Dirichlet series
| L(s) = 1 | + 2-s − 9·3-s − 6·4-s − 9·5-s − 9·6-s − 9·7-s − 5·8-s + 45·9-s − 9·10-s + 11-s + 54·12-s − 9·14-s + 81·15-s + 18·16-s + 11·17-s + 45·18-s − 7·19-s + 54·20-s + 81·21-s + 22-s + 22·23-s + 45·24-s + 14·25-s − 165·27-s + 54·28-s + 11·29-s + 81·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5.19·3-s − 3·4-s − 4.02·5-s − 3.67·6-s − 3.40·7-s − 1.76·8-s + 15·9-s − 2.84·10-s + 0.301·11-s + 15.5·12-s − 2.40·14-s + 20.9·15-s + 9/2·16-s + 2.66·17-s + 10.6·18-s − 1.60·19-s + 12.0·20-s + 17.6·21-s + 0.213·22-s + 4.58·23-s + 9.18·24-s + 14/5·25-s − 31.7·27-s + 10.2·28-s + 2.04·29-s + 14.7·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 7^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 7^{9} \cdot 13^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(18\) |
| Conductor: | \(3^{9} \cdot 7^{9} \cdot 13^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.17882\times 10^{13}\) |
| Root analytic conductor: | \(5.32343\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((18,\ 3^{9} \cdot 7^{9} \cdot 13^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.1429436145\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1429436145\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + T )^{9} \) |
| 7 | \( ( 1 + T )^{9} \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - T + 7 T^{2} - p^{3} T^{3} + 27 T^{4} - 17 p T^{5} + 77 T^{6} - 25 p^{2} T^{7} + 11 p^{4} T^{8} - 225 T^{9} + 11 p^{5} T^{10} - 25 p^{4} T^{11} + 77 p^{3} T^{12} - 17 p^{5} T^{13} + 27 p^{5} T^{14} - p^{9} T^{15} + 7 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) |
| 5 | \( 1 + 9 T + 67 T^{2} + 68 p T^{3} + 1529 T^{4} + 5617 T^{5} + 18774 T^{6} + 10819 p T^{7} + 143354 T^{8} + 333681 T^{9} + 143354 p T^{10} + 10819 p^{3} T^{11} + 18774 p^{3} T^{12} + 5617 p^{4} T^{13} + 1529 p^{5} T^{14} + 68 p^{7} T^{15} + 67 p^{7} T^{16} + 9 p^{8} T^{17} + p^{9} T^{18} \) | |
| 11 | \( 1 - T + 62 T^{2} - 98 T^{3} + 1855 T^{4} - 3833 T^{5} + 36076 T^{6} - 83423 T^{7} + 514622 T^{8} - 1139285 T^{9} + 514622 p T^{10} - 83423 p^{2} T^{11} + 36076 p^{3} T^{12} - 3833 p^{4} T^{13} + 1855 p^{5} T^{14} - 98 p^{6} T^{15} + 62 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 17 | \( 1 - 11 T + 140 T^{2} - 1070 T^{3} + 8375 T^{4} - 50149 T^{5} + 299946 T^{6} - 1482815 T^{7} + 7254666 T^{8} - 30108579 T^{9} + 7254666 p T^{10} - 1482815 p^{2} T^{11} + 299946 p^{3} T^{12} - 50149 p^{4} T^{13} + 8375 p^{5} T^{14} - 1070 p^{6} T^{15} + 140 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 19 | \( 1 + 7 T + 116 T^{2} + 876 T^{3} + 7093 T^{4} + 47955 T^{5} + 289544 T^{6} + 82937 p T^{7} + 8090752 T^{8} + 35451339 T^{9} + 8090752 p T^{10} + 82937 p^{3} T^{11} + 289544 p^{3} T^{12} + 47955 p^{4} T^{13} + 7093 p^{5} T^{14} + 876 p^{6} T^{15} + 116 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 23 | \( 1 - 22 T + 349 T^{2} - 3993 T^{3} + 38395 T^{4} - 310121 T^{5} + 2209850 T^{6} - 13856063 T^{7} + 78154952 T^{8} - 394125399 T^{9} + 78154952 p T^{10} - 13856063 p^{2} T^{11} + 2209850 p^{3} T^{12} - 310121 p^{4} T^{13} + 38395 p^{5} T^{14} - 3993 p^{6} T^{15} + 349 p^{7} T^{16} - 22 p^{8} T^{17} + p^{9} T^{18} \) | |
