Dirichlet series
| L(s) = 1 | − 3·3-s + 5-s − 9·7-s − 3·9-s + 9·11-s − 9·13-s − 3·15-s + 7·17-s − 17·19-s + 27·21-s + 11·23-s − 13·25-s + 18·27-s + 9·29-s − 8·31-s − 27·33-s − 9·35-s + 2·37-s + 27·39-s + 18·41-s + 7·43-s − 3·45-s + 15·47-s + 45·49-s − 21·51-s − 4·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s − 3.40·7-s − 9-s + 2.71·11-s − 2.49·13-s − 0.774·15-s + 1.69·17-s − 3.90·19-s + 5.89·21-s + 2.29·23-s − 2.59·25-s + 3.46·27-s + 1.67·29-s − 1.43·31-s − 4.70·33-s − 1.52·35-s + 0.328·37-s + 4.32·39-s + 2.81·41-s + 1.06·43-s − 0.447·45-s + 2.18·47-s + 45/7·49-s − 2.94·51-s − 0.549·53-s + 1.21·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.78735\times 10^{16}\) |
| Root analytic conductor: | \(7.99651\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 2^{27} \cdot 7^{9} \cdot 11^{9} \cdot 13^{9} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.409481665\) |
| \(L(\frac12)\) | \(\approx\) | \(4.409481665\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + T )^{9} \) | |
| 11 | \( ( 1 - T )^{9} \) | |
| 13 | \( ( 1 + T )^{9} \) | |
| good | 3 | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + 73 T^{4} + 49 p T^{5} + 110 p T^{6} + 197 p T^{7} + 409 p T^{8} + 1988 T^{9} + 409 p^{2} T^{10} + 197 p^{3} T^{11} + 110 p^{4} T^{12} + 49 p^{5} T^{13} + 73 p^{5} T^{14} + p^{9} T^{15} + 4 p^{8} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - T + 14 T^{2} - 13 T^{3} + 27 p T^{4} - 137 T^{5} + 998 T^{6} - 1043 T^{7} + 5921 T^{8} - 5886 T^{9} + 5921 p T^{10} - 1043 p^{2} T^{11} + 998 p^{3} T^{12} - 137 p^{4} T^{13} + 27 p^{6} T^{14} - 13 p^{6} T^{15} + 14 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 7 T + 94 T^{2} - 469 T^{3} + 3935 T^{4} - 16357 T^{5} + 111908 T^{6} - 418759 T^{7} + 145713 p T^{8} - 8244238 T^{9} + 145713 p^{2} T^{10} - 418759 p^{2} T^{11} + 111908 p^{3} T^{12} - 16357 p^{4} T^{13} + 3935 p^{5} T^{14} - 469 p^{6} T^{15} + 94 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 17 T + 250 T^{2} + 2543 T^{3} + 22649 T^{4} + 166717 T^{5} + 1096064 T^{6} + 6273913 T^{7} + 32423845 T^{8} + 148492824 T^{9} + 32423845 p T^{10} + 6273913 p^{2} T^{11} + 1096064 p^{3} T^{12} + 166717 p^{4} T^{13} + 22649 p^{5} T^{14} + 2543 p^{6} T^{15} + 250 p^{7} T^{16} + 17 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 11 T + 133 T^{2} - 918 T^{3} + 6479 T^{4} - 32786 T^{5} + 171365 T^{6} - 682762 T^{7} + 3177562 T^{8} - 12706630 T^{9} + 3177562 p T^{10} - 682762 p^{2} T^{11} + 171365 p^{3} T^{12} - 32786 p^{4} T^{13} + 6479 p^{5} T^{14} - 918 p^{6} T^{15} + 133 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 9 T + 151 T^{2} - 1528 T^{3} + 13853 T^{4} - 114984 T^{5} + 851637 T^{6} - 5649704 T^{7} + 34782342 T^{8} - 196302174 T^{9} + 34782342 p T^{10} - 5649704 p^{2} T^{11} + 851637 p^{3} T^{12} - 114984 p^{4} T^{13} + 13853 p^{5} T^{14} - 1528 p^{6} T^{15} + 151 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 8 T + 185 T^{2} + 1552 T^{3} + 17539 T^{4} + 137544 T^{5} + 1085741 T^{6} + 7483048 T^{7} + 47024194 T^{8} + 277210160 T^{9} + 47024194 p T^{10} + 7483048 p^{2} T^{11} + 1085741 p^{3} T^{12} + 137544 p^{4} T^{13} + 17539 p^{5} T^{14} + 1552 p^{6} T^{15} + 