Dirichlet series
| L(s) = 1 | − 2-s + 7·3-s − 3·4-s − 5·5-s − 7·6-s − 5·7-s + 5·8-s + 28·9-s + 5·10-s − 12·11-s − 21·12-s − 13·13-s + 5·14-s − 35·15-s − 4·16-s − 12·17-s − 28·18-s − 5·19-s + 15·20-s − 35·21-s + 12·22-s − 7·23-s + 35·24-s − 7·25-s + 13·26-s + 84·27-s + 15·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4.04·3-s − 3/2·4-s − 2.23·5-s − 2.85·6-s − 1.88·7-s + 1.76·8-s + 28/3·9-s + 1.58·10-s − 3.61·11-s − 6.06·12-s − 3.60·13-s + 1.33·14-s − 9.03·15-s − 16-s − 2.91·17-s − 6.59·18-s − 1.14·19-s + 3.35·20-s − 7.63·21-s + 2.55·22-s − 1.45·23-s + 7.14·24-s − 7/5·25-s + 2.54·26-s + 16.1·27-s + 2.83·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 23^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 23^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 23^{7} \cdot 29^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.65870\times 10^{8}\) |
| Root analytic conductor: | \(3.99725\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 23^{7} \cdot 29^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
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 | 3 | \( ( 1 - T )^{7} \) |
| 23 | \( ( 1 + T )^{7} \) | |
| 29 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + p^{2} T^{2} + p T^{3} + 13 T^{4} + 9 T^{5} + 15 p T^{6} + 13 T^{7} + 15 p^{2} T^{8} + 9 p^{2} T^{9} + 13 p^{3} T^{10} + p^{5} T^{11} + p^{7} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + p T + 32 T^{2} + 22 p T^{3} + 87 p T^{4} + 1162 T^{5} + 3436 T^{6} + 7342 T^{7} + 3436 p T^{8} + 1162 p^{2} T^{9} + 87 p^{4} T^{10} + 22 p^{5} T^{11} + 32 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 5 T + 44 T^{2} + 172 T^{3} + 851 T^{4} + 2679 T^{5} + 9529 T^{6} + 24079 T^{7} + 9529 p T^{8} + 2679 p^{2} T^{9} + 851 p^{3} T^{10} + 172 p^{4} T^{11} + 44 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 12 T + 116 T^{2} + 783 T^{3} + 4536 T^{4} + 21382 T^{5} + 88915 T^{6} + 312885 T^{7} + 88915 p T^{8} + 21382 p^{2} T^{9} + 4536 p^{3} T^{10} + 783 p^{4} T^{11} + 116 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + p T + 118 T^{2} + 720 T^{3} + 3455 T^{4} + 13353 T^{5} + 45817 T^{6} + 159073 T^{7} + 45817 p T^{8} + 13353 p^{2} T^{9} + 3455 p^{3} T^{10} + 720 p^{4} T^{11} + 118 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 12 T + 117 T^{2} + 781 T^{3} + 4831 T^{4} + 25246 T^{5} + 7410 p T^{6} + 537609 T^{7} + 7410 p^{2} T^{8} + 25246 p^{2} T^{9} + 4831 p^{3} T^{10} + 781 p^{4} T^{11} + 117 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 5 T + 71 T^{2} + 264 T^{3} + 2600 T^{4} + 7812 T^{5} + 64394 T^{6} + 167594 T^{7} + 64394 p T^{8} + 7812 p^{2} T^{9} + 2600 p^{3} T^{10} + 264 p^{4} T^{11} + 71 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 8 T + 163 T^{2} + 973 T^{3} + 12089 T^{4} + 58707 T^{5} + 556093 T^{6} + 2238228 T^{7} + 556093 p T^{8} + 58707 p^{2} T^{9} + 12089 p^{3} T^{10} + 973 p^{4} T^{11} + 163 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 24 T + 405 T^{2} + 4845 T^{3} + 49327 T^{4} + 416707 T^{5} + 3121913 T^{6} + 20087224 T^{7} + 3121913 p T^{8} + 416707 p^{2} T^{9} + 49327 p^{3} T^{10} + 4845 p^{4} T^{11} + 405 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 9 T + 100 T^{2} + 232 T^{3} - 1699 T^{4} + 37702 T^{5} + 118878 T^{6} + 134 p T^{7} + 118878 p T^{8} + 37702 p^{2} T^{9} - 1699 p^{3} T^{10} + 232 p^{4} T^{11} + 100 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + T + 186 T^{2} - 122 T^{3} + 16897 T^{4} - 23864 T^{5} + 1024474 T^{6} - 1457402 T^{7} + 1024474 p T^{8} - 23864 p^{2} T^{9} + 16897 p^{3} T^{10} - 122 p^{4} T^{11} + 186 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 27 T + 555 T^{2} - 7940 T^{3} + 95954 T^{4} - 943823 T^{5} + 8122004 T^{6} - 59256915 T^{7} + 8122004 p T^{8} - 943823 p^{2} T^{9} + 95954 p^{3} T^{10} - 7940 p^{4} T^{11} + 555 p^{5} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + T + 266 T^{2} + 824 T^{3} + 30639 T^{4} + 148452 T^{5} + 2179032 T^{6} + 11405030 T^{7} + 2179032 p T^{8} + 148452 p^{2} T^{9} + 30639 p^{3} T^{10} + 