Dirichlet series
| L(s) = 1 | + 3-s − 5·5-s − 15·9-s + 3·11-s − 3·13-s − 5·15-s − 11·17-s − 4·19-s + 9·23-s − 16·25-s − 14·27-s − 9·29-s − 3·31-s + 3·33-s − 10·37-s − 3·39-s − 23·41-s + 75·45-s + 11·47-s − 42·49-s − 11·51-s − 21·53-s − 15·55-s − 4·57-s + 4·59-s − 11·61-s + 15·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.23·5-s − 5·9-s + 0.904·11-s − 0.832·13-s − 1.29·15-s − 2.66·17-s − 0.917·19-s + 1.87·23-s − 3.19·25-s − 2.69·27-s − 1.67·29-s − 0.538·31-s + 0.522·33-s − 1.64·37-s − 0.480·39-s − 3.59·41-s + 11.1·45-s + 1.60·47-s − 6·49-s − 1.54·51-s − 2.88·53-s − 2.02·55-s − 0.529·57-s + 0.520·59-s − 1.40·61-s + 1.86·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 251^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 251^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{36} \cdot 251^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.58622\times 10^{13}\) |
| Root analytic conductor: | \(5.66285\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{36} \cdot 251^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( 1 \) |
| 251 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 - T + 16 T^{2} - 17 T^{3} + 133 T^{4} - 137 T^{5} + 736 T^{6} - 701 T^{7} + 2947 T^{8} - 2495 T^{9} + 2947 p T^{10} - 701 p^{2} T^{11} + 736 p^{3} T^{12} - 137 p^{4} T^{13} + 133 p^{5} T^{14} - 17 p^{6} T^{15} + 16 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + p T + 41 T^{2} + 156 T^{3} + 744 T^{4} + 2286 T^{5} + 8057 T^{6} + 20562 T^{7} + 2324 p^{2} T^{8} + 124071 T^{9} + 2324 p^{3} T^{10} + 20562 p^{2} T^{11} + 8057 p^{3} T^{12} + 2286 p^{4} T^{13} + 744 p^{5} T^{14} + 156 p^{6} T^{15} + 41 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 + 6 p T^{2} + 17 T^{3} + 846 T^{4} + 549 T^{5} + 11048 T^{6} + 7998 T^{7} + 103972 T^{8} + 69809 T^{9} + 103972 p T^{10} + 7998 p^{2} T^{11} + 11048 p^{3} T^{12} + 549 p^{4} T^{13} + 846 p^{5} T^{14} + 17 p^{6} T^{15} + 6 p^{8} T^{16} + p^{9} T^{18} \) | |
| 11 | \( 1 - 3 T + 68 T^{2} - 217 T^{3} + 2336 T^{4} - 648 p T^{5} + 51619 T^{6} - 142123 T^{7} + 72142 p T^{8} - 1890298 T^{9} + 72142 p^{2} T^{10} - 142123 p^{2} T^{11} + 51619 p^{3} T^{12} - 648 p^{5} T^{13} + 2336 p^{5} T^{14} - 217 p^{6} T^{15} + 68 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + 3 T + 90 T^{2} + 229 T^{3} + 3771 T^{4} + 8049 T^{5} + 97678 T^{6} + 175667 T^{7} + 1750123 T^{8} + 2681615 T^{9} + 1750123 p T^{10} + 175667 p^{2} T^{11} + 97678 p^{3} T^{12} + 8049 p^{4} T^{13} + 3771 p^{5} T^{14} + 229 p^{6} T^{15} + 90 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 11 T + 159 T^{2} + 1158 T^{3} + 9870 T^{4} + 55444 T^{5} + 356037 T^{6} + 1649486 T^{7} + 8633360 T^{8} + 33582711 T^{9} + 8633360 p T^{10} + 1649486 p^{2} T^{11} + 356037 p^{3} T^{12} + 55444 p^{4} T^{13} + 9870 p^{5} T^{14} + 1158 p^{6} T^{15} + 159 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 4 T + 108 T^{2} + 299 T^{3} + 4778 T^{4} + 8024 T^{5} + 118817 T^{6} + 4239 p T^{7} + 2161610 T^{8} + 326408 T^{9} + 2161610 p T^{10} + 4239 p^{3} T^{11} + 118817 p^{3} T^{12} + 8024 p^{4} T^{13} + 4778 p^{5} T^{14} + 299 p^{6} T^{15} + 108 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 9 T + 171 T^{2} - 1185 T^{3} + 12954 T^{4} - 74062 T^{5} + 600031 T^{6} - 2917424 T^{7} + 19157887 T^{8} - 79632349 T^{9} + 19157887 p T^{10} - 2917424 p^{2} T^{11} + 600031 p^{3} T^{12} - 74062 p^{4} T^{13} + 12954 p^{5} T^{14} - 1185 p^{6} T^{15} + 171 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 9 T + 193 T^{2} + 1425 T^{3} + 