Dirichlet series
| L(s) = 1 | + 5·3-s + 3·5-s − 9·7-s + 3·9-s − 9·11-s − 9·13-s + 15·15-s − 7·17-s + 13·19-s − 45·21-s + 9·23-s − 19·25-s − 30·27-s + 29-s + 10·31-s − 45·33-s − 27·35-s + 14·37-s − 45·39-s − 2·41-s + 5·43-s + 9·45-s + 13·47-s + 45·49-s − 35·51-s + 22·53-s − 27·55-s + ⋯ |
| L(s) = 1 | + 2.88·3-s + 1.34·5-s − 3.40·7-s + 9-s − 2.71·11-s − 2.49·13-s + 3.87·15-s − 1.69·17-s + 2.98·19-s − 9.81·21-s + 1.87·23-s − 3.79·25-s − 5.77·27-s + 0.185·29-s + 1.79·31-s − 7.83·33-s − 4.56·35-s + 2.30·37-s − 7.20·39-s − 0.312·41-s + 0.762·43-s + 1.34·45-s + 1.89·47-s + 45/7·49-s − 4.90·51-s + 3.02·53-s − 3.64·55-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
| \(L(1)\) | \(\approx\) | \(71.55828549\) |
| \(L(\frac12)\) | \(\approx\) | \(71.55828549\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - 5 T + 22 T^{2} - 65 T^{3} + 19 p^{2} T^{4} - 367 T^{5} + 722 T^{6} - 1261 T^{7} + 727 p T^{8} - 3668 T^{9} + 727 p^{2} T^{10} - 1261 p^{2} T^{11} + 722 p^{3} T^{12} - 367 p^{4} T^{13} + 19 p^{7} T^{14} - 65 p^{6} T^{15} + 22 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 - 3 T + 28 T^{2} - 71 T^{3} + 363 T^{4} - 787 T^{5} + 3016 T^{6} - 5699 T^{7} + 18739 T^{8} - 31646 T^{9} + 18739 p T^{10} - 5699 p^{2} T^{11} + 3016 p^{3} T^{12} - 787 p^{4} T^{13} + 363 p^{5} T^{14} - 71 p^{6} T^{15} + 28 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 + 7 T + 82 T^{2} + 259 T^{3} + 2045 T^{4} + 1873 T^{5} + 2172 p T^{6} + 24031 T^{7} + 942421 T^{8} + 1108142 T^{9} + 942421 p T^{10} + 24031 p^{2} T^{11} + 2172 p^{4} T^{12} + 1873 p^{4} T^{13} + 2045 p^{5} T^{14} + 259 p^{6} T^{15} + 82 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 - 13 T + 178 T^{2} - 1405 T^{3} + 10931 T^{4} - 60977 T^{5} + 343000 T^{6} - 1500671 T^{7} + 7197879 T^{8} - 28886252 T^{9} + 7197879 p T^{10} - 1500671 p^{2} T^{11} + 343000 p^{3} T^{12} - 60977 p^{4} T^{13} + 10931 p^{5} T^{14} - 1405 p^{6} T^{15} + 178 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 9 T + 169 T^{2} - 1328 T^{3} + 13931 T^{4} - 90824 T^{5} + 701323 T^{6} - 3805908 T^{7} + 23406036 T^{8} - 106261174 T^{9} + 23406036 p T^{10} - 3805908 p^{2} T^{11} + 701323 p^{3} T^{12} - 90824 p^{4} T^{13} + 13931 p^{5} T^{14} - 1328 p^{6} T^{15} + 169 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - T + 173 T^{2} - 346 T^{3} + 14427 T^{4} - 37748 T^{5} + 780989 T^{6} - 2179586 T^{7} + 30433390 T^{8} - 78376534 T^{9} + 30433390 p T^{10} - 2179586 p^{2} T^{11} + 780989 p^{3} T^{12} - 37748 p^{4} T^{13} + 14427 p^{5} T^{14} - 346 p^{6} T^{15} + 173 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 - 10 T + 229 T^{2} - 1956 T^{3} + 24783 T^{4} - 180686 T^{5} + 1644385 T^{6} - 10194632 T^{7} + 73319162 T^{8} - 382975896 T^{9} + 73319162 p T^{10} - 10194632 p^{2} T^{11} + 1644385 p^{3} T^{12} - 180686 p^{4} T^{13} + 24783 p^{5} T^{14} - 1956 p^{6} T^{15} + 229 p^{7} T^{16} - 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - 14 T + 337 T^{2} - 3471 T^{3} + 47889 T^{4} - 392212 T^{5} + 3960123 T^{6} - 721589 p T^{7} + 214442638 T^{8} - 1201000340 T^{9} + 214442638 p T^{10} - 