Dirichlet series
| L(s) = 1 | − 5-s + 2·7-s + 9·11-s + 10·13-s − 7·17-s − 2·19-s + 3·23-s − 13·25-s − 5·29-s + 12·31-s − 2·35-s + 15·37-s − 14·41-s + 6·43-s + 3·47-s − 16·49-s − 9·53-s − 9·55-s + 9·59-s + 30·61-s − 10·65-s + 16·67-s + 3·71-s + 32·73-s + 18·77-s + 24·79-s + 3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 2.71·11-s + 2.77·13-s − 1.69·17-s − 0.458·19-s + 0.625·23-s − 2.59·25-s − 0.928·29-s + 2.15·31-s − 0.338·35-s + 2.46·37-s − 2.18·41-s + 0.914·43-s + 0.437·47-s − 2.28·49-s − 1.23·53-s − 1.21·55-s + 1.17·59-s + 3.84·61-s − 1.24·65-s + 1.95·67-s + 0.356·71-s + 3.74·73-s + 2.05·77-s + 2.70·79-s + 0.329·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{18} \cdot 3^{18} \cdot 167^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.35414\times 10^{15}\) |
| Root analytic conductor: | \(6.92864\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{18} \cdot 3^{18} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(37.71541992\) |
| \(L(\frac12)\) | \(\approx\) | \(37.71541992\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 167 | \( ( 1 - T )^{9} \) | |
| good | 5 | \( 1 + T + 14 T^{2} + 16 T^{3} + 108 T^{4} + 81 T^{5} + 686 T^{6} + 298 T^{7} + 3439 T^{8} + 1678 T^{9} + 3439 p T^{10} + 298 p^{2} T^{11} + 686 p^{3} T^{12} + 81 p^{4} T^{13} + 108 p^{5} T^{14} + 16 p^{6} T^{15} + 14 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) |
| 7 | \( 1 - 2 T + 20 T^{2} - 31 T^{3} + 268 T^{4} - 467 T^{5} + 394 p T^{6} - 4405 T^{7} + 3323 p T^{8} - 36366 T^{9} + 3323 p^{2} T^{10} - 4405 p^{2} T^{11} + 394 p^{4} T^{12} - 467 p^{4} T^{13} + 268 p^{5} T^{14} - 31 p^{6} T^{15} + 20 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - 9 T + 56 T^{2} - 300 T^{3} + 148 p T^{4} - 7585 T^{5} + 2930 p T^{6} - 124292 T^{7} + 467121 T^{8} - 1604664 T^{9} + 467121 p T^{10} - 124292 p^{2} T^{11} + 2930 p^{4} T^{12} - 7585 p^{4} T^{13} + 148 p^{6} T^{14} - 300 p^{6} T^{15} + 56 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 - 10 T + 92 T^{2} - 48 p T^{3} + 3559 T^{4} - 17541 T^{5} + 78209 T^{6} - 312916 T^{7} + 1202007 T^{8} - 4377322 T^{9} + 1202007 p T^{10} - 312916 p^{2} T^{11} + 78209 p^{3} T^{12} - 17541 p^{4} T^{13} + 3559 p^{5} T^{14} - 48 p^{7} T^{15} + 92 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 7 T + 103 T^{2} + 692 T^{3} + 5651 T^{4} + 32089 T^{5} + 198783 T^{6} + 946082 T^{7} + 4759334 T^{8} + 19249834 T^{9} + 4759334 p T^{10} + 946082 p^{2} T^{11} + 198783 p^{3} T^{12} + 32089 p^{4} T^{13} + 5651 p^{5} T^{14} + 692 p^{6} T^{15} + 103 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 2 T + 82 T^{2} + 142 T^{3} + 3811 T^{4} + 6287 T^{5} + 124015 T^{6} + 184422 T^{7} + 3042753 T^{8} + 4042502 T^{9} + 3042753 p T^{10} + 184422 p^{2} T^{11} + 124015 p^{3} T^{12} + 6287 p^{4} T^{13} + 3811 p^{5} T^{14} + 142 p^{6} T^{15} + 82 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 3 T + 146 T^{2} - 480 T^{3} + 9952 T^{4} - 35395 T^{5} + 426894 T^{6} - 1554404 T^{7} + 13049775 T^{8} - 44055692 T^{9} + 13049775 p T^{10} - 1554404 p^{2} T^{11} + 426894 p^{3} T^{12} - 35395 p^{4} T^{13} + 9952 p^{5} T^{14} - 480 p^{6} T^{15} + 146 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 5 T + 140 T^{2} + 1018 T^{3} + 10260 T^{4} + 83219 T^{5} + 568490 T^{6} + 3901674 T^{7} + 23850973 T^{8} + 128905912 T^{9} + 23850973 p T^{10} + 3901674 p^{2} T^{11} + 568490 p^{3} T^{12} + 83219 p^{4} T^{13} + 