Dirichlet series
| L(s) = 1 | − 3-s − 8·5-s + 2·7-s − 13·9-s − 10·11-s − 13·13-s + 8·15-s − 8·17-s − 19-s − 2·21-s − 3·23-s + 11·25-s + 11·27-s − 25·29-s − 31-s + 10·33-s − 16·35-s − 35·37-s + 13·39-s − 16·41-s + 9·43-s + 104·45-s − 47-s − 30·49-s + 8·51-s − 29·53-s + 80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3.57·5-s + 0.755·7-s − 4.33·9-s − 3.01·11-s − 3.60·13-s + 2.06·15-s − 1.94·17-s − 0.229·19-s − 0.436·21-s − 0.625·23-s + 11/5·25-s + 2.11·27-s − 4.64·29-s − 0.179·31-s + 1.74·33-s − 2.70·35-s − 5.75·37-s + 2.08·39-s − 2.49·41-s + 1.37·43-s + 15.5·45-s − 0.145·47-s − 4.28·49-s + 1.12·51-s − 3.98·53-s + 10.7·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{27} \cdot 167^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(2^{27} \cdot 167^{9}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(1.78958\times 10^{9}\) |
| Root analytic conductor: | \(3.26619\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(9\) |
| Selberg data: | \((18,\ 2^{27} \cdot 167^{9} ,\ ( \ : [1/2]^{9} ),\ -1 )\) |
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 | \( 1 \) |
| 167 | \( ( 1 - T )^{9} \) | |
| good | 3 | \( 1 + T + 14 T^{2} + 16 T^{3} + 107 T^{4} + 41 p T^{5} + 190 p T^{6} + 610 T^{7} + 2245 T^{8} + 2147 T^{9} + 2245 p T^{10} + 610 p^{2} T^{11} + 190 p^{4} T^{12} + 41 p^{5} T^{13} + 107 p^{5} T^{14} + 16 p^{6} T^{15} + 14 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 8 T + 53 T^{2} + 254 T^{3} + 1063 T^{4} + 3774 T^{5} + 2407 p T^{6} + 34178 T^{7} + 88336 T^{8} + 206612 T^{9} + 88336 p T^{10} + 34178 p^{2} T^{11} + 2407 p^{4} T^{12} + 3774 p^{4} T^{13} + 1063 p^{5} T^{14} + 254 p^{6} T^{15} + 53 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 7 | \( 1 - 2 T + 34 T^{2} - 72 T^{3} + 81 p T^{4} - 1231 T^{5} + 913 p T^{6} - 13376 T^{7} + 55371 T^{8} - 106443 T^{9} + 55371 p T^{10} - 13376 p^{2} T^{11} + 913 p^{4} T^{12} - 1231 p^{4} T^{13} + 81 p^{6} T^{14} - 72 p^{6} T^{15} + 34 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 + 10 T + 93 T^{2} + 590 T^{3} + 3378 T^{4} + 16365 T^{5} + 72894 T^{6} + 292609 T^{7} + 1094660 T^{8} + 3753996 T^{9} + 1094660 p T^{10} + 292609 p^{2} T^{11} + 72894 p^{3} T^{12} + 16365 p^{4} T^{13} + 3378 p^{5} T^{14} + 590 p^{6} T^{15} + 93 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 13 | \( 1 + p T + 133 T^{2} + 953 T^{3} + 6225 T^{4} + 33638 T^{5} + 169261 T^{6} + 741887 T^{7} + 3086116 T^{8} + 878050 p T^{9} + 3086116 p T^{10} + 741887 p^{2} T^{11} + 169261 p^{3} T^{12} + 33638 p^{4} T^{13} + 6225 p^{5} T^{14} + 953 p^{6} T^{15} + 133 p^{7} T^{16} + p^{9} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 8 T + 92 T^{2} + 500 T^{3} + 3438 T^{4} + 14774 T^{5} + 76420 T^{6} + 279916 T^{7} + 1273329 T^{8} + 4593380 T^{9} + 1273329 p T^{10} + 279916 p^{2} T^{11} + 76420 p^{3} T^{12} + 14774 p^{4} T^{13} + 3438 p^{5} T^{14} + 500 p^{6} T^{15} + 92 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + T + 74 T^{2} + 124 T^{3} + 3178 T^{4} + 6606 T^{5} + 98399 T^{6} + 201683 T^{7} + 2392182 T^{8} + 4452264 T^{9} + 2392182 p T^{10} + 201683 p^{2} T^{11} + 98399 p^{3} T^{12} + 6606 p^{4} T^{13} + 3178 p^{5} T^{14} + 124 p^{6} T^{15} + 74 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 + 3 T + 59 T^{2} - 39 T^{3} + 1183 T^{4} - 8810 T^{5} + 24729 T^{6} - 233961 T^{7} + 1236788 T^{8} - 3999570 T^{9} + 1236788 p T^{10} - 233961 p^{2} T^{11} + 24729 p^{3} T^{12} - 8810 p^{4} T^{13} + 1183 p^{5} T^{14} - 39 p^{6} T^{15} + 59 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 + 