185 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 2 T + 183 T^{2} - 305 T^{3} + 15447 T^{4} - 26690 T^{5} + 831383 T^{6} - 1647047 T^{7} + 34407826 T^{8} - 72172888 T^{9} + 34407826 p T^{10} - 1647047 p^{2} T^{11} + 831383 p^{3} T^{12} - 26690 p^{4} T^{13} + 15447 p^{5} T^{14} - 305 p^{6} T^{15} + 183 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 - 18 T + 405 T^{2} - 5187 T^{3} + 68489 T^{4} - 678734 T^{5} + 6603241 T^{6} - 52672845 T^{7} + 406456472 T^{8} - 2650507104 T^{9} + 406456472 p T^{10} - 52672845 p^{2} T^{11} + 6603241 p^{3} T^{12} - 678734 p^{4} T^{13} + 68489 p^{5} T^{14} - 5187 p^{6} T^{15} + 405 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 7 T + 191 T^{2} - 706 T^{3} + 15597 T^{4} - 17440 T^{5} + 811461 T^{6} + 888253 T^{7} + 35770279 T^{8} + 73478904 T^{9} + 35770279 p T^{10} + 888253 p^{2} T^{11} + 811461 p^{3} T^{12} - 17440 p^{4} T^{13} + 15597 p^{5} T^{14} - 706 p^{6} T^{15} + 191 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 15 T + 257 T^{2} - 2386 T^{3} + 24847 T^{4} - 170378 T^{5} + 1469333 T^{6} - 8622622 T^{7} + 71632534 T^{8} - 404371246 T^{9} + 71632534 p T^{10} - 8622622 p^{2} T^{11} + 1469333 p^{3} T^{12} - 170378 p^{4} T^{13} + 24847 p^{5} T^{14} - 2386 p^{6} T^{15} + 257 p^{7} T^{16} - 15 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 4 T + 433 T^{2} + 1608 T^{3} + 85377 T^{4} + 287138 T^{5} + 10060393 T^{6} + 29749398 T^{7} + 780717777 T^{8} + 1955958154 T^{9} + 780717777 p T^{10} + 29749398 p^{2} T^{11} + 10060393 p^{3} T^{12} + 287138 p^{4} T^{13} + 85377 p^{5} T^{14} + 1608 p^{6} T^{15} + 433 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 23 T + 624 T^{2} + 9259 T^{3} + 146388 T^{4} + 1629728 T^{5} + 19041742 T^{6} + 170883333 T^{7} + 1611774001 T^{8} + 12064362626 T^{9} + 1611774001 p T^{10} + 170883333 p^{2} T^{11} + 19041742 p^{3} T^{12} + 1629728 p^{4} T^{13} + 146388 p^{5} T^{14} + 9259 p^{6} T^{15} + 624 p^{7} T^{16} + 23 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 - 12 T + 272 T^{2} - 3187 T^{3} + 44467 T^{4} - 447587 T^{5} + 4792264 T^{6} - 43060656 T^{7} + 382986649 T^{8} - 3016756990 T^{9} + 382986649 p T^{10} - 43060656 p^{2} T^{11} + 4792264 p^{3} T^{12} - 447587 p^{4} T^{13} + 44467 p^{5} T^{14} - 3187 p^{6} T^{15} + 272 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 + 16 T + 408 T^{2} + 5156 T^{3} + 80279 T^{4} + 839402 T^{5} + 10079582 T^{6} + 90178442 T^{7} + 905262375 T^{8} + 7006843812 T^{9} + 905262375 p T^{10} + 90178442 p^{2} T^{11} + 10079582 p^{3} T^{12} + 839402 p^{4} T^{13} + 80279 p^{5} T^{14} + 5156 p^{6} T^{15} + 408 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 6 T + 370 T^{2} + 2194 T^{3} + 70740 T^{4} + 395820 T^{5} + 9084418 T^{6} + 46344734 T^{7} + 852168895 T^{8} + 3855170876 T^{9} + 852168895 p T^{10} + 46344734 p^{2} T^{11} + 9084418 p^{3} T^{12} + 395820 p^{4} T^{13} + 70740 p^{5} T^{14} + 2194 p^{6} T^{15} + 370 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 - 4 T + 491 T^{2} - 2057 T^{3} + 116707 T^{4} - 475090 T^{5} + 17415531 T^{6} - 65174039 T^{7} + 1787196518 T^{8} - 5823984148 T^{9} + 1787196518 p T^{10} - 65174039 p^{2} T^{11} + 17415531 