824 p^{4} T^{11} + 266 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 8 T + 377 T^{2} - 2640 T^{3} + 62673 T^{4} - 373048 T^{5} + 5969217 T^{6} - 28925920 T^{7} + 5969217 p T^{8} - 373048 p^{2} T^{9} + 62673 p^{3} T^{10} - 2640 p^{4} T^{11} + 377 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - T + 293 T^{2} - 629 T^{3} + 40490 T^{4} - 112038 T^{5} + 3530870 T^{6} - 9478320 T^{7} + 3530870 p T^{8} - 112038 p^{2} T^{9} + 40490 p^{3} T^{10} - 629 p^{4} T^{11} + 293 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 16 T + 389 T^{2} + 3875 T^{3} + 53383 T^{4} + 375776 T^{5} + 4164638 T^{6} + 25313577 T^{7} + 4164638 p T^{8} + 375776 p^{2} T^{9} + 53383 p^{3} T^{10} + 3875 p^{4} T^{11} + 389 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 13 T + 521 T^{2} + 5337 T^{3} + 113902 T^{4} + 926278 T^{5} + 13634702 T^{6} + 87029512 T^{7} + 13634702 p T^{8} + 926278 p^{2} T^{9} + 113902 p^{3} T^{10} + 5337 p^{4} T^{11} + 521 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 23 T + 452 T^{2} + 6121 T^{3} + 70148 T^{4} + 672782 T^{5} + 6097489 T^{6} + 50887564 T^{7} + 6097489 p T^{8} + 672782 p^{2} T^{9} + 70148 p^{3} T^{10} + 6121 p^{4} T^{11} + 452 p^{5} T^{12} + 23 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 44 T + 1245 T^{2} + 25061 T^{3} + 405173 T^{4} + 5369531 T^{5} + 60465031 T^{6} + 579089508 T^{7} + 60465031 p T^{8} + 5369531 p^{2} T^{9} + 405173 p^{3} T^{10} + 25061 p^{4} T^{11} + 1245 p^{5} T^{12} + 44 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 21 T + 540 T^{2} - 8096 T^{3} + 119997 T^{4} - 1418842 T^{5} + 15465914 T^{6} - 148208926 T^{7} + 15465914 p T^{8} - 1418842 p^{2} T^{9} + 119997 p^{3} T^{10} - 8096 p^{4} T^{11} + 540 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 5 T + 428 T^{2} + 2396 T^{3} + 90651 T^{4} + 477799 T^{5} + 12139591 T^{6} + 54121615 T^{7} + 12139591 p T^{8} + 477799 p^{2} T^{9} + 90651 p^{3} T^{10} + 2396 p^{4} T^{11} + 428 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 55 T + 1849 T^{2} + 43681 T^{3} + 811452 T^{4} + 12227790 T^{5} + 154536730 T^{6} + 1648617028 T^{7} + 154536730 p T^{8} + 12227790 p^{2} T^{9} + 811452 p^{3} T^{10} + 43681 p^{4} T^{11} + 1849 p^{5} T^{12} + 55 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.50030500749884654998510872217, −4.48977406172041110571684228508, −4.44779874476558578327167044742, −4.30953130498294012319441422752, −4.10899390878836019766944822112, −3.87046681834299117226543085567, −3.82981513196037708608951912559, −3.82246299426895757573130600217, −3.74073554656721584808731227771, −3.72574072111506657014122739012, −3.19613419599613677369348436318, −3.16731961325277590139019571727, −3.10598278694345872956242237109, −2.67389853080316596280155605773, −2.62457899772300937635548164489, −2.61695067341154117391605470205, −2.56902408649765190830487222917, −2.43247447572355268133773713351, −2.42940199002026156043149289279, −2.16668558272558830836989764920, −2.08664394595949765359949195160, −1.63836418820017179196332876904, −1.56548099159779790386630272480, −1.37725616078221464571945029065, −1.35311682646038305861999327272, 0, 0, 0, 0, 0, 0, 0, 1.35311682646038305861999327272, 1.37725616078221464571945029065, 1.56548099159779790386630272480, 1.63836418820017179196332876904, 2.08664394595949765359949195160, 2.16668558272558830836989764920, 2.42940199002026156043149289279, 2.43247447572355268133773713351, 2.56902408649765190830487222917, 2.61695067341154117391605470205, 2.62457899772300937635548164489, 2.67389853080316596280155605773, 3.10598278694345872956242237109, 3.16731961325277590139019571727, 3.19613419599613677369348436318, 3.72574072111506657014122739012, 3.74073554656721584808731227771, 3.82246299426895757573130600217, 3.82981513196037708608951912559, 3.87046681834299117226543085567, 4.10899390878836019766944822112, 4.30953130498294012319441422752, 4.44779874476558578327167044742, 4.48977406172041110571684228508, 4.50030500749884654998510872217
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.