25 T + 395 T^{2} + 4479 T^{3} + 40930 T^{4} + 309446 T^{5} + 2033900 T^{6} + 11929004 T^{7} + 65912049 T^{8} + 354500133 T^{9} + 65912049 p T^{10} + 11929004 p^{2} T^{11} + 2033900 p^{3} T^{12} + 309446 p^{4} T^{13} + 40930 p^{5} T^{14} + 4479 p^{6} T^{15} + 395 p^{7} T^{16} + 25 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + T + 156 T^{2} + 46 T^{3} + 12646 T^{4} - 64 T^{5} + 695121 T^{6} - 73599 T^{7} + 28337228 T^{8} - 2988116 T^{9} + 28337228 p T^{10} - 73599 p^{2} T^{11} + 695121 p^{3} T^{12} - 64 p^{4} T^{13} + 12646 p^{5} T^{14} + 46 p^{6} T^{15} + 156 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 + 35 T + 763 T^{2} + 11917 T^{3} + 150667 T^{4} + 1593626 T^{5} + 14655455 T^{6} + 118318739 T^{7} + 851803426 T^{8} + 5469266198 T^{9} + 851803426 p T^{10} + 118318739 p^{2} T^{11} + 14655455 p^{3} T^{12} + 1593626 p^{4} T^{13} + 150667 p^{5} T^{14} + 11917 p^{6} T^{15} + 763 p^{7} T^{16} + 35 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 16 T + 303 T^{2} + 2844 T^{3} + 31747 T^{4} + 213788 T^{5} + 1872807 T^{6} + 10023732 T^{7} + 81015990 T^{8} + 398200168 T^{9} + 81015990 p T^{10} + 10023732 p^{2} T^{11} + 1872807 p^{3} T^{12} + 213788 p^{4} T^{13} + 31747 p^{5} T^{14} + 2844 p^{6} T^{15} + 303 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 9 T + 277 T^{2} - 1841 T^{3} + 33599 T^{4} - 176736 T^{5} + 2548969 T^{6} - 11255519 T^{7} + 141367302 T^{8} - 545391838 T^{9} + 141367302 p T^{10} - 11255519 p^{2} T^{11} + 2548969 p^{3} T^{12} - 176736 p^{4} T^{13} + 33599 p^{5} T^{14} - 1841 p^{6} T^{15} + 277 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 + T + 272 T^{2} - 136 T^{3} + 32074 T^{4} - 76640 T^{5} + 2223103 T^{6} - 10058089 T^{7} + 113020750 T^{8} - 644899588 T^{9} + 113020750 p T^{10} - 10058089 p^{2} T^{11} + 2223103 p^{3} T^{12} - 76640 p^{4} T^{13} + 32074 p^{5} T^{14} - 136 p^{6} T^{15} + 272 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 + 29 T + 726 T^{2} + 12562 T^{3} + 188263 T^{4} + 2341888 T^{5} + 25766913 T^{6} + 247956318 T^{7} + 2138364493 T^{8} + 16409384230 T^{9} + 2138364493 p T^{10} + 247956318 p^{2} T^{11} + 25766913 p^{3} T^{12} + 2341888 p^{4} T^{13} + 188263 p^{5} T^{14} + 12562 p^{6} T^{15} + 726 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 14 T + 4 p T^{2} + 2756 T^{3} + 33441 T^{4} + 327888 T^{5} + 3255193 T^{6} + 28063996 T^{7} + 240319503 T^{8} + 1866179140 T^{9} + 240319503 p T^{10} + 28063996 p^{2} T^{11} + 3255193 p^{3} T^{12} + 327888 p^{4} T^{13} + 33441 p^{5} T^{14} + 2756 p^{6} T^{15} + 4 p^{8} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 28 T + 8 p T^{2} + 6082 T^{3} + 61351 T^{4} + 542901 T^{5} + 71873 p T^{6} + 33042120 T^{7} + 240205165 T^{8} + 1799569925 T^{9} + 240205165 p T^{10} + 33042120 p^{2} T^{11} + 71873 p^{4} T^{12} + 542901 p^{4} T^{13} + 61351 p^{5} T^{14} + 6082 p^{6} T^{15} + 8 p^{8} T^{16} + 28 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 19 T + 373 T^{2} - 4527 T^{3} + 55745 T^{4} - 463082 T^{5} + 4103863 T^{6} - 24884529 T^{7} + 197147226 T^{8} - 1121891190 T^{9} + 197147226 p T^{10} - 24884529 p^{2} T^{11} + 4103863 p^{3} T^{12} - 463082 p^{4} T^{13} + 55745 p^{5} T^{14} - 4527 p^{6} T^{15} + 373 p^{7} T^{16} - 19 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 + 9 T + 336 T^{2} + 3512 T^{3} + 59605 T^{4} + 653252 T^{5} + 7286033 T^{6} + 77748744 T^{7} + 665144243 T^{8} + 6510156198 T^{9} + 665144243 p T^{10} + 77748744 p^{2} T^{11} + 7286033 p^{3} T^{12} + 653252 p^{4} T^{13} + 59605 p^{5} T^{14} + 3512 p^{6} T^{15} + 336 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 160 T^{2} - 1178 T^{3} + 13812 T^{4} - 174332 T^{5} + 1312718 T^{6} - 13146518 T^{7} + 138980941 T^{8} - 746538888 T^{9} + 138980941 p T^{10} - 13146518 p^{2} T^{11} + 1312718 p^{3} T^{12} - 174332 p^{4} T^{13} + 13812 p^{5} T^{14} - 1178 p^{6} T^{15} + 160 p^{7} T^{16} + p^{9} T^{18} \) | |