| 29 | \( 1 - 11 T + 7 p T^{2} - 1277 T^{3} + 12846 T^{4} - 33910 T^{5} + 237194 T^{6} + 1376549 T^{7} - 5088840 T^{8} + 90178562 T^{9} - 5088840 p T^{10} + 1376549 p^{2} T^{11} + 237194 p^{3} T^{12} - 33910 p^{4} T^{13} + 12846 p^{5} T^{14} - 1277 p^{6} T^{15} + 7 p^{8} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 31 | \( 1 + 7 T + 180 T^{2} + 990 T^{3} + 14805 T^{4} + 68589 T^{5} + 788636 T^{6} + 3245229 T^{7} + 31413468 T^{8} + 115441497 T^{9} + 31413468 p T^{10} + 3245229 p^{2} T^{11} + 788636 p^{3} T^{12} + 68589 p^{4} T^{13} + 14805 p^{5} T^{14} + 990 p^{6} T^{15} + 180 p^{7} T^{16} + 7 p^{8} T^{17} + p^{9} T^{18} \) | |
| 37 | \( 1 - T + 139 T^{2} - 120 T^{3} + 7239 T^{4} + 119 p T^{5} + 3096 p T^{6} + 1276235 T^{7} - 4229252 T^{8} + 74114065 T^{9} - 4229252 p T^{10} + 1276235 p^{2} T^{11} + 3096 p^{4} T^{12} + 119 p^{5} T^{13} + 7239 p^{5} T^{14} - 120 p^{6} T^{15} + 139 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} \) | |
| 41 | \( 1 + 16 T + 367 T^{2} + 4213 T^{3} + 56801 T^{4} + 513619 T^{5} + 5143454 T^{6} + 38182969 T^{7} + 306801982 T^{8} + 1894474467 T^{9} + 306801982 p T^{10} + 38182969 p^{2} T^{11} + 5143454 p^{3} T^{12} + 513619 p^{4} T^{13} + 56801 p^{5} T^{14} + 4213 p^{6} T^{15} + 367 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18} \) | |
| 43 | \( 1 - 32 T + 671 T^{2} - 9760 T^{3} + 113888 T^{4} - 1079744 T^{5} + 8872696 T^{6} - 64272096 T^{7} + 438008954 T^{8} - 2871977024 T^{9} + 438008954 p T^{10} - 64272096 p^{2} T^{11} + 8872696 p^{3} T^{12} - 1079744 p^{4} T^{13} + 113888 p^{5} T^{14} - 9760 p^{6} T^{15} + 671 p^{7} T^{16} - 32 p^{8} T^{17} + p^{9} T^{18} \) | |
| 47 | \( 1 - 12 T + 214 T^{2} - 1749 T^{3} + 17443 T^{4} - 94618 T^{5} + 571839 T^{6} - 888331 T^{7} - 1187615 T^{8} + 79790732 T^{9} - 1187615 p T^{10} - 888331 p^{2} T^{11} + 571839 p^{3} T^{12} - 94618 p^{4} T^{13} + 17443 p^{5} T^{14} - 1749 p^{6} T^{15} + 214 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 53 | \( 1 - 13 T + 404 T^{2} - 4099 T^{3} + 71825 T^{4} - 11262 p T^{5} + 144539 p T^{6} - 53721461 T^{7} + 559046783 T^{8} - 3356965562 T^{9} + 559046783 p T^{10} - 53721461 p^{2} T^{11} + 144539 p^{4} T^{12} - 11262 p^{5} T^{13} + 71825 p^{5} T^{14} - 4099 p^{6} T^{15} + 404 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18} \) | |
| 59 | \( 1 + 29 T + 694 T^{2} + 11211 T^{3} + 159983 T^{4} + 1870322 T^{5} + 340781 p T^{6} + 189194029 T^{7} + 1670101617 T^{8} + 13209434082 T^{9} + 1670101617 p T^{10} + 189194029 p^{2} T^{11} + 340781 p^{4} T^{12} + 1870322 p^{4} T^{13} + 159983 p^{5} T^{14} + 11211 p^{6} T^{15} + 694 p^{7} T^{16} + 29 p^{8} T^{17} + p^{9} T^{18} \) | |
| 61 | \( 1 + 12 T + 202 T^{2} + 2217 T^{3} + 29467 T^{4} + 254058 T^{5} + 2662757 T^{6} + 22016239 T^{7} + 201270905 T^{8} + 1435693300 T^{9} + 201270905 p T^{10} + 22016239 p^{2} T^{11} + 2662757 p^{3} T^{12} + 254058 p^{4} T^{13} + 29467 p^{5} T^{14} + 2217 p^{6} T^{15} + 202 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18} \) | |
| 67 | \( 1 - 20 T + 490 T^{2} - 6843 T^{3} + 104799 T^{4} - 1168054 T^{5} + 13834799 T^{6} - 129550685 T^{7} + 1274059293 T^{8} - 10196246828 T^{9} + 1274059293 p T^{10} - 129550685 p^{2} T^{11} + 13834799 p^{3} T^{12} - 1168054 p^{4} T^{13} + 104799 p^{5} T^{14} - 6843 p^{6} T^{15} + 490 p^{7} T^{16} - 20 p^{8} T^{17} + p^{9} T^{18} \) | |
| 71 | \( 1 - 2 T + 257 T^{2} - 1135 T^{3} + 37661 T^{4} - 206923 T^{5} + 3993156 T^{6} - 23957569 T^{7} + 341807796 T^{8} - 1945647231 T^{9} + 341807796 p T^{10} - 23957569 p^{2} T^{11} + 3993156 p^{3} T^{12} - 206923 p^{4} T^{13} + 37661 p^{5} T^{14} - 1135 p^{6} T^{15} + 257 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18} \) | |
