Dirichlet series
| L(s) = 1 | − 2-s + 2·3-s − 4-s + 7·5-s − 2·6-s − 3·7-s − 3·9-s − 7·10-s + 15·11-s − 2·12-s + 5·13-s + 3·14-s + 14·15-s + 2·16-s − 5·17-s + 3·18-s + 13·19-s − 7·20-s − 6·21-s − 15·22-s − 5·23-s + 28·25-s − 5·26-s − 27-s + 3·28-s − 3·29-s − 14·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 3.13·5-s − 0.816·6-s − 1.13·7-s − 9-s − 2.21·10-s + 4.52·11-s − 0.577·12-s + 1.38·13-s + 0.801·14-s + 3.61·15-s + 1/2·16-s − 1.21·17-s + 0.707·18-s + 2.98·19-s − 1.56·20-s − 1.30·21-s − 3.19·22-s − 1.04·23-s + 28/5·25-s − 0.980·26-s − 0.192·27-s + 0.566·28-s − 0.557·29-s − 2.55·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 73^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 73^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 73^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1786.44\) |
| Root analytic conductor: | \(1.70720\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{7} \cdot 73^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(7.558923749\) |
| \(L(\frac12)\) | \(\approx\) | \(7.558923749\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{7} \) |
| 73 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + T + p T^{2} + 3 T^{3} + 3 T^{4} + 7 T^{5} + 9 p T^{6} + 17 T^{7} + 9 p^{2} T^{8} + 7 p^{2} T^{9} + 3 p^{3} T^{10} + 3 p^{4} T^{11} + p^{6} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 2 T + 7 T^{2} - 19 T^{3} + 43 T^{4} - 97 T^{5} + 184 T^{6} - 331 T^{7} + 184 p T^{8} - 97 p^{2} T^{9} + 43 p^{3} T^{10} - 19 p^{4} T^{11} + 7 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 3 T + 27 T^{2} + 73 T^{3} + 407 T^{4} + 131 p T^{5} + 3981 T^{6} + 7790 T^{7} + 3981 p T^{8} + 131 p^{3} T^{9} + 407 p^{3} T^{10} + 73 p^{4} T^{11} + 27 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 15 T + 155 T^{2} - 1140 T^{3} + 6859 T^{4} - 33713 T^{5} + 12898 p T^{6} - 505739 T^{7} + 12898 p^{2} T^{8} - 33713 p^{2} T^{9} + 6859 p^{3} T^{10} - 1140 p^{4} T^{11} + 155 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 5 T + 54 T^{2} - 220 T^{3} + 1371 T^{4} - 4919 T^{5} + 23954 T^{6} - 75531 T^{7} + 23954 p T^{8} - 4919 p^{2} T^{9} + 1371 p^{3} T^{10} - 220 p^{4} T^{11} + 54 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 5 T + 90 T^{2} + 376 T^{3} + 3771 T^{4} + 13391 T^{5} + 97266 T^{6} + 287171 T^{7} + 97266 p T^{8} + 13391 p^{2} T^{9} + 3771 p^{3} T^{10} + 376 p^{4} T^{11} + 90 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 13 T + 177 T^{2} - 1417 T^{3} + 11157 T^{4} - 64511 T^{5} + 364189 T^{6} - 1606742 T^{7} + 364189 p T^{8} - 64511 p^{2} T^{9} + 11157 p^{3} T^{10} - 1417 p^{4} T^{11} + 177 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 5 T + 126 T^{2} + 538 T^{3} + 7425 T^{4} + 26705 T^{5} + 263322 T^{6} + 779991 T^{7} + 263322 p T^{8} + 26705 p^{2} T^{9} + 7425 p^{3} T^{10} + 538 p^{4} T^{11} + 126 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 3 T + 123 T^{2} + 255 T^{3} + 7269 T^{4} + 13433 T^{5} + 295511 T^{6} + 500890 T^{7} + 295511 p T^{8} + 13433 p^{2} T^{9} + 7269 p^{3} T^{10} + 255 p^{4} T^{11} + 123 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 11 T + 156 T^{2} - 1138 T^{3} + 10783 T^{4} - 65287 T^{5} + 488162 T^{6} - 2460011 T^{7} + 488162 p T^{8} - 65287 p^{2} T^{9} + 10783 p^{3} T^{10} - 1138 p^{4} T^{11} + 156 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 26 T + 374 T^{2} + 3933 T^{3} + 34028 T^{4} + 255610 T^{5} + 1748385 T^{6} + 11054126 T^{7} + 1748385 p T^{8} + 255610 p^{2} T^{9} + 34028 p^{3} T^{10} + 3933 p^{4} T^{11} + 374 p^{5} T^{12} + 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 7 T + 204 T^{2} - 1112 T^{3} + 19679 T^{4} - 89389 T^{5} + 1194186 T^{6} - 4527293 T^{7} + 1194186 p T^{8} - 89389 p^{2} T^{9} + 19679 p^{3} T^{10} - 1112 p^{4} T^{11} + 204 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 16 T + 254 T^{2} - 2805 T^{3} + 29348 T^{4} - 241848 T^{5} + 1935619 T^{6} - 13001894 T^{7} + 1935619 p T^{8} - 241848 p^{2} T^{9} + 29348 p^{3} T^{10} - 2805 p^{4} T^{11} + 254 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 3 T + 68 T^{2} - 249 T^{3} + 5318 