Dirichlet series
| L(s) = 1 | − 9·2-s + 3-s + 45·4-s − 5-s − 9·6-s + 13·7-s − 165·8-s − 14·9-s + 9·10-s + 3·11-s + 45·12-s − 117·14-s − 15-s + 495·16-s − 8·17-s + 126·18-s + 9·19-s − 45·20-s + 13·21-s − 27·22-s − 10·23-s − 165·24-s − 18·25-s − 12·27-s + 585·28-s − 20·29-s + 9·30-s + ⋯ |
| L(s) = 1 | − 6.36·2-s + 0.577·3-s + 45/2·4-s − 0.447·5-s − 3.67·6-s + 4.91·7-s − 58.3·8-s − 4.66·9-s + 2.84·10-s + 0.904·11-s + 12.9·12-s − 31.2·14-s − 0.258·15-s + 123.·16-s − 1.94·17-s + 29.6·18-s + 2.06·19-s − 10.0·20-s + 2.83·21-s − 5.75·22-s − 2.08·23-s − 33.6·24-s − 3.59·25-s − 2.30·27-s + 110.·28-s − 3.71·29-s + 1.64·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{18} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{18} \cdot 19^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{9} \cdot 13^{18} \cdot 19^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.45203\times 10^{15}\) |
| Root analytic conductor: | \(7.16100\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{9} \cdot 13^{18} \cdot 19^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.559829670\) |
| \(L(\frac12)\) | \(\approx\) | \(1.559829670\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{9} \) |
| 13 | \( 1 \) | |
| 19 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 - T + 5 p T^{2} - 17 T^{3} + 121 T^{4} - 140 T^{5} + 658 T^{6} - 725 T^{7} + 874 p T^{8} - 2587 T^{9} + 874 p^{2} T^{10} - 725 p^{2} T^{11} + 658 p^{3} T^{12} - 140 p^{4} T^{13} + 121 p^{5} T^{14} - 17 p^{6} T^{15} + 5 p^{8} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + T + 19 T^{2} + 19 T^{3} + 172 T^{4} + 238 T^{5} + 1029 T^{6} + 2083 T^{7} + 1013 p T^{8} + 12599 T^{9} + 1013 p^{2} T^{10} + 2083 p^{2} T^{11} + 1029 p^{3} T^{12} + 238 p^{4} T^{13} + 172 p^{5} T^{14} + 19 p^{6} T^{15} + 19 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 - 13 T + 107 T^{2} - 94 p T^{3} + 478 p T^{4} - 2077 p T^{5} + 7947 p T^{6} - 189549 T^{7} + 582362 T^{8} - 1617293 T^{9} + 582362 p T^{10} - 189549 p^{2} T^{11} + 7947 p^{4} T^{12} - 2077 p^{5} T^{13} + 478 p^{6} T^{14} - 94 p^{7} T^{15} + 107 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 3 T + 40 T^{2} - 75 T^{3} + 954 T^{4} - 1470 T^{5} + 16037 T^{6} - 17869 T^{7} + 214516 T^{8} - 222359 T^{9} + 214516 p T^{10} - 17869 p^{2} T^{11} + 16037 p^{3} T^{12} - 1470 p^{4} T^{13} + 954 p^{5} T^{14} - 75 p^{6} T^{15} + 40 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 8 T + 141 T^{2} + 892 T^{3} + 8864 T^{4} + 45997 T^{5} + 331163 T^{6} + 1433534 T^{7} + 8177502 T^{8} + 29584405 T^{9} + 8177502 p T^{10} + 1433534 p^{2} T^{11} + 331163 p^{3} T^{12} + 45997 p^{4} T^{13} + 8864 p^{5} T^{14} + 892 p^{6} T^{15} + 141 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 10 T + 134 T^{2} + 902 T^{3} + 7585 T^{4} + 42777 T^{5} + 290591 T^{6} + 1472385 T^{7} + 8604275 T^{8} + 38866483 T^{9} + 8604275 p T^{10} + 1472385 p^{2} T^{11} + 290591 p^{3} T^{12} + 42777 p^{4} T^{13} + 7585 p^{5} T^{14} + 902 p^{6} T^{15} + 134 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 20 T + 312 T^{2} + 3450 T^{3} + 33580 T^{4} + 274407 T^{5} + 2046226 T^{6} + 13528469 T^{7} + 83212602 T^{8} + 15979465 p T^{9} + 83212602 p T^{10} + 13528469 p^{2} T^{11} + 2046226 p^{3} T^{12} + 274407 p^{4} T^{13} + 33580 p^{5} T^{14} + 3450 p^{6} T^{15} + 312 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - T + 85 T^{2} + 277 T^{3} + 3600 T^{4} + 27391 T^{5} + 168077 T^{6} + 1050395 T^{7} + 8391968 T^{8} + 29555711 T^{9} + 8391968 p T^{10} + 1050395 p^{2} T^{11} + 168077 p^{3} T^{12} + 27391 p^{4} T^{13} + 3600 p^{5} T^{14} + 277 p^{6} T^{15} + 85 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 15 T + 314 T^{2} - 3383 T^{3} + 41227 T^{4} - 347746 T^{5} + 3154886 T^{6} - 21950513 T^{7} + 162441451 T^{8} - 958691303 T^{9} + 162441451 p T^{10} - 21950513 p^{2} T^{11} + 3154886 p^{3} T^{12} - 347746 p^{4} T^{13} + 41227 p^{5} T^{14} - 3383 p^{6} T^{15} + 314 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 19 T + 433 T^{2} - 5770 T^{3} + 76658 T^{4} - 782005 T^{5} + 7606956 T^{6} - 1510419 p T^{7} + 474824016 T^{8} - 3136899389 T^{9} + 474824016 p T^{10} - 1510419 p^{3} T^{11} + 7606956 p^{3} T^{12} - 782005 p^{4} T^{13} + 76658 p^{5} T^{14} - 5770 p^{6} T^{15} + 433 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 16 T + 339 T^{2} + 4244 T^{3} + 53180 T^{4} + 536459 T^{5} + 5015332 T^{6} + 41707632 T^{7} + 313586296 T^{8} + 2168590665 T^{9} + 313586296 p T^{10} + 41707632 p^{2} T^{11} + 5015332 p^{3} T^{12} + 536459 p^{4} T^{13} + 53180 p^{5} T^{14} + 4244 p^{6} T^{15} + 339 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 18 T + 469 T^{2} - 5977 T^{3} + 89800 T^{4} - 889336 T^{5} + 9764321 T^{6} - 78401843 T^{7} + 681995364 T^{8} - 4506165209 T^{9} + 681995364 p T^{10} - 78401843 p^{2} T^{11} + 9764321 p^{3} T^{12} - 889336 p^{4} T^{13} + 89800 p^{5} T^{14} - 5977 p^{6} T^{15} + 469 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 17 T + 432 T^{2} - 5143 T^{3} + 76113 T^{4} - 705145 T^{5} + 7816361 T^{6} - 60008226 T^{7} + 552278217 T^{8} - 3661797017 T^{9} + 552278217 p T^{10} - 60008226 p^{2} T^{11} + 7816361 p^{3} T^{12} - 705145 p^{4} T^{13} + 76113 p^{5} T^{14} - 5143 p^{6} T^{15} + 432 p^{7} T^{16} - 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 24 T + 507 T^{2} - 7779 T^{3} + 107298 T^{4} - 1245671 T^{5} + 13351289 T^{6} - 126541581 T^{7} + 1117148621 T^{8} - 8876291257 T^{9} + 1117148621 p T^{10} - 126541581 p^{2} T^{11} + 13351289 p^{3} T^{12} - 1245671 p^{4} T^{13} + 107298 p^{5} T^{14} - 7779 p^{6} T^{15} + 507 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 6 T + 390 T^{2} - 2389 T^{3} + 71089 T^{4} - 429695 T^{5} + 8121372 T^{6} - 46740711 T^{7} + 658559563 T^{8} - 3422965169 T^{9} + 658559563 p T^{10} - 46740711 p^{2} T^{11} + 8121372 p^{3} T^{12} - 429695 p^{4} T^{13} + 71089 p^{5} T^{14} - 2389 p^{6} T^{15} + 390 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 29 T + 780 T^{2} - 14004 T^{3} + 227131 T^{4} - 3006472 T^{5} + 36316287 T^{6} - 380057017 T^{7} + 3662151343 T^{8} - 31189767747 T^{9} + 3662151343 p T^{10} - 380057017 p^{2} T^{11} + 36316287 p^{3} T^{12} - 3006472 p^{4} T^{13} + 227131 p^{5} T^{14} - 14004 p^{6} T^{15} + 780 p^{7} T^{16} - 29 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 23 T + 542 T^{2} - 7821 T^{3} + 112554 T^{4} - 1245937 T^{5} + 13952393 T^{6} - 130633427 T^{7} + 1255589337 T^{8} - 10422863517 T^{9} + 1255589337 p T^{10} - 130633427 p^{2} T^{11} + 13952393 p^{3} T^{12} - 1245937 p^{4} T^{13} + 112554 p^{5} T^{14} - 7821 p^{6} T^{15} + 542 p^{7} T^{16} - 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 38 T + 1144 T^{2} - 24250 T^{3} + 437817 T^{4} - 6542249 T^{5} + 86123240 T^{6} - 