10260 p^{5} T^{14} + 1018 p^{6} T^{15} + 140 p^{7} T^{16} + 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 12 T + 193 T^{2} - 1789 T^{3} + 600 p T^{4} - 141873 T^{5} + 1141452 T^{6} - 7333927 T^{7} + 48912708 T^{8} - 268589310 T^{9} + 48912708 p T^{10} - 7333927 p^{2} T^{11} + 1141452 p^{3} T^{12} - 141873 p^{4} T^{13} + 600 p^{6} T^{14} - 1789 p^{6} T^{15} + 193 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 15 T + 362 T^{2} - 3972 T^{3} + 54830 T^{4} - 472427 T^{5} + 4734840 T^{6} - 33133904 T^{7} + 261735687 T^{8} - 1504564436 T^{9} + 261735687 p T^{10} - 33133904 p^{2} T^{11} + 4734840 p^{3} T^{12} - 472427 p^{4} T^{13} + 54830 p^{5} T^{14} - 3972 p^{6} T^{15} + 362 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 14 T + 327 T^{2} + 3205 T^{3} + 44586 T^{4} + 348111 T^{5} + 3681236 T^{6} + 24097457 T^{7} + 209386044 T^{8} + 1166849428 T^{9} + 209386044 p T^{10} + 24097457 p^{2} T^{11} + 3681236 p^{3} T^{12} + 348111 p^{4} T^{13} + 44586 p^{5} T^{14} + 3205 p^{6} T^{15} + 327 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 6 T + 319 T^{2} - 1839 T^{3} + 47906 T^{4} - 252725 T^{5} + 4413570 T^{6} - 20520067 T^{7} + 273224160 T^{8} - 1081545560 T^{9} + 273224160 p T^{10} - 20520067 p^{2} T^{11} + 4413570 p^{3} T^{12} - 252725 p^{4} T^{13} + 47906 p^{5} T^{14} - 1839 p^{6} T^{15} + 319 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 3 T + 278 T^{2} - 1171 T^{3} + 38707 T^{4} - 177397 T^{5} + 3551703 T^{6} - 15372301 T^{7} + 231098509 T^{8} - 875034452 T^{9} + 231098509 p T^{10} - 15372301 p^{2} T^{11} + 3551703 p^{3} T^{12} - 177397 p^{4} T^{13} + 38707 p^{5} T^{14} - 1171 p^{6} T^{15} + 278 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 9 T + 172 T^{2} + 1209 T^{3} + 14217 T^{4} + 51649 T^{5} + 463831 T^{6} - 1181855 T^{7} - 894743 T^{8} - 208766930 T^{9} - 894743 p T^{10} - 1181855 p^{2} T^{11} + 463831 p^{3} T^{12} + 51649 p^{4} T^{13} + 14217 p^{5} T^{14} + 1209 p^{6} T^{15} + 172 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 9 T + 210 T^{2} - 1695 T^{3} + 26299 T^{4} - 206235 T^{5} + 2400067 T^{6} - 17846869 T^{7} + 172863813 T^{8} - 1191970608 T^{9} + 172863813 p T^{10} - 17846869 p^{2} T^{11} + 2400067 p^{3} T^{12} - 206235 p^{4} T^{13} + 26299 p^{5} T^{14} - 1695 p^{6} T^{15} + 210 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 30 T + 786 T^{2} - 13574 T^{3} + 210383 T^{4} - 2611429 T^{5} + 29858725 T^{6} - 291974242 T^{7} + 2675679737 T^{8} - 21519130886 T^{9} + 2675679737 p T^{10} - 291974242 p^{2} T^{11} + 29858725 p^{3} T^{12} - 2611429 p^{4} T^{13} + 210383 p^{5} T^{14} - 13574 p^{6} T^{15} + 786 p^{7} T^{16} - 30 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 16 T + 454 T^{2} - 6618 T^{3} + 104959 T^{4} - 1279883 T^{5} + 15122569 T^{6} - 153614690 T^{7} + 1458313809 T^{8} - 12428852816 T^{9} + 1458313809 p T^{10} - 153614690 p^{2} T^{11} + 15122569 p^{3} T^{12} - 1279883 p^{4} T^{13} + 104959 p^{5} T^{14} - 6618 p^{6} T^{15} + 454 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 3 T + 389 T^{2} - 385 T^{3} + 72285 T^{4} + 22199 T^{5} + 9030607 T^{6} + 7099821 T^{7} + 847401670 T^{8} + 661160352 T^{9} + 847401670 p T^{10} + 7099821 p^{2} T^{11} + 9030607 p^{3} T^{12} + 22199 p^{4} T^{13} + 72285 p^{5} T^{14} - 385 p^{6} T^{15} + 389 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 32 T + 664 T^{2} - 136 p T^{3} + 127791 T^{4} - 1457675 T^{5} + 