| 73 | \( 1 + T + 251 T^{2} + 273 T^{3} + 39710 T^{4} + 44458 T^{5} + 4512122 T^{6} + 3924751 T^{7} + 408933204 T^{8} + 329944074 T^{9} + 408933204 p T^{10} + 3924751 p^{2} T^{11} + 4512122 p^{3} T^{12} + 44458 p^{4} T^{13} + 39710 p^{5} T^{14} + 273 p^{6} T^{15} + 251 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18} \) | |
| 79 | \( 1 - 3 T + 351 T^{2} - 9 p T^{3} + 68936 T^{4} - 118814 T^{5} + 9492776 T^{6} - 14212553 T^{7} + 973700846 T^{8} - 1259632414 T^{9} + 973700846 p T^{10} - 14212553 p^{2} T^{11} + 9492776 p^{3} T^{12} - 118814 p^{4} T^{13} + 68936 p^{5} T^{14} - 9 p^{7} T^{15} + 351 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18} \) | |
| 83 | \( 1 + 24 T + 612 T^{2} + 11051 T^{3} + 178697 T^{4} + 2479414 T^{5} + 31603753 T^{6} + 355979085 T^{7} + 3737596007 T^{8} + 35290745636 T^{9} + 3737596007 p T^{10} + 355979085 p^{2} T^{11} + 31603753 p^{3} T^{12} + 2479414 p^{4} T^{13} + 178697 p^{5} T^{14} + 11051 p^{6} T^{15} + 612 p^{7} T^{16} + 24 p^{8} T^{17} + p^{9} T^{18} \) | |
| 89 | \( 1 + 11 T + 463 T^{2} + 50 p T^{3} + 92803 T^{4} + 814799 T^{5} + 10879056 T^{6} + 94642331 T^{7} + 947631952 T^{8} + 8831887241 T^{9} + 947631952 p T^{10} + 94642331 p^{2} T^{11} + 10879056 p^{3} T^{12} + 814799 p^{4} T^{13} + 92803 p^{5} T^{14} + 50 p^{7} T^{15} + 463 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18} \) | |
| 97 | \( 1 + 20 T + 676 T^{2} + 9305 T^{3} + 157949 T^{4} + 1512578 T^{5} + 14867943 T^{6} + 91386519 T^{7} + 418508311 T^{8} + 2709196404 T^{9} + 418508311 p T^{10} + 91386519 p^{2} T^{11} + 14867943 p^{3} T^{12} + 1512578 p^{4} T^{13} + 157949 p^{5} T^{14} + 9305 p^{6} T^{15} + 676 p^{7} T^{16} + 20 p^{8} T^{17} + p^{9} T^{18} \) | |
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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
−3.27804776965463905262742928138, −3.17833150446390415721565625233, −3.17294307055690495076610215020, −3.14473230843182922620395825715, −3.10456887296203496877972712263, −2.84495953247700579946033400634, −2.80008185553775102516968277993, −2.60630318261803059938219132984, −2.35226775679900831600469597089, −2.29178855665367447169691805832, −1.97912583170320817993923651398, −1.88658718603194916639427147836, −1.67026932771570483956781515649, −1.65366928945394174546471427719, −1.48337139677062981704276590571, −1.47313895437470080101977913380, −1.04717610594165875993501620301, −0.76353995893802135008619710114, −0.74123019530914175773919584634, −0.66485514074211196962407266911, −0.58402048596228891002918675683, −0.56994860030203061491952043006, −0.44619932091872179528723609212, −0.44026045152808842644955927199, −0.17379618286753217049952657468, 0.17379618286753217049952657468, 0.44026045152808842644955927199, 0.44619932091872179528723609212, 0.56994860030203061491952043006, 0.58402048596228891002918675683, 0.66485514074211196962407266911, 0.74123019530914175773919584634, 0.76353995893802135008619710114, 1.04717610594165875993501620301, 1.47313895437470080101977913380, 1.48337139677062981704276590571, 1.65366928945394174546471427719, 1.67026932771570483956781515649, 1.88658718603194916639427147836, 1.97912583170320817993923651398, 2.29178855665367447169691805832, 2.35226775679900831600469597089, 2.60630318261803059938219132984, 2.80008185553775102516968277993, 2.84495953247700579946033400634, 3.10456887296203496877972712263, 3.14473230843182922620395825715, 3.17294307055690495076610215020, 3.17833150446390415721565625233, 3.27804776965463905262742928138
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.