Dirichlet series
| L(s) = 1 | − 7·2-s + 28·4-s − 2·5-s + 6·7-s − 84·8-s + 14·10-s − 9·11-s − 2·13-s − 42·14-s + 210·16-s + 7·17-s − 2·19-s − 56·20-s + 63·22-s − 15·23-s − 9·25-s + 14·26-s + 168·28-s − 9·29-s − 2·31-s − 462·32-s − 49·34-s − 12·35-s + 5·37-s + 14·38-s + 168·40-s + 33·41-s + ⋯ |
| L(s) = 1 | − 4.94·2-s + 14·4-s − 0.894·5-s + 2.26·7-s − 29.6·8-s + 4.42·10-s − 2.71·11-s − 0.554·13-s − 11.2·14-s + 52.5·16-s + 1.69·17-s − 0.458·19-s − 12.5·20-s + 13.4·22-s − 3.12·23-s − 9/5·25-s + 2.74·26-s + 31.7·28-s − 1.67·29-s − 0.359·31-s − 81.6·32-s − 8.40·34-s − 2.02·35-s + 0.821·37-s + 2.27·38-s + 26.5·40-s + 5.15·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 223^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 223^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{7} \cdot 3^{14} \cdot 223^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.47521\times 10^{10}\) |
| Root analytic conductor: | \(5.66144\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{7} \cdot 3^{14} \cdot 223^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7980936804\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7980936804\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T )^{7} \) |
| 3 | \( 1 \) | |
| 223 | \( ( 1 + T )^{7} \) | |
| good | 5 | \( 1 + 2 T + 13 T^{2} + 18 T^{3} + 67 T^{4} + 166 T^{5} + 429 T^{6} + 1296 T^{7} + 429 p T^{8} + 166 p^{2} T^{9} + 67 p^{3} T^{10} + 18 p^{4} T^{11} + 13 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - 6 T + 41 T^{2} - 164 T^{3} + 701 T^{4} - 310 p T^{5} + 7157 T^{6} - 18328 T^{7} + 7157 p T^{8} - 310 p^{3} T^{9} + 701 p^{3} T^{10} - 164 p^{4} T^{11} + 41 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 9 T + 63 T^{2} + 340 T^{3} + 1713 T^{4} + 7187 T^{5} + 28125 T^{6} + 96298 T^{7} + 28125 p T^{8} + 7187 p^{2} T^{9} + 1713 p^{3} T^{10} + 340 p^{4} T^{11} + 63 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 2 T + 27 T^{2} + 6 p T^{3} + 749 T^{4} + 110 p T^{5} + 12173 T^{6} + 26660 T^{7} + 12173 p T^{8} + 110 p^{3} T^{9} + 749 p^{3} T^{10} + 6 p^{5} T^{11} + 27 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 7 T + 75 T^{2} - 422 T^{3} + 2657 T^{4} - 13017 T^{5} + 64695 T^{6} - 268656 T^{7} + 64695 p T^{8} - 13017 p^{2} T^{9} + 2657 p^{3} T^{10} - 422 p^{4} T^{11} + 75 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 2 T + 77 T^{2} + 132 T^{3} + 3061 T^{4} + 4686 T^{5} + 4267 p T^{6} + 107896 T^{7} + 4267 p^{2} T^{8} + 4686 p^{2} T^{9} + 3061 p^{3} T^{10} + 132 p^{4} T^{11} + 77 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 15 T + 205 T^{2} + 1902 T^{3} + 15529 T^{4} + 103889 T^{5} + 615469 T^{6} + 3131844 T^{7} + 615469 p T^{8} + 103889 p^{2} T^{9} + 15529 p^{3} T^{10} + 1902 p^{4} T^{11} + 205 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 9 T + 107 T^{2} + 722 T^{3} + 4829 T^{4} + 23967 T^{5} + 4795 p T^{6} + 602588 T^{7} + 4795 p^{2} T^{8} + 23967 p^{2} T^{9} + 4829 p^{3} T^{10} + 722 p^{4} T^{11} + 107 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 2 T + 125 T^{2} - 100 T^{3} + 6369 T^{4} - 22362 T^{5} + 216297 T^{6} - 1064192 T^{7} + 216297 p T^{8} - 22362 p^{2} T^{9} + 6369 p^{3} T^{10} - 100 p^{4} T^{11} + 125 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 5 T + 51 T^{2} - 402 T^{3} + 3485 T^{4} - 16691 T^{5} + 132407 T^{6} - 755612 T^{7} + 132407 p T^{8} - 16691 p^{2} T^{9} + 3485 p^{3} T^{10} - 402 p^{4} T^{11} + 51 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 33 T + 655 T^{2} - 9442 T^{3} + 107781 T^{4} - 1018431 T^{5} + 8167163 T^{6} - 56270876 T^{7} + 8167163 p T^{8} - 1018431 p^{2} T^{9} + 107781 p^{3} T^{10} - 9442 p^{4} T^{11} + 655 p^{5} T^{12} - 33 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 20 T + 373 T^{2} - 4328 T^{3} + 47717 T^{4} - 402636 T^{5} + 3292073 T^{6} - 21857328 T^{7} + 3292073 p T^{8} - 402636 p^{2} T^{9} + 47717 p^{3} T^{10} - 4328 p^{4} T^{11} + 373 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 2 T + 113 T^{2} + 20 T^{3} + 8501 T^{4} + 906 T^{5} + 518921 T^{6} + 118368 T^{7} + 518921 p T^{8} + 906 p^{2} T^{9} + 8501 p^{3} T^{10} + 20 p^{4} T^{11} + 113 