15824861 T^{6} - 160721244 T^{7} + 1549062291 T^{8} - 13708341698 T^{9} + 1549062291 p T^{10} - 160721244 p^{2} T^{11} + 15824861 p^{3} T^{12} - 1457675 p^{4} T^{13} + 127791 p^{5} T^{14} - 136 p^{7} T^{15} + 664 p^{7} T^{16} - 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 24 T + 693 T^{2} - 12373 T^{3} + 216700 T^{4} - 2985079 T^{5} + 39461338 T^{6} - 437526161 T^{7} + 4632269092 T^{8} - 42200922052 T^{9} + 4632269092 p T^{10} - 437526161 p^{2} T^{11} + 39461338 p^{3} T^{12} - 2985079 p^{4} T^{13} + 216700 p^{5} T^{14} - 12373 p^{6} T^{15} + 693 p^{7} T^{16} - 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 3 T + 312 T^{2} - 843 T^{3} + 47477 T^{4} - 78457 T^{5} + 4769641 T^{6} - 1897577 T^{7} + 385296895 T^{8} + 92865088 T^{9} + 385296895 p T^{10} - 1897577 p^{2} T^{11} + 4769641 p^{3} T^{12} - 78457 p^{4} T^{13} + 47477 p^{5} T^{14} - 843 p^{6} T^{15} + 312 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 46 T + 1334 T^{2} + 27744 T^{3} + 484163 T^{4} + 7267645 T^{5} + 97930165 T^{6} + 1174238724 T^{7} + 12786451321 T^{8} + 125889131394 T^{9} + 12786451321 p T^{10} + 1174238724 p^{2} T^{11} + 97930165 p^{3} T^{12} + 7267645 p^{4} T^{13} + 484163 p^{5} T^{14} + 27744 p^{6} T^{15} + 1334 p^{7} T^{16} + 46 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 43 T + 1385 T^{2} - 31873 T^{3} + 627071 T^{4} - 10318687 T^{5} + 151150379 T^{6} - 1939378947 T^{7} + 22511211436 T^{8} - 232559681172 T^{9} + 22511211436 p T^{10} - 1939378947 p^{2} T^{11} + 151150379 p^{3} T^{12} - 10318687 p^{4} T^{13} + 627071 p^{5} T^{14} - 31873 p^{6} T^{15} + 1385 p^{7} T^{16} - 43 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.02738333368407561668146345090, −2.94809382749922121961845169926, −2.70455700422211634082656897020, −2.68877899538291244773999087722, −2.67196917184876615173240175882, −2.49268555859594015051450025688, −2.48215525152398914552305103720, −2.08629490655831793258730578935, −1.99659460861224642678507002968, −1.95401077481412090195048121957, −1.89664872909963619031455913252, −1.84280456603011009775255852882, −1.71943554521531121787282913663, −1.70627983744025703731735245679, −1.68610441344605381793480807037, −1.64512870007334456560401244242, −1.21156871219022255999254237124, −0.978147470439231979520231035755, −0.865522664180294114035797643981, −0.820788894604101841275666140544, −0.78740410928748943784629315284, −0.69745094246454166693836091092, −0.64410373298850437470337297996, −0.47616921993962698163603373649, −0.17026090522041742175688739933, 0.17026090522041742175688739933, 0.47616921993962698163603373649, 0.64410373298850437470337297996, 0.69745094246454166693836091092, 0.78740410928748943784629315284, 0.820788894604101841275666140544, 0.865522664180294114035797643981, 0.978147470439231979520231035755, 1.21156871219022255999254237124, 1.64512870007334456560401244242, 1.68610441344605381793480807037, 1.70627983744025703731735245679, 1.71943554521531121787282913663, 1.84280456603011009775255852882, 1.89664872909963619031455913252, 1.95401077481412090195048121957, 1.99659460861224642678507002968, 2.08629490655831793258730578935, 2.48215525152398914552305103720, 2.49268555859594015051450025688, 2.67196917184876615173240175882, 2.68877899538291244773999087722, 2.70455700422211634082656897020, 2.94809382749922121961845169926, 3.02738333368407561668146345090
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.