p^{3} T^{12} - 475090 p^{4} T^{13} + 116707 p^{5} T^{14} - 2057 p^{6} T^{15} + 491 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 21 T + 693 T^{2} - 11140 T^{3} + 211527 T^{4} - 2728732 T^{5} + 37906651 T^{6} - 401639805 T^{7} + 4422828729 T^{8} - 38747122180 T^{9} + 4422828729 p T^{10} - 401639805 p^{2} T^{11} + 37906651 p^{3} T^{12} - 2728732 p^{4} T^{13} + 211527 p^{5} T^{14} - 11140 p^{6} T^{15} + 693 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 16 T + 360 T^{2} + 4569 T^{3} + 66641 T^{4} + 736949 T^{5} + 9003496 T^{6} + 87090600 T^{7} + 930887019 T^{8} + 8038020744 T^{9} + 930887019 p T^{10} + 87090600 p^{2} T^{11} + 9003496 p^{3} T^{12} + 736949 p^{4} T^{13} + 66641 p^{5} T^{14} + 4569 p^{6} T^{15} + 360 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 5 T + 204 T^{2} - 679 T^{3} + 36443 T^{4} - 121843 T^{5} + 4398946 T^{6} - 10639421 T^{7} + 472479025 T^{8} - 1329308246 T^{9} + 472479025 p T^{10} - 10639421 p^{2} T^{11} + 4398946 p^{3} T^{12} - 121843 p^{4} T^{13} + 36443 p^{5} T^{14} - 679 p^{6} T^{15} + 204 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 18 T + 675 T^{2} - 9315 T^{3} + 194149 T^{4} - 2183092 T^{5} + 33162181 T^{6} - 320173045 T^{7} + 4028505266 T^{8} - 34845875524 T^{9} + 4028505266 p T^{10} - 320173045 p^{2} T^{11} + 33162181 p^{3} T^{12} - 2183092 p^{4} T^{13} + 194149 p^{5} T^{14} - 9315 p^{6} T^{15} + 675 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.79092782569540392730747516631, −2.75345059090556870366203915664, −2.73279087154492495719502824886, −2.69174868277475657021902573956, −2.65188517407452467510325753386, −2.44510121976464074044270484972, −2.38959730801423049334865395645, −2.25989157905654449193559247134, −1.99712889425619858512111131672, −1.96697049031867533096641116206, −1.90556099116131800586494793652, −1.65173006740566129486036786416, −1.64518336704685956304102483762, −1.64102078710959124031392224990, −1.63673732219487163474573030837, −1.26138472878929005896051748070, −1.12030375127877512008095934409, −0.896102107191924041388679186407, −0.75783315802334772827118144531, −0.73890267027712775087779390446, −0.54862557925401252143866036642, −0.40772237337830975635546737568, −0.37938414777856202988291825376, −0.35703921593915835108244291141, −0.27926487709821426737903592466, 0.27926487709821426737903592466, 0.35703921593915835108244291141, 0.37938414777856202988291825376, 0.40772237337830975635546737568, 0.54862557925401252143866036642, 0.73890267027712775087779390446, 0.75783315802334772827118144531, 0.896102107191924041388679186407, 1.12030375127877512008095934409, 1.26138472878929005896051748070, 1.63673732219487163474573030837, 1.64102078710959124031392224990, 1.64518336704685956304102483762, 1.65173006740566129486036786416, 1.90556099116131800586494793652, 1.96697049031867533096641116206, 1.99712889425619858512111131672, 2.25989157905654449193559247134, 2.38959730801423049334865395645, 2.44510121976464074044270484972, 2.65188517407452467510325753386, 2.69174868277475657021902573956, 2.73279087154492495719502824886, 2.75345059090556870366203915664, 2.79092782569540392730747516631
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.