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 13 T + 259 T^{2} - 2626 T^{3} + 33677 T^{4} - 278539 T^{5} + 2685015 T^{6} - 18247356 T^{7} + 2685015 p T^{8} - 278539 p^{2} T^{9} + 33677 p^{3} T^{10} - 2626 p^{4} T^{11} + 259 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 9 T + 203 T^{2} + 1630 T^{3} + 23925 T^{4} + 2897 p T^{5} + 1964329 T^{6} + 12171514 T^{7} + 1964329 p T^{8} + 2897 p^{3} T^{9} + 23925 p^{3} T^{10} + 1630 p^{4} T^{11} + 203 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 8 T + 239 T^{2} - 2446 T^{3} + 30153 T^{4} - 299532 T^{5} + 2695533 T^{6} - 21956832 T^{7} + 2695533 p T^{8} - 299532 p^{2} T^{9} + 30153 p^{3} T^{10} - 2446 p^{4} T^{11} + 239 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 29 T + 547 T^{2} - 7824 T^{3} + 93989 T^{4} - 1001607 T^{5} + 9666465 T^{6} - 83193754 T^{7} + 9666465 p T^{8} - 1001607 p^{2} T^{9} + 93989 p^{3} T^{10} - 7824 p^{4} T^{11} + 547 p^{5} T^{12} - 29 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 73 T^{2} + 48 T^{3} + 7885 T^{4} - 42944 T^{5} + 476501 T^{6} - 2121440 T^{7} + 476501 p T^{8} - 42944 p^{2} T^{9} + 7885 p^{3} T^{10} + 48 p^{4} T^{11} + 73 p^{5} T^{12} + p^{7} T^{14} \) | |
| 73 | \( 1 + 37 T + 983 T^{2} + 18018 T^{3} + 274469 T^{4} + 3384747 T^{5} + 36363747 T^{6} + 330718412 T^{7} + 36363747 p T^{8} + 3384747 p^{2} T^{9} + 274469 p^{3} T^{10} + 18018 p^{4} T^{11} + 983 p^{5} T^{12} + 37 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 32 T + 825 T^{2} - 14640 T^{3} + 224645 T^{4} - 2807072 T^{5} + 31109933 T^{6} - 292707104 T^{7} + 31109933 p T^{8} - 2807072 p^{2} T^{9} + 224645 p^{3} T^{10} - 14640 p^{4} T^{11} + 825 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 6 T + 381 T^{2} - 2108 T^{3} + 73989 T^{4} - 4334 p T^{5} + 9067089 T^{6} - 37267848 T^{7} + 9067089 p T^{8} - 4334 p^{3} T^{9} + 73989 p^{3} T^{10} - 2108 p^{4} T^{11} + 381 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 17 T + 447 T^{2} - 7166 T^{3} + 104309 T^{4} - 1338359 T^{5} + 14938191 T^{6} - 148859160 T^{7} + 14938191 p T^{8} - 1338359 p^{2} T^{9} + 104309 p^{3} T^{10} - 7166 p^{4} T^{11} + 447 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 12 T + 251 T^{2} - 2024 T^{3} + 40393 T^{4} - 296116 T^{5} + 5234683 T^{6} - 36247216 T^{7} + 5234683 p T^{8} - 296116 p^{2} T^{9} + 40393 p^{3} T^{10} - 2024 p^{4} T^{11} + 251 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.77966885821998555392943002772, −3.72396810734122748735942188883, −3.58487096595638161525680445390, −3.49146698341575216169158487000, −3.09485880205996195714066071804, −3.01866130045936704452810239656, −2.79917806950955075894138892031, −2.71547289768375995695091910908, −2.70402052516242545643271379698, −2.66379588211572041602563066577, −2.32323267076984549157249059962, −2.07883231118365935514476766449, −2.04062212049707104656044750934, −1.97458335718549929429175228531, −1.94543818755445071592966626596, −1.81325517058211431969361441153, −1.75277769287851410695756871004, −1.68133347259820368418004461284, −1.00524815863611790209033080791, −0.945424236987175721142637123696, −0.888318930650294845475192220116, −0.63980567483382581177600725725, −0.57093366483578033172605006334, −0.43052389911182868387624310114, −0.29570279429190020382977894129, 0.29570279429190020382977894129, 0.43052389911182868387624310114, 0.57093366483578033172605006334, 0.63980567483382581177600725725, 0.888318930650294845475192220116, 0.945424236987175721142637123696, 1.00524815863611790209033080791, 1.68133347259820368418004461284, 1.75277769287851410695756871004, 1.81325517058211431969361441153, 1.94543818755445071592966626596, 1.97458335718549929429175228531, 2.04062212049707104656044750934, 2.07883231118365935514476766449, 2.32323267076984549157249059962, 2.66379588211572041602563066577, 2.70402052516242545643271379698, 2.71547289768375995695091910908, 2.79917806950955075894138892031, 3.01866130045936704452810239656, 3.09485880205996195714066071804, 3.49146698341575216169158487000, 3.58487096595638161525680445390, 3.72396810734122748735942188883, 3.77966885821998555392943002772
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.