16900 T^{4} + 108679 T^{5} + 931247 T^{6} + 5328603 T^{7} + 36527203 T^{8} + 183039488 T^{9} + 36527203 p T^{10} + 5328603 p^{2} T^{11} + 931247 p^{3} T^{12} + 108679 p^{4} T^{13} + 16900 p^{5} T^{14} + 1425 p^{6} T^{15} + 193 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 3 T + 181 T^{2} + 710 T^{3} + 16192 T^{4} + 69857 T^{5} + 950879 T^{6} + 3994573 T^{7} + 40050775 T^{8} + 150151518 T^{9} + 40050775 p T^{10} + 3994573 p^{2} T^{11} + 950879 p^{3} T^{12} + 69857 p^{4} T^{13} + 16192 p^{5} T^{14} + 710 p^{6} T^{15} + 181 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 10 T + 293 T^{2} + 2525 T^{3} + 39913 T^{4} + 292183 T^{5} + 3273928 T^{6} + 20233887 T^{7} + 177240541 T^{8} + 915619550 T^{9} + 177240541 p T^{10} + 20233887 p^{2} T^{11} + 3273928 p^{3} T^{12} + 292183 p^{4} T^{13} + 39913 p^{5} T^{14} + 2525 p^{6} T^{15} + 293 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 23 T + 518 T^{2} + 7507 T^{3} + 100089 T^{4} + 1066086 T^{5} + 10400915 T^{6} + 86647669 T^{7} + 662535855 T^{8} + 4425715811 T^{9} + 662535855 p T^{10} + 86647669 p^{2} T^{11} + 10400915 p^{3} T^{12} + 1066086 p^{4} T^{13} + 100089 p^{5} T^{14} + 7507 p^{6} T^{15} + 518 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 + 229 T^{2} - 505 T^{3} + 25033 T^{4} - 95093 T^{5} + 1850150 T^{6} - 8235541 T^{7} + 103295925 T^{8} - 437280466 T^{9} + 103295925 p T^{10} - 8235541 p^{2} T^{11} + 1850150 p^{3} T^{12} - 95093 p^{4} T^{13} + 25033 p^{5} T^{14} - 505 p^{6} T^{15} + 229 p^{7} T^{16} + p^{9} T^{18} \) | |
| 47 | \( 1 - 11 T + 304 T^{2} - 2379 T^{3} + 35917 T^{4} - 196156 T^{5} + 2190202 T^{6} - 7822829 T^{7} + 89098332 T^{8} - 255370170 T^{9} + 89098332 p T^{10} - 7822829 p^{2} T^{11} + 2190202 p^{3} T^{12} - 196156 p^{4} T^{13} + 35917 p^{5} T^{14} - 2379 p^{6} T^{15} + 304 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 21 T + 571 T^{2} + 8372 T^{3} + 131947 T^{4} + 1483676 T^{5} + 17016111 T^{6} + 152931868 T^{7} + 1376311042 T^{8} + 10036793022 T^{9} + 1376311042 p T^{10} + 152931868 p^{2} T^{11} + 17016111 p^{3} T^{12} + 1483676 p^{4} T^{13} + 131947 p^{5} T^{14} + 8372 p^{6} T^{15} + 571 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 4 T + 400 T^{2} - 1232 T^{3} + 76098 T^{4} - 190184 T^{5} + 9093980 T^{6} - 18940360 T^{7} + 750071737 T^{8} - 1319289400 T^{9} + 750071737 p T^{10} - 18940360 p^{2} T^{11} + 9093980 p^{3} T^{12} - 190184 p^{4} T^{13} + 76098 p^{5} T^{14} - 1232 p^{6} T^{15} + 400 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 11 T + 363 T^{2} + 3178 T^{3} + 64388 T^{4} + 484001 T^{5} + 7421927 T^{6} + 48387836 T^{7} + 610501909 T^{8} + 3456902892 T^{9} + 610501909 p T^{10} + 48387836 p^{2} T^{11} + 7421927 p^{3} T^{12} + 484001 p^{4} T^{13} + 64388 p^{5} T^{14} + 3178 p^{6} T^{15} + 363 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 4 T + 463 T^{2} - 1319 T^{3} + 96965 T^{4} - 183954 T^{5} + 12423379 T^{6} - 15236907 T^{7} + 1114089690 T^{8} - 1022393523 T^{9} + 1114089690 p T^{10} - 15236907 p^{2} T^{11} + 12423379 p^{3} T^{12} - 183954 p^{4} T^{13} + 96965 p^{5} T^{14} - 1319 p^{6} T^{15} + 463 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 19 T + 476 T^{2} - 5215 T^{3} + 76213 T^{4} - 550470 T^{5} + 6501366 T^{6} - 32177773 T^{7} + 411733912 T^{8} - 1744233622 T^{9} + 411733912 p T^{10} - 32177773 p^{2} T^{11} + 6501366 p^{3} T^{12} - 550470 p^{4} T^{13} + 76213 p^{5} T^{14} - 5215 p^{6} T^{15} + 476 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 31 T + 975 T^{2} + 