721589 p^{3} T^{11} + 3960123 p^{3} T^{12} - 392212 p^{4} T^{13} + 47889 p^{5} T^{14} - 3471 p^{6} T^{15} + 337 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 2 T + 221 T^{2} + 223 T^{3} + 24657 T^{4} + 12100 T^{5} + 1839835 T^{6} + 407801 T^{7} + 99897770 T^{8} + 12312092 T^{9} + 99897770 p T^{10} + 407801 p^{2} T^{11} + 1839835 p^{3} T^{12} + 12100 p^{4} T^{13} + 24657 p^{5} T^{14} + 223 p^{6} T^{15} + 221 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 5 T + 123 T^{2} - 1154 T^{3} + 10921 T^{4} - 107984 T^{5} + 828763 T^{6} - 6379915 T^{7} + 50015045 T^{8} - 295475136 T^{9} + 50015045 p T^{10} - 6379915 p^{2} T^{11} + 828763 p^{3} T^{12} - 107984 p^{4} T^{13} + 10921 p^{5} T^{14} - 1154 p^{6} T^{15} + 123 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 13 T + 299 T^{2} - 3256 T^{3} + 43497 T^{4} - 405584 T^{5} + 4045121 T^{6} - 32451124 T^{7} + 262651918 T^{8} - 1807952110 T^{9} + 262651918 p T^{10} - 32451124 p^{2} T^{11} + 4045121 p^{3} T^{12} - 405584 p^{4} T^{13} + 43497 p^{5} T^{14} - 3256 p^{6} T^{15} + 299 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 22 T + 491 T^{2} - 7108 T^{3} + 99627 T^{4} - 1104726 T^{5} + 11770375 T^{6} - 105785560 T^{7} + 911903901 T^{8} - 6777044994 T^{9} + 911903901 p T^{10} - 105785560 p^{2} T^{11} + 11770375 p^{3} T^{12} - 1104726 p^{4} T^{13} + 99627 p^{5} T^{14} - 7108 p^{6} T^{15} + 491 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 - 43 T + 1232 T^{2} - 25283 T^{3} + 422904 T^{4} - 5872392 T^{5} + 70486296 T^{6} - 735093349 T^{7} + 114910849 p T^{8} - 55202302762 T^{9} + 114910849 p^{2} T^{10} - 735093349 p^{2} T^{11} + 70486296 p^{3} T^{12} - 5872392 p^{4} T^{13} + 422904 p^{5} T^{14} - 25283 p^{6} T^{15} + 1232 p^{7} T^{16} - 43 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 10 T + 236 T^{2} + 1473 T^{3} + 27947 T^{4} + 138579 T^{5} + 2411530 T^{6} + 9032442 T^{7} + 164334623 T^{8} + 540869862 T^{9} + 164334623 p T^{10} + 9032442 p^{2} T^{11} + 2411530 p^{3} T^{12} + 138579 p^{4} T^{13} + 27947 p^{5} T^{14} + 1473 p^{6} T^{15} + 236 p^{7} T^{16} + 10 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 26 T + 566 T^{2} - 7610 T^{3} + 97755 T^{4} - 966248 T^{5} + 10180100 T^{6} - 89786202 T^{7} + 864941829 T^{8} - 6868878016 T^{9} + 864941829 p T^{10} - 89786202 p^{2} T^{11} + 10180100 p^{3} T^{12} - 966248 p^{4} T^{13} + 97755 p^{5} T^{14} - 7610 p^{6} T^{15} + 566 p^{7} T^{16} - 26 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 18 T + 458 T^{2} - 6234 T^{3} + 102872 T^{4} - 1164860 T^{5} + 14660798 T^{6} - 139687702 T^{7} + 1450238575 T^{8} - 11784537732 T^{9} + 1450238575 p T^{10} - 139687702 p^{2} T^{11} + 14660798 p^{3} T^{12} - 1164860 p^{4} T^{13} + 102872 p^{5} T^{14} - 6234 p^{6} T^{15} + 458 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + 8 T + 397 T^{2} + 2345 T^{3} + 79739 T^{4} + 390610 T^{5} + 10694461 T^{6} + 44434099 T^{7} + 1040954774 T^{8} + 3746189660 T^{9} + 1040954774 p T^{10} + 44434099 p^{2} T^{11} + 10694461 p^{3} T^{12} + 390610 p^{4} T^{13} + 79739 p^{5} T^{14} + 2345 p^{6} T^{15} + 397 