Dirichlet series
| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 8·11-s + 10·13-s + 3·15-s + 7·17-s + 13·19-s − 3·23-s − 4·25-s − 5·27-s + 4·29-s − 9·31-s + 8·33-s + 2·37-s + 10·39-s + 8·41-s + 13·43-s − 6·45-s − 30·47-s − 13·49-s + 7·51-s − 7·53-s + 24·55-s + 13·57-s + 18·61-s + 30·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 2.41·11-s + 2.77·13-s + 0.774·15-s + 1.69·17-s + 2.98·19-s − 0.625·23-s − 4/5·25-s − 0.962·27-s + 0.742·29-s − 1.61·31-s + 1.39·33-s + 0.328·37-s + 1.60·39-s + 1.24·41-s + 1.98·43-s − 0.894·45-s − 4.37·47-s − 1.85·49-s + 0.980·51-s − 0.961·53-s + 3.23·55-s + 1.72·57-s + 2.30·61-s + 3.72·65-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(2^{14} \cdot 89^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1499.99\) |
| Root analytic conductor: | \(1.68602\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 2^{14} \cdot 89^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.756430745\) |
| \(L(\frac12)\) | \(\approx\) | \(8.756430745\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 89 | \( ( 1 + T )^{7} \) | |
| good | 3 | \( 1 - T + p T^{2} + 4 p T^{4} - 14 T^{5} + 4 p^{2} T^{6} - 4 T^{7} + 4 p^{3} T^{8} - 14 p^{2} T^{9} + 4 p^{4} T^{10} + p^{6} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 - 3 T + 13 T^{2} - 36 T^{3} + 92 T^{4} - 52 p T^{5} + 726 T^{6} - 1562 T^{7} + 726 p T^{8} - 52 p^{3} T^{9} + 92 p^{3} T^{10} - 36 p^{4} T^{11} + 13 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 13 T^{2} + 8 T^{3} + 129 T^{4} - 20 T^{5} + 139 p T^{6} - 192 T^{7} + 139 p^{2} T^{8} - 20 p^{2} T^{9} + 129 p^{3} T^{10} + 8 p^{4} T^{11} + 13 p^{5} T^{12} + p^{7} T^{14} \) | |
| 11 | \( 1 - 8 T + 49 T^{2} - 224 T^{3} + 1017 T^{4} - 3944 T^{5} + 14385 T^{6} - 46944 T^{7} + 14385 p T^{8} - 3944 p^{2} T^{9} + 1017 p^{3} T^{10} - 224 p^{4} T^{11} + 49 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 10 T + 59 T^{2} - 204 T^{3} + 621 T^{4} - 2742 T^{5} + 16815 T^{6} - 72328 T^{7} + 16815 p T^{8} - 2742 p^{2} T^{9} + 621 p^{3} T^{10} - 204 p^{4} T^{11} + 59 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 7 T + 53 T^{2} - 368 T^{3} + 2300 T^{4} - 10592 T^{5} + 53866 T^{6} - 235742 T^{7} + 53866 p T^{8} - 10592 p^{2} T^{9} + 2300 p^{3} T^{10} - 368 p^{4} T^{11} + 53 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 13 T + 137 T^{2} - 862 T^{3} + 4986 T^{4} - 20566 T^{5} + 92924 T^{6} - 348304 T^{7} + 92924 p T^{8} - 20566 p^{2} T^{9} + 4986 p^{3} T^{10} - 862 p^{4} T^{11} + 137 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 3 T + 81 T^{2} + 154 T^{3} + 3938 T^{4} + 6858 T^{5} + 126848 T^{6} + 169576 T^{7} + 126848 p T^{8} + 6858 p^{2} T^{9} + 3938 p^{3} T^{10} + 154 p^{4} T^{11} + 81 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 4 T + 103 T^{2} - 192 T^{3} + 4809 T^{4} - 4268 T^{5} + 173655 T^{6} - 126816 T^{7} + 173655 p T^{8} - 4268 p^{2} T^{9} + 4809 p^{3} T^{10} - 192 p^{4} T^{11} + 103 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 9 T + 137 T^{2} + 30 p T^{3} + 8964 T^{4} + 51896 T^{5} + 380038 T^{6} + 1887192 T^{7} + 380038 p T^{8} + 51896 p^{2} T^{9} + 8964 p^{3} T^{10} + 30 p^{5} T^{11} + 137 p^{5} T^{12} + 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 2 T + 159 T^{2} - 372 T^{3} + 12281 T^{4} - 29870 T^{5} + 620767 T^{6} - 1399704 T^{7} + 620767 p T^{8} - 29870 p^{2} T^{9} + 12281 p^{3} T^{10} - 372 p^{4} T^{11} + 159 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 8 T + 175 T^{2} - 1176 T^{3} + 16293 T^{4} - 95080 T^{5} + 977403 T^{6} - 4746096 T^{7} + 977403 p T^{8} - 95080 p^{2} T^{9} + 16293 p^{3} T^{10} - 1176 p^{4} T^{11} + 175 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 13 T + 203 T^{2} - 1412 T^{3} + 15090 T^{4} - 93080 T^{5} + 954766 T^{6} - 5283988 T^{7} + 954766 p T^{8} - 93080 p^{2} T^{9} + 15090 p^{3} T^{10} - 1412 p^{4} T^{11} + 203 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 30 T + 589 T^{2} + 8252 T^{3} + 95441 T^{4} + 19598 p T^{5} + 7771749 T^{6} + 56698152 T^{7} + 7771749 p T^{8} + 19598 p^{3} T^{9} + 95441 p^{3} T^{10} + 8252 p^{4} T^{11} + 589 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 7 T + 297 T^{2} + 1848 T^{3} + 39840 T^{4} + 217436 T^{5} + 3207422 T^{6} + 14750862 T^{7} + 3207422 p T^{8} + 217436 p^{2} T^{9} + 39840 p^{3} T^{10} + 1848 p^{4} T^{11} + 297 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 153 T^{2} + 188 T^{3} + 16577 T^{4} + 22116 T^{5} + 1308329 T^{6} + 1332776 T^{7} + 1308329 p T^{8} + 22116 p^{2} T^{9} + 16577 p^{3} T^{10} + 188 p^{4} T^{11} + 153 p^{5} T^{12} + p^{7} T^{14} \) | |