18708 T^{3} + 343818 T^{4} + 4818451 T^{5} + 64286029 T^{6} + 702746801 T^{7} + 7317435477 T^{8} + 64040954134 T^{9} + 7317435477 p T^{10} + 702746801 p^{2} T^{11} + 64286029 p^{3} T^{12} + 4818451 p^{4} T^{13} + 343818 p^{5} T^{14} + 18708 p^{6} T^{15} + 975 p^{7} T^{16} + 31 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 4 T + 497 T^{2} + 1179 T^{3} + 116109 T^{4} + 148892 T^{5} + 17277791 T^{6} + 11543951 T^{7} + 1835589466 T^{8} + 818930763 T^{9} + 1835589466 p T^{10} + 11543951 p^{2} T^{11} + 17277791 p^{3} T^{12} + 148892 p^{4} T^{13} + 116109 p^{5} T^{14} + 1179 p^{6} T^{15} + 497 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 22 T + 697 T^{2} - 10747 T^{3} + 202141 T^{4} - 2458553 T^{5} + 34682944 T^{6} - 349803141 T^{7} + 4025119603 T^{8} - 34396711778 T^{9} + 4025119603 p T^{10} - 349803141 p^{2} T^{11} + 34682944 p^{3} T^{12} - 2458553 p^{4} T^{13} + 202141 p^{5} T^{14} - 10747 p^{6} T^{15} + 697 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 36 T + 1250 T^{2} + 27317 T^{3} + 556616 T^{4} + 8813319 T^{5} + 130176166 T^{6} + 1596565568 T^{7} + 18315953568 T^{8} + 178565859993 T^{9} + 18315953568 p T^{10} + 1596565568 p^{2} T^{11} + 130176166 p^{3} T^{12} + 8813319 p^{4} T^{13} + 556616 p^{5} T^{14} + 27317 p^{6} T^{15} + 1250 p^{7} T^{16} + 36 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 38 T + 1014 T^{2} + 20302 T^{3} + 362890 T^{4} + 5556782 T^{5} + 76897379 T^{6} + 945841020 T^{7} + 10761247144 T^{8} + 110399206676 T^{9} + 10761247144 p T^{10} + 945841020 p^{2} T^{11} + 76897379 p^{3} T^{12} + 5556782 p^{4} T^{13} + 362890 p^{5} T^{14} + 20302 p^{6} T^{15} + 1014 p^{7} T^{16} + 38 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.52267986444287483957580007714, −3.43334478112417892376635122469, −3.37401747657471847259398509579, −3.33675961602317576771429768214, −3.30346402801647674540511447943, −3.27231403610749200434011371944, −3.22234190745770537848432430698, −2.79569335219899066369078428277, −2.73791357387176462340552822556, −2.64555782655818501480037179199, −2.58984623454233148748873879679, −2.51950151747918728674352320091, −2.35817137180055013603923168171, −2.35539121261291524789932581362, −2.31746350888857263957991276420, −2.30925870945889337798916802168, −1.87768551529521598739930767093, −1.69198375870896413039719910978, −1.67250431315552910192768142669, −1.65290158108869069712290694451, −1.52302152439771810471341643329, −1.30175152792620122801089248799, −1.23913746287921926948979661302, −1.15940390701803361073731014400, −0.920858679290003401890488966094, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.920858679290003401890488966094, 1.15940390701803361073731014400, 1.23913746287921926948979661302, 1.30175152792620122801089248799, 1.52302152439771810471341643329, 1.65290158108869069712290694451, 1.67250431315552910192768142669, 1.69198375870896413039719910978, 1.87768551529521598739930767093, 2.30925870945889337798916802168, 2.31746350888857263957991276420, 2.35539121261291524789932581362, 2.35817137180055013603923168171, 2.51950151747918728674352320091, 2.58984623454233148748873879679, 2.64555782655818501480037179199, 2.73791357387176462340552822556, 2.79569335219899066369078428277, 3.22234190745770537848432430698, 3.27231403610749200434011371944, 3.30346402801647674540511447943, 3.33675961602317576771429768214, 3.37401747657471847259398509579, 3.43334478112417892376635122469, 3.52267986444287483957580007714
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.