Dirichlet series
| L(s) = 1 | − 3·2-s + 7·3-s + 2·4-s − 6·5-s − 21·6-s − 6·7-s + 28·9-s + 18·10-s − 7·11-s + 14·12-s + 18·14-s − 42·15-s + 4·16-s − 2·17-s − 84·18-s − 8·19-s − 12·20-s − 42·21-s + 21·22-s + 4·23-s + 7·25-s + 84·27-s − 12·28-s − 12·29-s + 126·30-s + 10·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 4.04·3-s + 4-s − 2.68·5-s − 8.57·6-s − 2.26·7-s + 28/3·9-s + 5.69·10-s − 2.11·11-s + 4.04·12-s + 4.81·14-s − 10.8·15-s + 16-s − 0.485·17-s − 19.7·18-s − 1.83·19-s − 2.68·20-s − 9.16·21-s + 4.47·22-s + 0.834·23-s + 7/5·25-s + 16.1·27-s − 2.26·28-s − 2.22·29-s + 23.0·30-s + 1.79·31-s − 1.06·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 11^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 11^{7} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 11^{7} \cdot 13^{14}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(3.47331\times 10^{11}\) |
| Root analytic conductor: | \(6.67327\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 11^{7} \cdot 13^{14} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 3 | \( ( 1 - T )^{7} \) |
| 11 | \( ( 1 + T )^{7} \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + 3 T + 7 T^{2} + 15 T^{3} + 27 T^{4} + 45 T^{5} + 71 T^{6} + 101 T^{7} + 71 p T^{8} + 45 p^{2} T^{9} + 27 p^{3} T^{10} + 15 p^{4} T^{11} + 7 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 6 T + 29 T^{2} + 106 T^{3} + 343 T^{4} + 968 T^{5} + 2539 T^{6} + 5888 T^{7} + 2539 p T^{8} + 968 p^{2} T^{9} + 343 p^{3} T^{10} + 106 p^{4} T^{11} + 29 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 6 T + 31 T^{2} + 116 T^{3} + 383 T^{4} + 1126 T^{5} + 3009 T^{6} + 8016 T^{7} + 3009 p T^{8} + 1126 p^{2} T^{9} + 383 p^{3} T^{10} + 116 p^{4} T^{11} + 31 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 2 T + 69 T^{2} + 172 T^{3} + 2517 T^{4} + 6628 T^{5} + 60381 T^{6} + 144332 T^{7} + 60381 p T^{8} + 6628 p^{2} T^{9} + 2517 p^{3} T^{10} + 172 p^{4} T^{11} + 69 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 8 T + 111 T^{2} + 744 T^{3} + 5735 T^{4} + 31188 T^{5} + 174057 T^{6} + 757800 T^{7} + 174057 p T^{8} + 31188 p^{2} T^{9} + 5735 p^{3} T^{10} + 744 p^{4} T^{11} + 111 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 4 T + 85 T^{2} - 504 T^{3} + 3817 T^{4} - 25280 T^{5} + 123013 T^{6} - 728008 T^{7} + 123013 p T^{8} - 25280 p^{2} T^{9} + 3817 p^{3} T^{10} - 504 p^{4} T^{11} + 85 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 12 T + 205 T^{2} + 1730 T^{3} + 17161 T^{4} + 111022 T^{5} + 804745 T^{6} + 4114856 T^{7} + 804745 p T^{8} + 111022 p^{2} T^{9} + 17161 p^{3} T^{10} + 1730 p^{4} T^{11} + 205 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 10 T + 5 p T^{2} - 1266 T^{3} + 11031 T^{4} - 74908 T^{5} + 483933 T^{6} - 2807256 T^{7} + 483933 p T^{8} - 74908 p^{2} T^{9} + 11031 p^{3} T^{10} - 1266 p^{4} T^{11} + 5 p^{6} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 6 T + 123 T^{2} + 540 T^{3} + 8341 T^{4} + 33890 T^{5} + 429655 T^{6} + 1511352 T^{7} + 429655 p T^{8} + 33890 p^{2} T^{9} + 8341 p^{3} T^{10} + 540 p^{4} T^{11} + 123 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 2 T + 157 T^{2} + 312 T^{3} + 13071 T^{4} + 24834 T^{5} + 731507 T^{6} + 1223368 T^{7} + 731507 p T^{8} + 24834 p^{2} T^{9} + 13071 p^{3} T^{10} + 312 p^{4} T^{11} + 157 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 16 T + 211 T^{2} + 2288 T^{3} + 22441 T^{4} + 191142 T^{5} + 1465639 T^{6} + 9828708 T^{7} + 1465639 p T^{8} + 191142 p^{2} T^{9} + 22441 p^{3} T^{10} + 2288 p^{4} T^{11} + 211 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 18 T + 285 T^{2} + 2340 T^{3} + 15929 T^{4} + 31606 T^{5} - 173731 T^{6} - 3507160 T^{7} - 173731 p T^{8} + 31606 p^{2} T^{9} + 15929 p^{3} T^{10} + 2340 p^{4} T^{11} + 285 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 10 T + 283 T^{2} - 1932 T^{3} + 32981 T^{4} - 164822 T^{5} + 2330967 T^{6} - 9583240 T^{7} + 2330967 p T^{8} - 164822 p^{2} T^{9} + 32981 p^{3} T^{10} - 1932 p^{4} T^{11} + 