Dirichlet series
| L(s) = 1 | − 2-s − 7·3-s − 5·4-s − 2·5-s + 7·6-s − 2·7-s + 6·8-s + 15·9-s + 2·10-s − 13·11-s + 35·12-s + 7·13-s + 2·14-s + 14·15-s + 8·16-s − 10·17-s − 15·18-s − 2·19-s + 10·20-s + 14·21-s + 13·22-s − 24·23-s − 42·24-s − 12·25-s − 7·26-s + 12·27-s + 10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4.04·3-s − 5/2·4-s − 0.894·5-s + 2.85·6-s − 0.755·7-s + 2.12·8-s + 5·9-s + 0.632·10-s − 3.91·11-s + 10.1·12-s + 1.94·13-s + 0.534·14-s + 3.61·15-s + 2·16-s − 2.42·17-s − 3.53·18-s − 0.458·19-s + 2.23·20-s + 3.05·21-s + 2.77·22-s − 5.00·23-s − 8.57·24-s − 2.39·25-s − 1.37·26-s + 2.30·27-s + 1.88·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 37^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{7} \cdot 37^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(13^{7} \cdot 37^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(12329.7\) |
| Root analytic conductor: | \(1.95979\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 13^{7} \cdot 37^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 13 | \( ( 1 - T )^{7} \) |
| 37 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + 3 p T^{2} + 5 T^{3} + 21 T^{4} + p^{4} T^{5} + 53 T^{6} + 17 p T^{7} + 53 p T^{8} + p^{6} T^{9} + 21 p^{3} T^{10} + 5 p^{4} T^{11} + 3 p^{6} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + 7 T + 34 T^{2} + 121 T^{3} + 355 T^{4} + 875 T^{5} + 1864 T^{6} + 3451 T^{7} + 1864 p T^{8} + 875 p^{2} T^{9} + 355 p^{3} T^{10} + 121 p^{4} T^{11} + 34 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 2 T + 16 T^{2} + 22 T^{3} + 123 T^{4} + 37 p T^{5} + 33 p^{2} T^{6} + 1262 T^{7} + 33 p^{3} T^{8} + 37 p^{3} T^{9} + 123 p^{3} T^{10} + 22 p^{4} T^{11} + 16 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 2 T + 34 T^{2} + 73 T^{3} + 568 T^{4} + 1135 T^{5} + 122 p^{2} T^{6} + 10119 T^{7} + 122 p^{3} T^{8} + 1135 p^{2} T^{9} + 568 p^{3} T^{10} + 73 p^{4} T^{11} + 34 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 13 T + 117 T^{2} + 771 T^{3} + 4280 T^{4} + 19827 T^{5} + 80807 T^{6} + 284641 T^{7} + 80807 p T^{8} + 19827 p^{2} T^{9} + 4280 p^{3} T^{10} + 771 p^{4} T^{11} + 117 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 10 T + 100 T^{2} + 704 T^{3} + 4339 T^{4} + 22855 T^{5} + 110667 T^{6} + 468484 T^{7} + 110667 p T^{8} + 22855 p^{2} T^{9} + 4339 p^{3} T^{10} + 704 p^{4} T^{11} + 100 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 2 T + 48 T^{2} + 65 T^{3} + 998 T^{4} + 43 p T^{5} + 19292 T^{6} + 11680 T^{7} + 19292 p T^{8} + 43 p^{3} T^{9} + 998 p^{3} T^{10} + 65 p^{4} T^{11} + 48 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 24 T + 363 T^{2} + 3854 T^{3} + 32657 T^{4} + 226567 T^{5} + 1347956 T^{6} + 6905418 T^{7} + 1347956 p T^{8} + 226567 p^{2} T^{9} + 32657 p^{3} T^{10} + 3854 p^{4} T^{11} + 363 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 17 T + 253 T^{2} + 2656 T^{3} + 24177 T^{4} + 181967 T^{5} + 1209330 T^{6} + 6915074 T^{7} + 1209330 p T^{8} + 181967 p^{2} T^{9} + 24177 p^{3} T^{10} + 2656 p^{4} T^{11} + 253 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 11 T + 142 T^{2} + 958 T^{3} + 7453 T^{4} + 35035 T^{5} + 7307 p T^{6} + 956874 T^{7} + 7307 p^{2} T^{8} + 35035 p^{2} T^{9} + 7453 p^{3} T^{10} + 958 p^{4} T^{11} + 142 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 17 T + 257 T^{2} + 2268 T^{3} + 19451 T^{4} + 122334 T^{5} + 863557 T^{6} + 5008487 T^{7} + 863557 p T^{8} + 122334 p^{2} T^{9} + 19451 p^{3} T^{10} + 2268 p^{4} T^{11} + 257 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 3 T + 67 T^{2} - 84 T^{3} + 3983 T^{4} + 1809 T^{5} + 241186 T^{6} + 1010 T^{7} + 241186 p T^{8} + 1809 p^{2} T^{9} + 3983 p^{3} T^{10} - 84 p^{4} T^{11} + 67 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 21 T + 394 T^{2} + 4137 T^{3} + 40498 T^{4} + 260301 T^{5} + 1861770 T^{6} + 10356195 T^{7} + 1861770 p T^{8} + 260301 p^{2} T^{9} + 40498 p^{3} T^{10} + 4137 p^{4} T^{11} + 394 p^{5} T^{12} + 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 16 T + 252 T^{2} + 2751 T^{3} + 30878 T^{4} + 264423 