983915224 T^{7} + 10055017274 T^{8} - 90633327835 T^{9} + 10055017274 p T^{10} - 983915224 p^{2} T^{11} + 86123240 p^{3} T^{12} - 6542249 p^{4} T^{13} + 437817 p^{5} T^{14} - 24250 p^{6} T^{15} + 1144 p^{7} T^{16} - 38 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 20 T + 601 T^{2} + 8027 T^{3} + 145569 T^{4} + 1526336 T^{5} + 21650972 T^{6} + 192756574 T^{7} + 2309289000 T^{8} + 17753997419 T^{9} + 2309289000 p T^{10} + 192756574 p^{2} T^{11} + 21650972 p^{3} T^{12} + 1526336 p^{4} T^{13} + 145569 p^{5} T^{14} + 8027 p^{6} T^{15} + 601 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 20 T + 672 T^{2} - 9731 T^{3} + 192949 T^{4} - 2258375 T^{5} + 33512045 T^{6} - 328765125 T^{7} + 3942673291 T^{8} - 32653941943 T^{9} + 3942673291 p T^{10} - 328765125 p^{2} T^{11} + 33512045 p^{3} T^{12} - 2258375 p^{4} T^{13} + 192949 p^{5} T^{14} - 9731 p^{6} T^{15} + 672 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 7 T + 479 T^{2} + 2779 T^{3} + 120964 T^{4} + 619472 T^{5} + 20137369 T^{6} + 89986442 T^{7} + 2413665898 T^{8} + 9431458351 T^{9} + 2413665898 p T^{10} + 89986442 p^{2} T^{11} + 20137369 p^{3} T^{12} + 619472 p^{4} T^{13} + 120964 p^{5} T^{14} + 2779 p^{6} T^{15} + 479 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 28 T + 999 T^{2} - 19299 T^{3} + 4192 p T^{4} - 6045683 T^{5} + 93624447 T^{6} - 1118615148 T^{7} + 13693190854 T^{8} - 133533474823 T^{9} + 13693190854 p T^{10} - 1118615148 p^{2} T^{11} + 93624447 p^{3} T^{12} - 6045683 p^{4} T^{13} + 4192 p^{6} T^{14} - 19299 p^{6} T^{15} + 999 p^{7} T^{16} - 28 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.65354044510241021977351231007, −2.62924359557619844153358544546, −2.60479886280351994666813987879, −2.46363628466044924139561703493, −2.43076610808118839351044852542, −2.26334711564455946325951468541, −2.18114246982512537707698954698, −2.14848829908865401841508340334, −2.06268637061304900643775013977, −2.00261558319115123779769504560, −1.87619790338032507319560702322, −1.76244368266079034480042945223, −1.75651123707194871249523126094, −1.63688634909538246786972336446, −1.49754370380150364617892807350, −1.33074610523139312641670450761, −1.22796330186211769604980057560, −0.950531374296183422854036313302, −0.73699684317271086378026760037, −0.70242700785001156504091770629, −0.66792143329379907432758296821, −0.58942988568268433531400093651, −0.50135207082468331632941056864, −0.37652529803565193535058129365, −0.23669527309720202602567776507, 0.23669527309720202602567776507, 0.37652529803565193535058129365, 0.50135207082468331632941056864, 0.58942988568268433531400093651, 0.66792143329379907432758296821, 0.70242700785001156504091770629, 0.73699684317271086378026760037, 0.950531374296183422854036313302, 1.22796330186211769604980057560, 1.33074610523139312641670450761, 1.49754370380150364617892807350, 1.63688634909538246786972336446, 1.75651123707194871249523126094, 1.76244368266079034480042945223, 1.87619790338032507319560702322, 2.00261558319115123779769504560, 2.06268637061304900643775013977, 2.14848829908865401841508340334, 2.18114246982512537707698954698, 2.26334711564455946325951468541, 2.43076610808118839351044852542, 2.46363628466044924139561703493, 2.60479886280351994666813987879, 2.62924359557619844153358544546, 2.65354044510241021977351231007
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.