283 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 2 T + 253 T^{2} - 12 T^{3} + 30397 T^{4} - 41266 T^{5} + 2394233 T^{6} - 4109736 T^{7} + 2394233 p T^{8} - 41266 p^{2} T^{9} + 30397 p^{3} T^{10} - 12 p^{4} T^{11} + 253 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 10 T + 439 T^{2} + 3544 T^{3} + 81553 T^{4} + 529902 T^{5} + 8345663 T^{6} + 42813248 T^{7} + 8345663 p T^{8} + 529902 p^{2} T^{9} + 81553 p^{3} T^{10} + 3544 p^{4} T^{11} + 439 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 8 T + 247 T^{2} + 1606 T^{3} + 34415 T^{4} + 2822 p T^{5} + 3206121 T^{6} + 15229160 T^{7} + 3206121 p T^{8} + 2822 p^{3} T^{9} + 34415 p^{3} T^{10} + 1606 p^{4} T^{11} + 247 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 36 T + 849 T^{2} + 14216 T^{3} + 197573 T^{4} + 2288676 T^{5} + 23379653 T^{6} + 208029408 T^{7} + 23379653 p T^{8} + 2288676 p^{2} T^{9} + 197573 p^{3} T^{10} + 14216 p^{4} T^{11} + 849 p^{5} T^{12} + 36 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 20 T + 397 T^{2} + 4216 T^{3} + 42675 T^{4} + 266264 T^{5} + 1819487 T^{6} + 10056872 T^{7} + 1819487 p T^{8} + 266264 p^{2} T^{9} + 42675 p^{3} T^{10} + 4216 p^{4} T^{11} + 397 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 6 T + 399 T^{2} - 2284 T^{3} + 76961 T^{4} - 389888 T^{5} + 9195259 T^{6} - 38954460 T^{7} + 9195259 p T^{8} - 389888 p^{2} T^{9} + 76961 p^{3} T^{10} - 2284 p^{4} T^{11} + 399 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 30 T + 813 T^{2} + 14236 T^{3} + 224949 T^{4} + 2797346 T^{5} + 31904401 T^{6} + 302487208 T^{7} + 31904401 p T^{8} + 2797346 p^{2} T^{9} + 224949 p^{3} T^{10} + 14236 p^{4} T^{11} + 813 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 34 T + 953 T^{2} + 18368 T^{3} + 304551 T^{4} + 4090348 T^{5} + 48525519 T^{6} + 485121860 T^{7} + 48525519 p T^{8} + 4090348 p^{2} T^{9} + 304551 p^{3} T^{10} + 18368 p^{4} T^{11} + 953 p^{5} T^{12} + 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 16 T + 467 T^{2} + 4096 T^{3} + 65633 T^{4} + 203976 T^{5} + 3647819 T^{6} - 8084944 T^{7} + 3647819 p T^{8} + 203976 p^{2} T^{9} + 65633 p^{3} T^{10} + 4096 p^{4} T^{11} + 467 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.04863030248324384072391837104, −4.02290741327592396094541255163, −3.90221985717832957547559929245, −3.55278166525376832067745283360, −3.40505675446766046204309156637, −3.36994325057554322143589480833, −3.31318899305983818402999414100, −3.24023713438942321790912897443, −3.18652240596425331539302509698, −3.00827588802802444326164293418, −2.85814861412478761648171002941, −2.84095805239973846171021404679, −2.75358532228487045328198558265, −2.54680799886424568042636875669, −2.52766947556852410395817094310, −2.25217355790317773516925233489, −2.05733419985698124414175194443, −2.00420351263148829246891068586, −1.84441975559563422547491155623, −1.60907330158909417244385847311, −1.49070717794361119798046986645, −1.37045985446278621408605905177, −1.32014839304408894187199495511, −0.988714589671065127402674126138, −0.827590777756021901164862561951, 0, 0, 0, 0, 0, 0, 0, 0.827590777756021901164862561951, 0.988714589671065127402674126138, 1.32014839304408894187199495511, 1.37045985446278621408605905177, 1.49070717794361119798046986645, 1.60907330158909417244385847311, 1.84441975559563422547491155623, 2.00420351263148829246891068586, 2.05733419985698124414175194443, 2.25217355790317773516925233489, 2.52766947556852410395817094310, 2.54680799886424568042636875669, 2.75358532228487045328198558265, 2.84095805239973846171021404679, 2.85814861412478761648171002941, 3.00827588802802444326164293418, 3.18652240596425331539302509698, 3.24023713438942321790912897443, 3.31318899305983818402999414100, 3.36994325057554322143589480833, 3.40505675446766046204309156637, 3.55278166525376832067745283360, 3.90221985717832957547559929245, 4.02290741327592396094541255163, 4.04863030248324384072391837104
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.