T^{5} + 2331890 T^{6} + 16858495 T^{7} + 2331890 p T^{8} + 264423 p^{2} T^{9} + 30878 p^{3} T^{10} + 2751 p^{4} T^{11} + 252 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 18 T + 330 T^{2} + 3588 T^{3} + 43899 T^{4} + 382467 T^{5} + 3673733 T^{6} + 26502258 T^{7} + 3673733 p T^{8} + 382467 p^{2} T^{9} + 43899 p^{3} T^{10} + 3588 p^{4} T^{11} + 330 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 20 T + 335 T^{2} + 3204 T^{3} + 26819 T^{4} + 110665 T^{5} + 380210 T^{6} - 1267472 T^{7} + 380210 p T^{8} + 110665 p^{2} T^{9} + 26819 p^{3} T^{10} + 3204 p^{4} T^{11} + 335 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 7 T + 310 T^{2} - 1658 T^{3} + 46408 T^{4} - 3122 p T^{5} + 4535576 T^{6} - 17300046 T^{7} + 4535576 p T^{8} - 3122 p^{3} T^{9} + 46408 p^{3} T^{10} - 1658 p^{4} T^{11} + 310 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 35 T + 789 T^{2} + 12065 T^{3} + 151898 T^{4} + 1586239 T^{5} + 15235229 T^{6} + 131541155 T^{7} + 15235229 p T^{8} + 1586239 p^{2} T^{9} + 151898 p^{3} T^{10} + 12065 p^{4} T^{11} + 789 p^{5} T^{12} + 35 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 8 T + 400 T^{2} - 2997 T^{3} + 75506 T^{4} - 490305 T^{5} + 8580796 T^{6} - 45886359 T^{7} + 8580796 p T^{8} - 490305 p^{2} T^{9} + 75506 p^{3} T^{10} - 2997 p^{4} T^{11} + 400 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 18 T + 400 T^{2} + 4755 T^{3} + 69218 T^{4} + 670773 T^{5} + 7626730 T^{6} + 62053274 T^{7} + 7626730 p T^{8} + 670773 p^{2} T^{9} + 69218 p^{3} T^{10} + 4755 p^{4} T^{11} + 400 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 19 T + 413 T^{2} - 5540 T^{3} + 83434 T^{4} - 896074 T^{5} + 10322407 T^{6} - 90798361 T^{7} + 10322407 p T^{8} - 896074 p^{2} T^{9} + 83434 p^{3} T^{10} - 5540 p^{4} T^{11} + 413 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + T + 108 T^{2} - 8 T^{3} + 17873 T^{4} + 44447 T^{5} + 1792625 T^{6} + 1317142 T^{7} + 1792625 p T^{8} + 44447 p^{2} T^{9} + 17873 p^{3} T^{10} - 8 p^{4} T^{11} + 108 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 22 T + 681 T^{2} - 11326 T^{3} + 195623 T^{4} - 2537231 T^{5} + 31274594 T^{6} - 319248170 T^{7} + 31274594 p T^{8} - 2537231 p^{2} T^{9} + 195623 p^{3} T^{10} - 11326 p^{4} T^{11} + 681 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.88540634948445483922441218830, −5.68981471147806808962460823908, −5.62924534751133921613187360978, −5.41055942900877277645987240947, −5.26403911740962584781373260391, −5.10892922961568186972000846682, −4.94433245276165536832342562350, −4.82386188786775588368711220303, −4.77528091603297390046934022575, −4.62002101350231475005048858294, −4.51706859831967813581224705389, −4.17121161941006069488875395066, −4.07313726748432627588070699999, −3.75332709502046677067680917154, −3.71539155039070121787047223527, −3.66438443978671055409262331765, −3.28521910507751493236487106106, −3.13817899277373725112991748332, −2.97705006164817882239841190746, −2.80572371471283034678120927299, −2.18031748816396507603960631286, −2.04615716078789646422591795472, −1.87207431198315367587987011933, −1.70880693694479774587298214227, −1.56970140513842038242971126360, 0, 0, 0, 0, 0, 0, 0, 1.56970140513842038242971126360, 1.70880693694479774587298214227, 1.87207431198315367587987011933, 2.04615716078789646422591795472, 2.18031748816396507603960631286, 2.80572371471283034678120927299, 2.97705006164817882239841190746, 3.13817899277373725112991748332, 3.28521910507751493236487106106, 3.66438443978671055409262331765, 3.71539155039070121787047223527, 3.75332709502046677067680917154, 4.07313726748432627588070699999, 4.17121161941006069488875395066, 4.51706859831967813581224705389, 4.62002101350231475005048858294, 4.77528091603297390046934022575, 4.82386188786775588368711220303, 4.94433245276165536832342562350, 5.10892922961568186972000846682, 5.26403911740962584781373260391, 5.41055942900877277645987240947, 5.62924534751133921613187360978, 5.68981471147806808962460823908, 5.88540634948445483922441218830
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.