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 + 9 T + 557 T^{2} + 4916 T^{3} + 144985 T^{4} + 1224674 T^{5} + 23345681 T^{6} + 181746505 T^{7} + 2580593467 T^{8} + 17576249308 T^{9} + 2580593467 p T^{10} + 181746505 p^{2} T^{11} + 23345681 p^{3} T^{12} + 1224674 p^{4} T^{13} + 144985 p^{5} T^{14} + 4916 p^{6} T^{15} + 557 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 - 4 T + 480 T^{2} - 2095 T^{3} + 113607 T^{4} - 509061 T^{5} + 17525726 T^{6} - 75538206 T^{7} + 1947321779 T^{8} - 7539999324 T^{9} + 1947321779 p T^{10} - 75538206 p^{2} T^{11} + 17525726 p^{3} T^{12} - 509061 p^{4} T^{13} + 113607 p^{5} T^{14} - 2095 p^{6} T^{15} + 480 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 - 7 T + 618 T^{2} - 3859 T^{3} + 179101 T^{4} - 1003609 T^{5} + 32276296 T^{6} - 160578387 T^{7} + 4008678339 T^{8} - 17246000590 T^{9} + 4008678339 p T^{10} - 160578387 p^{2} T^{11} + 32276296 p^{3} T^{12} - 1003609 p^{4} T^{13} + 179101 p^{5} T^{14} - 3859 p^{6} T^{15} + 618 p^{7} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 - 2 T + 505 T^{2} - 51 T^{3} + 128739 T^{4} + 115864 T^{5} + 22391131 T^{6} + 27065123 T^{7} + 2878949816 T^{8} + 3334138196 T^{9} + 2878949816 p T^{10} + 27065123 p^{2} T^{11} + 22391131 p^{3} T^{12} + 115864 p^{4} T^{13} + 128739 p^{5} T^{14} - 51 p^{6} T^{15} + 505 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−2.69781812663843756237497738903, −2.68834704815619502466939971392, −2.63320329502732504302736872953, −2.63074083589357490517897669427, −2.61544995376478860245528665299, −2.58890976283764601774334717698, −2.50518601167129646199916838300, −2.20886128372682987377570286221, −2.09900063971503595807512328178, −2.03550098822337700211678181002, −2.01843488123534985120075748750, −1.97539117791992037674295431953, −1.97434061277933007173589502984, −1.89468280470516035912561077627, −1.61049068424112989083133459184, −1.48022394041595948593135674714, −1.04841440098051811227554984541, −0.921518304256721106596330092222, −0.68540187005075107433914056707, −0.68214287450473700532246326923, −0.67494517425230898193510373119, −0.58815486111319540821697317269, −0.46132554199478947113339310675, −0.37855907474488165649068681878, −0.37302375855318762267446774610, 0.37302375855318762267446774610, 0.37855907474488165649068681878, 0.46132554199478947113339310675, 0.58815486111319540821697317269, 0.67494517425230898193510373119, 0.68214287450473700532246326923, 0.68540187005075107433914056707, 0.921518304256721106596330092222, 1.04841440098051811227554984541, 1.48022394041595948593135674714, 1.61049068424112989083133459184, 1.89468280470516035912561077627, 1.97434061277933007173589502984, 1.97539117791992037674295431953, 2.01843488123534985120075748750, 2.03550098822337700211678181002, 2.09900063971503595807512328178, 2.20886128372682987377570286221, 2.50518601167129646199916838300, 2.58890976283764601774334717698, 2.61544995376478860245528665299, 2.63074083589357490517897669427, 2.63320329502732504302736872953, 2.68834704815619502466939971392, 2.69781812663843756237497738903
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.