| 79 | \( 1 + 18 T + 613 T^{2} + 8232 T^{3} + 162407 T^{4} + 1767190 T^{5} + 26081621 T^{6} + 239485944 T^{7} + 2879648274 T^{8} + 22496197200 T^{9} + 2879648274 p T^{10} + 239485944 p^{2} T^{11} + 26081621 p^{3} T^{12} + 1767190 p^{4} T^{13} + 162407 p^{5} T^{14} + 8232 p^{6} T^{15} + 613 p^{7} T^{16} + 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 13 T + 673 T^{2} + 7417 T^{3} + 205967 T^{4} + 1935852 T^{5} + 37738433 T^{6} + 301498679 T^{7} + 4568581146 T^{8} + 30624712542 T^{9} + 4568581146 p T^{10} + 301498679 p^{2} T^{11} + 37738433 p^{3} T^{12} + 1935852 p^{4} T^{13} + 205967 p^{5} T^{14} + 7417 p^{6} T^{15} + 673 p^{7} T^{16} + 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 21 T + 709 T^{2} + 11529 T^{3} + 229075 T^{4} + 3006419 T^{5} + 44380307 T^{6} + 482662319 T^{7} + 5716348292 T^{8} + 51903552752 T^{9} + 5716348292 p T^{10} + 482662319 p^{2} T^{11} + 44380307 p^{3} T^{12} + 3006419 p^{4} T^{13} + 229075 p^{5} T^{14} + 11529 p^{6} T^{15} + 709 p^{7} T^{16} + 21 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 2 T + 615 T^{2} - 1236 T^{3} + 171368 T^{4} - 352849 T^{5} + 29435302 T^{6} - 61397603 T^{7} + 3635999768 T^{8} - 7173227220 T^{9} + 3635999768 p T^{10} - 61397603 p^{2} T^{11} + 29435302 p^{3} T^{12} - 352849 p^{4} T^{13} + 171368 p^{5} T^{14} - 1236 p^{6} T^{15} + 615 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−4.16655740409495167825534522499, −4.14871390202699848428349110623, −4.07201623747804523757795484522, −3.87243769082786514398927008617, −3.86601576930200172431683344133, −3.65710054765632239302303546634, −3.59529958432571237117709391793, −3.35273600522843636133450009959, −3.31269604395047616232752788618, −3.08471518501592427061187269296, −3.07699200779516648207722651430, −3.06074648879592539524146224418, −2.99552604846120172756766724170, −2.81771297289894240158767617731, −2.80008509792233966460518414506, −2.37690816939709716191679771922, −2.30256538497453927538824944265, −2.28744903388417718466571993622, −2.07341581901262472635300042817, −1.96882514720437380541949957434, −1.82756722322965835551001378987, −1.73877678076197676665562785960, −1.56244122897092035302677151413, −1.52635882268899645968087467614, −1.19159359769122529833817146915, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.19159359769122529833817146915, 1.52635882268899645968087467614, 1.56244122897092035302677151413, 1.73877678076197676665562785960, 1.82756722322965835551001378987, 1.96882514720437380541949957434, 2.07341581901262472635300042817, 2.28744903388417718466571993622, 2.30256538497453927538824944265, 2.37690816939709716191679771922, 2.80008509792233966460518414506, 2.81771297289894240158767617731, 2.99552604846120172756766724170, 3.06074648879592539524146224418, 3.07699200779516648207722651430, 3.08471518501592427061187269296, 3.31269604395047616232752788618, 3.35273600522843636133450009959, 3.59529958432571237117709391793, 3.65710054765632239302303546634, 3.86601576930200172431683344133, 3.87243769082786514398927008617, 4.07201623747804523757795484522, 4.14871390202699848428349110623, 4.16655740409495167825534522499
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.