Dirichlet series
| L(s) = 1 | + 2·2-s + 4·3-s + 2·4-s − 7·5-s + 8·6-s + 10·7-s + 3·8-s + 3·9-s − 14·10-s + 6·11-s + 8·12-s + 4·13-s + 20·14-s − 28·15-s + 2·16-s + 17·17-s + 6·18-s + 3·19-s − 14·20-s + 40·21-s + 12·22-s + 5·23-s + 12·24-s + 28·25-s + 8·26-s − 14·27-s + 20·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 4-s − 3.13·5-s + 3.26·6-s + 3.77·7-s + 1.06·8-s + 9-s − 4.42·10-s + 1.80·11-s + 2.30·12-s + 1.10·13-s + 5.34·14-s − 7.22·15-s + 1/2·16-s + 4.12·17-s + 1.41·18-s + 0.688·19-s − 3.13·20-s + 8.72·21-s + 2.55·22-s + 1.04·23-s + 2.44·24-s + 28/5·25-s + 1.56·26-s − 2.69·27-s + 3.77·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 67^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(980.061\) |
| Root analytic conductor: | \(1.63553\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{7} \cdot 67^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.64194986\) |
| \(L(\frac12)\) | \(\approx\) | \(23.64194986\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{7} \) |
| 67 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 - p T + p T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + 13 T^{6} - 15 p T^{7} + 13 p T^{8} - p^{4} T^{9} + 3 p^{4} T^{10} - 3 p^{4} T^{11} + p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 4 T + 13 T^{2} - 26 T^{3} + 14 p T^{4} - 8 p T^{5} - 19 T^{6} + 110 T^{7} - 19 p T^{8} - 8 p^{3} T^{9} + 14 p^{4} T^{10} - 26 p^{4} T^{11} + 13 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 10 T + 61 T^{2} - 244 T^{3} + 692 T^{4} - 1286 T^{5} + 1315 T^{6} - 818 T^{7} + 1315 p T^{8} - 1286 p^{2} T^{9} + 692 p^{3} T^{10} - 244 p^{4} T^{11} + 61 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 6 T + 45 T^{2} - 212 T^{3} + 1021 T^{4} - 382 p T^{5} + 15209 T^{6} - 54040 T^{7} + 15209 p T^{8} - 382 p^{3} T^{9} + 1021 p^{3} T^{10} - 212 p^{4} T^{11} + 45 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 4 T + 46 T^{2} - 178 T^{3} + 1263 T^{4} - 4188 T^{5} + 22931 T^{6} - 67078 T^{7} + 22931 p T^{8} - 4188 p^{2} T^{9} + 1263 p^{3} T^{10} - 178 p^{4} T^{11} + 46 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - p T + 197 T^{2} - 1646 T^{3} + 11467 T^{4} - 66639 T^{5} + 338863 T^{6} - 1487332 T^{7} + 338863 p T^{8} - 66639 p^{2} T^{9} + 11467 p^{3} T^{10} - 1646 p^{4} T^{11} + 197 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 3 T + 70 T^{2} - 169 T^{3} + 2727 T^{4} - 5946 T^{5} + 71873 T^{6} - 132741 T^{7} + 71873 p T^{8} - 5946 p^{2} T^{9} + 2727 p^{3} T^{10} - 169 p^{4} T^{11} + 70 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 5 T + 101 T^{2} - 302 T^{3} + 4593 T^{4} - 10491 T^{5} + 148477 T^{6} - 292996 T^{7} + 148477 p T^{8} - 10491 p^{2} T^{9} + 4593 p^{3} T^{10} - 302 p^{4} T^{11} + 101 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + T + 77 T^{2} + 22 T^{3} + 3426 T^{4} - 2082 T^{5} + 114307 T^{6} - 141241 T^{7} + 114307 p T^{8} - 2082 p^{2} T^{9} + 3426 p^{3} T^{10} + 22 p^{4} T^{11} + 77 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 33 T^{2} + 40 T^{3} + 1501 T^{4} + 3904 T^{5} + 38973 T^{6} + 198832 T^{7} + 38973 p T^{8} + 3904 p^{2} T^{9} + 1501 p^{3} T^{10} + 40 p^{4} T^{11} + 33 p^{5} T^{12} + p^{7} T^{14} \) | |
| 37 | \( 1 - 15 T + 151 T^{2} - 1034 T^{3} + 7321 T^{4} - 47449 T^{5} + 324199 T^{6} - 1904108 T^{7} + 324199 p T^{8} - 47449 p^{2} T^{9} + 7321 p^{3} T^{10} - 1034 p^{4} T^{11} + 151 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 20 T + 211 T^{2} + 1608 T^{3} + 10057 T^{4} + 44908 T^{5} + 137811 T^{6} + 451824 T^{7} + 137811 p T^{8} + 44908 p^{2} T^{9} + 10057 p^{3} T^{10} + 1608 p^{4} T^{11} + 211 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 20 T + 335 T^{2} - 3678 T^{3} + 37846 T^{4} - 311362 T^{5} + 2475607 T^{6} - 16483114 T^{7} + 2475607 p T^{8} - 311362 p^{2} T^{9} + 37846 p^{3} T^{10} - 3678 p^{4} T^{11} + 335 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 19 T + 305 T^{2} - 3046 T^{3} + 30085 T^{4} - 243069 T^{5} + 2048533 T^{6} - 14193428 T^{7} + 2048533 p T^{8} - 243069 p^{2} T^{9} + 