653, −0.989421787597128177557098332874, −0.74587008656107827281545288313, −0.68071241412213144385753175564, −0.60083575397208972808881650325, −0.57701032895185045379696906840, −0.50269612982984683248568409238, −0.34699169338110983737723536800, −0.31599889151716148770017745970, 0.31599889151716148770017745970, 0.34699169338110983737723536800, 0.50269612982984683248568409238, 0.57701032895185045379696906840, 0.60083575397208972808881650325, 0.68071241412213144385753175564, 0.74587008656107827281545288313, 0.989421787597128177557098332874, 1.07198025537385256956213218653, 1.34645643144581499011267084480, 1.42201657527127267504581576758, 1.51576833862843940560399500814, 1.56438450508867976930619362418, 1.61688660085569852581291242326, 1.66530426499429613251523338973, 1.72312463618480209611960700353, 1.76419693301777717539836953657, 1.81739584912946824172333081621, 2.32508499707073854781336406735, 2.32758649515763948012257597169, 2.38694264051638749270700240973, 2.54857098065108725937764034718, 2.62055618991723166657816606241, 2.69601434949686808590367355545, 2.69743345209173527885264808550
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.