Dirichlet series
| L(s) = 1 | − 3·3-s + 3·5-s − 9·7-s − 6·9-s − 3·11-s − 6·13-s − 9·15-s − 3·17-s + 27·21-s − 24·23-s − 3·25-s + 26·27-s − 15·29-s − 6·31-s + 9·33-s − 27·35-s + 24·37-s + 18·39-s − 12·41-s − 9·43-s − 18·45-s + 12·47-s + 18·49-s + 9·51-s − 18·53-s − 9·55-s − 57·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 3.40·7-s − 2·9-s − 0.904·11-s − 1.66·13-s − 2.32·15-s − 0.727·17-s + 5.89·21-s − 5.00·23-s − 3/5·25-s + 5.00·27-s − 2.78·29-s − 1.07·31-s + 1.56·33-s − 4.56·35-s + 3.94·37-s + 2.88·39-s − 1.87·41-s − 1.37·43-s − 2.68·45-s + 1.75·47-s + 18/7·49-s + 1.26·51-s − 2.47·53-s − 1.21·55-s − 7.42·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{36} \cdot 19^{18}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(9.44362\times 10^{14}\) |
| Root analytic conductor: | \(6.79128\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{36} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( 1 \) |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T + 5 p T^{2} + 37 T^{3} + 13 p^{2} T^{4} + p^{5} T^{5} + 620 T^{6} + 373 p T^{7} + 805 p T^{8} + 3853 T^{9} + 805 p^{2} T^{10} + 373 p^{3} T^{11} + 620 p^{3} T^{12} + p^{9} T^{13} + 13 p^{7} T^{14} + 37 p^{6} T^{15} + 5 p^{8} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - 3 T + 12 T^{2} - 37 T^{3} + 114 T^{4} - 327 T^{5} + 899 T^{6} - 2217 T^{7} + 5634 T^{8} - 12032 T^{9} + 5634 p T^{10} - 2217 p^{2} T^{11} + 899 p^{3} T^{12} - 327 p^{4} T^{13} + 114 p^{5} T^{14} - 37 p^{6} T^{15} + 12 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 + 9 T + 9 p T^{2} + 283 T^{3} + 156 p T^{4} + 444 p T^{5} + 7915 T^{6} + 14439 T^{7} + 27609 T^{8} + 46906 T^{9} + 27609 p T^{10} + 14439 p^{2} T^{11} + 7915 p^{3} T^{12} + 444 p^{5} T^{13} + 156 p^{6} T^{14} + 283 p^{6} T^{15} + 9 p^{8} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 3 T + 30 T^{2} + 85 T^{3} + 642 T^{4} + 1449 T^{5} + 926 p T^{6} + 23445 T^{7} + 140436 T^{8} + 285627 T^{9} + 140436 p T^{10} + 23445 p^{2} T^{11} + 926 p^{4} T^{12} + 1449 p^{4} T^{13} + 642 p^{5} T^{14} + 85 p^{6} T^{15} + 30 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 6 T + 69 T^{2} + 274 T^{3} + 2199 T^{4} + 7590 T^{5} + 3904 p T^{6} + 151860 T^{7} + 65415 p T^{8} + 2213212 T^{9} + 65415 p^{2} T^{10} + 151860 p^{2} T^{11} + 3904 p^{4} T^{12} + 7590 p^{4} T^{13} + 2199 p^{5} T^{14} + 274 p^{6} T^{15} + 69 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 3 T + 96 T^{2} + 211 T^{3} + 4512 T^{4} + 7797 T^{5} + 140442 T^{6} + 203061 T^{7} + 187506 p T^{8} + 3970395 T^{9} + 187506 p^{2} T^{10} + 203061 p^{2} T^{11} + 140442 p^{3} T^{12} + 7797 p^{4} T^{13} + 4512 p^{5} T^{14} + 211 p^{6} T^{15} + 96 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 24 T + 378 T^{2} + 4249 T^{3} + 38733 T^{4} + 293142 T^{5} + 1935413 T^{6} + 11326215 T^{7} + 60732681 T^{8} + 300819908 T^{9} + 60732681 p T^{10} + 11326215 p^{2} T^{11} + 1935413 p^{3} T^{12} + 293142 p^{4} T^{13} + 38733 p^{5} T^{14} + 4249 p^{6} T^{15} + 378 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 15 T + 258 T^{2} + 2607 T^{3} + 27174 T^{4} + 212997 T^{5} + 1685183 T^{6} + 10850541 T^{7} + 69958308 T^{8} + 377001272 T^{9} + 69958308 p T^{10} + 10850541 p^{2} T^{11} + 1685183 p^{3} T^{12} + 212997 p^{4} T^{13} + 27174 p^{5} T^{14} + 2607 p^{6} T^{15} + 258 p^{7} T^{16} + 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 