T^{4} - 97 p T^{5} + 333059 T^{6} - 749110 T^{7} + 333059 p T^{8} - 97 p^{3} T^{9} + 5318 p^{3} T^{10} - 249 p^{4} T^{11} + 68 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 34 T + 729 T^{2} + 11293 T^{3} + 142615 T^{4} + 1497547 T^{5} + 13543332 T^{6} + 105510405 T^{7} + 13543332 p T^{8} + 1497547 p^{2} T^{9} + 142615 p^{3} T^{10} + 11293 p^{4} T^{11} + 729 p^{5} T^{12} + 34 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 27 T + 8 p T^{2} - 5903 T^{3} + 66350 T^{4} - 641693 T^{5} + 5724283 T^{6} - 45137410 T^{7} + 5724283 p T^{8} - 641693 p^{2} T^{9} + 66350 p^{3} T^{10} - 5903 p^{4} T^{11} + 8 p^{6} T^{12} - 27 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 15 T + 388 T^{2} - 4040 T^{3} + 61267 T^{4} - 494597 T^{5} + 5643906 T^{6} - 37142013 T^{7} + 5643906 p T^{8} - 494597 p^{2} T^{9} + 61267 p^{3} T^{10} - 4040 p^{4} T^{11} + 388 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 11 T + 232 T^{2} - 2368 T^{3} + 29249 T^{4} - 256117 T^{5} + 2421762 T^{6} - 19790711 T^{7} + 2421762 p T^{8} - 256117 p^{2} T^{9} + 29249 p^{3} T^{10} - 2368 p^{4} T^{11} + 232 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 26 T + 618 T^{2} - 8977 T^{3} + 123668 T^{4} - 1283482 T^{5} + 13223443 T^{6} - 110479830 T^{7} + 13223443 p T^{8} - 1283482 p^{2} T^{9} + 123668 p^{3} T^{10} - 8977 p^{4} T^{11} + 618 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 19 T + 504 T^{2} - 6615 T^{3} + 104554 T^{4} - 1053989 T^{5} + 12528803 T^{6} - 102660626 T^{7} + 12528803 p T^{8} - 1053989 p^{2} T^{9} + 104554 p^{3} T^{10} - 6615 p^{4} T^{11} + 504 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 17 T + 635 T^{2} + 8171 T^{3} + 166131 T^{4} + 1661103 T^{5} + 23498377 T^{6} + 182600442 T^{7} + 23498377 p T^{8} + 1661103 p^{2} T^{9} + 166131 p^{3} T^{10} + 8171 p^{4} T^{11} + 635 p^{5} T^{12} + 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 2 T + 405 T^{2} - 927 T^{3} + 84763 T^{4} - 175333 T^{5} + 11174168 T^{6} - 20184215 T^{7} + 11174168 p T^{8} - 175333 p^{2} T^{9} + 84763 p^{3} T^{10} - 927 p^{4} T^{11} + 405 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 4 T + 287 T^{2} + 712 T^{3} + 54013 T^{4} + 136060 T^{5} + 7030435 T^{6} + 13600624 T^{7} + 7030435 p T^{8} + 136060 p^{2} T^{9} + 54013 p^{3} T^{10} + 712 p^{4} T^{11} + 287 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.67471667989363668701550362469, −5.50972257352945544271262905360, −5.36081966735310362795293431269, −5.20245605454850567598364442018, −4.98773252400522843216992598424, −4.73721753980081582970668472797, −4.61744874562853637999465655155, −4.48571020910898275854871227489, −4.08216141321223985531125857289, −4.07487761409524249202692671258, −3.65765966901187121842617062852, −3.52212205565853420592498469259, −3.39361069915497714400437693931, −3.36044039662732703922609022400, −3.22945640069213805590126820314, −3.10120937868503236515555915340, −2.65821087693851236259141564868, −2.38378266546067569034490133316, −2.35518755159959083824785143034, −1.87716002538912264856367235305, −1.80000624773436518045144077204, −1.55890069704332068295714985159, −1.30335087817942449359472530612, −0.868516383919736682959803704261, −0.843236538919231621968087004009, 0.843236538919231621968087004009, 0.868516383919736682959803704261, 1.30335087817942449359472530612, 1.55890069704332068295714985159, 1.80000624773436518045144077204, 1.87716002538912264856367235305, 2.35518755159959083824785143034, 2.38378266546067569034490133316, 2.65821087693851236259141564868, 3.10120937868503236515555915340, 3.22945640069213805590126820314, 3.36044039662732703922609022400, 3.39361069915497714400437693931, 3.52212205565853420592498469259, 3.65765966901187121842617062852, 4.07487761409524249202692671258, 4.08216141321223985531125857289, 4.48571020910898275854871227489, 4.61744874562853637999465655155, 4.73721753980081582970668472797, 4.98773252400522843216992598424, 5.20245605454850567598364442018, 5.36081966735310362795293431269, 5.50972257352945544271262905360, 5.67471667989363668701550362469
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.