30085 p^{3} T^{10} - 3046 p^{4} T^{11} + 305 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 18 T + 363 T^{2} - 82 p T^{3} + 53700 T^{4} - 489044 T^{5} + 4523561 T^{6} - 32799886 T^{7} + 4523561 p T^{8} - 489044 p^{2} T^{9} + 53700 p^{3} T^{10} - 82 p^{5} T^{11} + 363 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 39 T + 999 T^{2} + 17908 T^{3} + 257608 T^{4} + 2999992 T^{5} + 29574113 T^{6} + 245281391 T^{7} + 29574113 p T^{8} + 2999992 p^{2} T^{9} + 257608 p^{3} T^{10} + 17908 p^{4} T^{11} + 999 p^{5} T^{12} + 39 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 6 T + 231 T^{2} + 1996 T^{3} + 30121 T^{4} + 265034 T^{5} + 2622263 T^{6} + 20730664 T^{7} + 2622263 p T^{8} + 265034 p^{2} T^{9} + 30121 p^{3} T^{10} + 1996 p^{4} T^{11} + 231 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 28 T + 766 T^{2} + 13072 T^{3} + 203615 T^{4} + 2444032 T^{5} + 26568885 T^{6} + 235303704 T^{7} + 26568885 p T^{8} + 2444032 p^{2} T^{9} + 203615 p^{3} T^{10} + 13072 p^{4} T^{11} + 766 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 41 T + 1003 T^{2} - 17142 T^{3} + 231601 T^{4} - 2586535 T^{5} + 25415699 T^{6} - 225623764 T^{7} + 25415699 p T^{8} - 2586535 p^{2} T^{9} + 231601 p^{3} T^{10} - 17142 p^{4} T^{11} + 1003 p^{5} T^{12} - 41 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 24 T + 457 T^{2} - 6184 T^{3} + 65397 T^{4} - 581256 T^{5} + 4757453 T^{6} - 38403696 T^{7} + 4757453 p T^{8} - 581256 p^{2} T^{9} + 65397 p^{3} T^{10} - 6184 p^{4} T^{11} + 457 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 38 T + 1061 T^{2} + 20684 T^{3} + 334813 T^{4} + 4430346 T^{5} + 50689825 T^{6} + 494006632 T^{7} + 50689825 p T^{8} + 4430346 p^{2} T^{9} + 334813 p^{3} T^{10} + 20684 p^{4} T^{11} + 1061 p^{5} T^{12} + 38 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + T + 5 p T^{2} + 442 T^{3} + 95718 T^{4} + 83766 T^{5} + 12760587 T^{6} + 9374795 T^{7} + 12760587 p T^{8} + 83766 p^{2} T^{9} + 95718 p^{3} T^{10} + 442 p^{4} T^{11} + 5 p^{6} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 18 T + 541 T^{2} - 8044 T^{3} + 137990 T^{4} - 1631768 T^{5} + 21022429 T^{6} - 198030280 T^{7} + 21022429 p T^{8} - 1631768 p^{2} T^{9} + 137990 p^{3} T^{10} - 8044 p^{4} T^{11} + 541 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.57871209019807585250506976150, −5.44104197576927158710792316646, −5.29977139370722696886793378437, −5.08040223269850065533241878100, −4.88939311799749259588592804478, −4.66963275533064205327253982322, −4.62790113088339858953041423855, −4.38562784686660066856829111019, −4.27628791201105322817175026394, −4.02445725992885264689706333540, −3.89529640294822208219462649528, −3.87884329127671543097478863850, −3.55836309525677952470943775696, −3.50438758043864314151962253901, −3.31945173423218320346344630761, −3.24157813209017349306842738473, −2.75351967876922352224941175985, −2.72333974671439839090794778912, −2.53646885910010873654397082697, −2.44972410678649735369384158700, −1.62432263996520490268335650298, −1.52756182898610704783046628135, −1.45367806651969974526685479497, −1.07513622420056400448697184310, −1.01120488902977534519832372397, 1.01120488902977534519832372397, 1.07513622420056400448697184310, 1.45367806651969974526685479497, 1.52756182898610704783046628135, 1.62432263996520490268335650298, 2.44972410678649735369384158700, 2.53646885910010873654397082697, 2.72333974671439839090794778912, 2.75351967876922352224941175985, 3.24157813209017349306842738473, 3.31945173423218320346344630761, 3.50438758043864314151962253901, 3.55836309525677952470943775696, 3.87884329127671543097478863850, 3.89529640294822208219462649528, 4.02445725992885264689706333540, 4.27628791201105322817175026394, 4.38562784686660066856829111019, 4.62790113088339858953041423855, 4.66963275533064205327253982322, 4.88939311799749259588592804478, 5.08040223269850065533241878100, 5.29977139370722696886793378437, 5.44104197576927158710792316646, 5.57871209019807585250506976150
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.