| 61 | \( 1 - 18 T + 399 T^{2} - 5404 T^{3} + 70801 T^{4} - 12006 p T^{5} + 7107207 T^{6} - 57281192 T^{7} + 7107207 p T^{8} - 12006 p^{3} T^{9} + 70801 p^{3} T^{10} - 5404 p^{4} T^{11} + 399 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 2 T + 205 T^{2} + 196 T^{3} + 19589 T^{4} + 8958 T^{5} + 1348449 T^{6} + 602008 T^{7} + 1348449 p T^{8} + 8958 p^{2} T^{9} + 19589 p^{3} T^{10} + 196 p^{4} T^{11} + 205 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 30 T + 769 T^{2} + 13068 T^{3} + 196677 T^{4} + 2338946 T^{5} + 25046805 T^{6} + 221881832 T^{7} + 25046805 p T^{8} + 2338946 p^{2} T^{9} + 196677 p^{3} T^{10} + 13068 p^{4} T^{11} + 769 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 19 T + 265 T^{2} - 2660 T^{3} + 34772 T^{4} - 368020 T^{5} + 3590758 T^{6} - 28732898 T^{7} + 3590758 p T^{8} - 368020 p^{2} T^{9} + 34772 p^{3} T^{10} - 2660 p^{4} T^{11} + 265 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 18 T + 7 p T^{2} + 8052 T^{3} + 130629 T^{4} + 1536238 T^{5} + 17157693 T^{6} + 159970200 T^{7} + 17157693 p T^{8} + 1536238 p^{2} T^{9} + 130629 p^{3} T^{10} + 8052 p^{4} T^{11} + 7 p^{6} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 20 T + 477 T^{2} + 6504 T^{3} + 92829 T^{4} + 970624 T^{5} + 10780385 T^{6} + 94383552 T^{7} + 10780385 p T^{8} + 970624 p^{2} T^{9} + 92829 p^{3} T^{10} + 6504 p^{4} T^{11} + 477 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 13 T + 529 T^{2} - 5732 T^{3} + 129956 T^{4} - 1164316 T^{5} + 19235998 T^{6} - 141745514 T^{7} + 19235998 p T^{8} - 1164316 p^{2} T^{9} + 129956 p^{3} T^{10} - 5732 p^{4} T^{11} + 529 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.65319374247628223845744208579, −5.48569583304130737919101034269, −5.38012872055393552528655114681, −5.32256030805130086313139047411, −5.13932243192463722055652314814, −4.73429003083055829057915334330, −4.48390527356892483396559640870, −4.28972064539951546675411052469, −4.27538932890044429654278842330, −4.16720492303027702212406569462, −3.70731447494994187282595624599, −3.69628022303070462780414522564, −3.46645029878532576282269930479, −3.41532571808466348302399888211, −3.16498209524460569878355254902, −2.99548740497748881484085837631, −2.84860993394282293350170137702, −2.79462114361419621210227485425, −2.08793538317703228428510641799, −1.88449086730846237852386043597, −1.77785904488526354367937645951, −1.51238033047633058312260542818, −1.24999144077871114340037283306, −1.24840868168447652847702305763, −0.69027475996634440520550749948, 0.69027475996634440520550749948, 1.24840868168447652847702305763, 1.24999144077871114340037283306, 1.51238033047633058312260542818, 1.77785904488526354367937645951, 1.88449086730846237852386043597, 2.08793538317703228428510641799, 2.79462114361419621210227485425, 2.84860993394282293350170137702, 2.99548740497748881484085837631, 3.16498209524460569878355254902, 3.41532571808466348302399888211, 3.46645029878532576282269930479, 3.69628022303070462780414522564, 3.70731447494994187282595624599, 4.16720492303027702212406569462, 4.27538932890044429654278842330, 4.28972064539951546675411052469, 4.48390527356892483396559640870, 4.73429003083055829057915334330, 5.13932243192463722055652314814, 5.32256030805130086313139047411, 5.38012872055393552528655114681, 5.48569583304130737919101034269, 5.65319374247628223845744208579
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.