6 T + 189 T^{2} + 879 T^{3} + 16812 T^{4} + 63807 T^{5} + 959511 T^{6} + 3076599 T^{7} + 39500235 T^{8} + 109519266 T^{9} + 39500235 p T^{10} + 3076599 p^{2} T^{11} + 959511 p^{3} T^{12} + 63807 p^{4} T^{13} + 16812 p^{5} T^{14} + 879 p^{6} T^{15} + 189 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 24 T + 372 T^{2} - 4065 T^{3} + 38085 T^{4} - 314670 T^{5} + 2456499 T^{6} - 17544471 T^{7} + 118070031 T^{8} - 734507188 T^{9} + 118070031 p T^{10} - 17544471 p^{2} T^{11} + 2456499 p^{3} T^{12} - 314670 p^{4} T^{13} + 38085 p^{5} T^{14} - 4065 p^{6} T^{15} + 372 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 12 T + 300 T^{2} + 2509 T^{3} + 34524 T^{4} + 206316 T^{5} + 2097130 T^{6} + 9146652 T^{7} + 87097788 T^{8} + 334416765 T^{9} + 87097788 p T^{10} + 9146652 p^{2} T^{11} + 2097130 p^{3} T^{12} + 206316 p^{4} T^{13} + 34524 p^{5} T^{14} + 2509 p^{6} T^{15} + 300 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 9 T + 6 p T^{2} + 2039 T^{3} + 30690 T^{4} + 221403 T^{5} + 2308482 T^{6} + 15456219 T^{7} + 126848832 T^{8} + 773392301 T^{9} + 126848832 p T^{10} + 15456219 p^{2} T^{11} + 2308482 p^{3} T^{12} + 221403 p^{4} T^{13} + 30690 p^{5} T^{14} + 2039 p^{6} T^{15} + 6 p^{8} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 12 T + 345 T^{2} - 2992 T^{3} + 49281 T^{4} - 324060 T^{5} + 4074240 T^{6} - 21366438 T^{7} + 236909019 T^{8} - 1080922236 T^{9} + 236909019 p T^{10} - 21366438 p^{2} T^{11} + 4074240 p^{3} T^{12} - 324060 p^{4} T^{13} + 49281 p^{5} T^{14} - 2992 p^{6} T^{15} + 345 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 18 T + 405 T^{2} + 4604 T^{3} + 62553 T^{4} + 545076 T^{5} + 5818198 T^{6} + 42882372 T^{7} + 395845407 T^{8} + 2570187708 T^{9} + 395845407 p T^{10} + 42882372 p^{2} T^{11} + 5818198 p^{3} T^{12} + 545076 p^{4} T^{13} + 62553 p^{5} T^{14} + 4604 p^{6} T^{15} + 405 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 57 T + 1950 T^{2} + 47125 T^{3} + 892800 T^{4} + 13774347 T^{5} + 178530094 T^{6} + 33409995 p T^{7} + 18767659578 T^{8} + 154686104319 T^{9} + 18767659578 p T^{10} + 33409995 p^{3} T^{11} + 178530094 p^{3} T^{12} + 13774347 p^{4} T^{13} + 892800 p^{5} T^{14} + 47125 p^{6} T^{15} + 1950 p^{7} T^{16} + 57 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 15 T + 411 T^{2} - 3747 T^{3} + 56838 T^{4} - 263046 T^{5} + 2980871 T^{6} + 6431745 T^{7} + 16235595 T^{8} + 1457141102 T^{9} + 16235595 p T^{10} + 6431745 p^{2} T^{11} + 2980871 p^{3} T^{12} - 263046 p^{4} T^{13} + 56838 p^{5} T^{14} - 3747 p^{6} T^{15} + 411 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 6 T + 369 T^{2} + 2466 T^{3} + 69771 T^{4} + 446952 T^{5} + 8727530 T^{6} + 50713632 T^{7} + 781816671 T^{8} + 4024967480 T^{9} + 781816671 p T^{10} + 50713632 p^{2} T^{11} + 8727530 p^{3} T^{12} + 446952 p^{4} T^{13} + 69771 p^{5} T^{14} + 2466 p^{6} T^{15} + 369 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 402 T^{2} - 621 T^{3} + 79089 T^{4} - 191430 T^{5} + 10334417 T^{6} - 26927667 T^{7} + 986521929 T^{8} - 2336546132 T^{9} + 986521929 p T^{10} - 26927667 p^{2} T^{11} + 10334417 p^{3} T^{12} - 191430 p^{4} T^{13} + 79089 p^{5} T^{14} - 621 p^{6} T^{15} + 402 p^{7} T^{16} + p^{9} T^{18} \) | |
| 73 | \( 1 - 36 T + 975 T^{2} - 17902 T^{3} + 277887 T^{4} - 3463506 T^{5} + 38702084 T^{6} - 374910186 T^{7} + 3471247977 T^{8} - 29708018820 T^{9} + 3471247977 p T^{10} - 374910186 p^{2} T^{11} + 38702084 p^{3} T^{12} - 3463506 p^{4} T^{13} + 277887 p^{5} T^{14} - 17902 p^{6} T^{15} + 975 p^{7} T^{16} - 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 3 T + 390 T^{2} + 137 T^{3} + 70740 T^{4} + 209247 T^{5} + 8697763 T^{6} + 39779109 T^{7} + 832608216 T^{8} + 4058261972 T^{9} + 832608216 p T^{10} + 39779109 p^{2} T^{11} + 8697763 p^{3} T^{12} + 209247 p^{4} T^{13} + 70740 p^{5} T^{14} + 137 p^{6} T^{15} + 390 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 27 T + 846 T^{2} + 14997 T^{3} + 280896 T^{4} + 3811401 T^{5} + 53229272 T^{6} + 583918251 T^{7} + 6526056546 T^{8} + 58979594089 T^{9} + 6526056546 p T^{10} + 583918251 p^{2} T^{11} + 53229272 p^{3} T^{12} + 3811401 p^{4} T^{13} + 280896 p^{5} T^{14} + 14997 p^{6} T^{15} + 846 p^{7} T^{16} + 27 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 30 T + 840 T^{2} + 15624 T^{3} + 270057 T^{4} + 3767964 T^{5} + 49617365 T^{6} + 564136665 T^{7} + 6127477839 T^{8} + 58986259711 T^{9} + 6127477839 p T^{10} + 564136665 p^{2} T^{11} + 49617365 p^{3} T^{12} + 3767964 p^{4} T^{13} + 270057 p^{5} T^{14} + 15624 p^{6} T^{15} + 840 p^{7} T^{16} + 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 3 T + 426 T^{2} + 281 T^{3} + 88470 T^{4} + 314493 T^{5} + 13046122 T^{6} + 67254849 T^{7} + 1540389942 T^{8} + 8178691697 T^{9} + 1540389942 p T^{10} + 67254849 p^{2} T^{11} + 13046122 p^{3} T^{12} + 314493 p^{4} T^{13} + 88470 p^{5} T^{14} + 281 p^{6} T^{15} + 426 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.41235231578198400154055032583, −3.32159062529716844350718436020, −3.25555446871014686651450454041, −3.11840879350905442317620348558, −2.98590761583601980739595614161, −2.94185501692544687417101756432, −2.89564508536218279060240946953, −2.65727482358655649136141346512, −2.59887793079982018697796392916, −2.57017525708669104531584846375, −2.46544851071841406795668733532, −2.43247556785008382833139185396, −2.33122722618368248552601710547, −2.27478649665374026848992550507, −2.07083922274082590277298192541, −1.87747086068834635526670618259, −1.81011820505833230892869243935, −1.71370658485329868719590393956, −1.63064471089501256707464361738, −1.55251220896072462339931316228, −1.35426870073069077670293284680, −1.20094037260526200543413804700, −1.14189777203939320669820041305, −0.884751310511882949089817681058, −0.69090980094758791337134891252, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.69090980094758791337134891252, 0.884751310511882949089817681058, 1.14189777203939320669820041305, 1.20094037260526200543413804700, 1.35426870073069077670293284680, 1.55251220896072462339931316228, 1.63064471089501256707464361738, 1.71370658485329868719590393956, 1.81011820505833230892869243935, 1.87747086068834635526670618259, 2.07083922274082590277298192541, 2.27478649665374026848992550507, 2.33122722618368248552601710547, 2.43247556785008382833139185396, 2.46544851071841406795668733532, 2.57017525708669104531584846375, 2.59887793079982018697796392916, 2.65727482358655649136141346512, 2.89564508536218279060240946953, 2.94185501692544687417101756432, 2.98590761583601980739595614161, 3.11840879350905442317620348558, 3.25555446871014686651450454041, 3.32159062529716844350718436020, 3.41235231578198400154055032583
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.