Dirichlet series
| L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s + 10·7-s − 3·8-s − 3·9-s − 2·10-s − 7·11-s + 2·12-s − 4·13-s − 10·14-s + 4·15-s + 2·16-s + 2·17-s + 3·18-s + 7·19-s + 2·20-s + 20·21-s + 7·22-s + 10·23-s − 6·24-s − 11·25-s + 4·26-s − 12·27-s + 10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 3.77·7-s − 1.06·8-s − 9-s − 0.632·10-s − 2.11·11-s + 0.577·12-s − 1.10·13-s − 2.67·14-s + 1.03·15-s + 1/2·16-s + 0.485·17-s + 0.707·18-s + 1.60·19-s + 0.447·20-s + 4.36·21-s + 1.49·22-s + 2.08·23-s − 1.22·24-s − 2.19·25-s + 0.784·26-s − 2.30·27-s + 1.88·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(11^{7} \cdot 19^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(36.0548\) |
| Root analytic conductor: | \(1.29184\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.548645330\) |
| \(L(\frac12)\) | \(\approx\) | \(2.548645330\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 + T )^{7} \) |
| 19 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + p T^{3} + 3 T^{4} + 7 T^{5} + p^{3} T^{6} - p T^{7} + p^{4} T^{8} + 7 p^{2} T^{9} + 3 p^{3} T^{10} + p^{5} T^{11} + p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 2 T + 7 T^{2} - 8 T^{3} + 25 T^{4} - 34 T^{5} + 82 T^{6} - 104 T^{7} + 82 p T^{8} - 34 p^{2} T^{9} + 25 p^{3} T^{10} - 8 p^{4} T^{11} + 7 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 2 T + 3 p T^{2} - 26 T^{3} + 113 T^{4} - 226 T^{5} + 752 T^{6} - 1466 T^{7} + 752 p T^{8} - 226 p^{2} T^{9} + 113 p^{3} T^{10} - 26 p^{4} T^{11} + 3 p^{6} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 10 T + 66 T^{2} - 334 T^{3} + 1439 T^{4} - 5258 T^{5} + 16844 T^{6} - 47228 T^{7} + 16844 p T^{8} - 5258 p^{2} T^{9} + 1439 p^{3} T^{10} - 334 p^{4} T^{11} + 66 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 4 T + 40 T^{2} + 118 T^{3} + 873 T^{4} + 2134 T^{5} + 13076 T^{6} + 27460 T^{7} + 13076 p T^{8} + 2134 p^{2} T^{9} + 873 p^{3} T^{10} + 118 p^{4} T^{11} + 40 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 2 T + 49 T^{2} - 160 T^{3} + 1671 T^{4} - 4814 T^{5} + 2199 p T^{6} - 107936 T^{7} + 2199 p^{2} T^{8} - 4814 p^{2} T^{9} + 1671 p^{3} T^{10} - 160 p^{4} T^{11} + 49 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 10 T + 110 T^{2} - 732 T^{3} + 4928 T^{4} - 24870 T^{5} + 5981 p T^{6} - 610984 T^{7} + 5981 p^{2} T^{8} - 24870 p^{2} T^{9} + 4928 p^{3} T^{10} - 732 p^{4} T^{11} + 110 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 18 T + 320 T^{2} + 3472 T^{3} + 35009 T^{4} + 266396 T^{5} + 1870372 T^{6} + 10488792 T^{7} + 1870372 p T^{8} + 266396 p^{2} T^{9} + 35009 p^{3} T^{10} + 3472 p^{4} T^{11} + 320 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 24 T + 431 T^{2} - 5368 T^{3} + 55269 T^{4} - 459990 T^{5} + 3278314 T^{6} - 19632048 T^{7} + 3278314 p T^{8} - 459990 p^{2} T^{9} + 55269 p^{3} T^{10} - 5368 p^{4} T^{11} + 431 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 138 T^{2} - 194 T^{3} + 9876 T^{4} - 19416 T^{5} + 504517 T^{6} - 914604 T^{7} + 504517 p T^{8} - 19416 p^{2} T^{9} + 9876 p^{3} T^{10} - 194 p^{4} T^{11} + 138 p^{5} T^{12} + p^{7} T^{14} \) | |
| 41 | \( 1 + 12 T + 282 T^{2} + 2426 T^{3} + 32453 T^{4} + 216142 T^{5} + 2107796 T^{6} + 11219712 T^{7} + 2107796 p T^{8} + 216142 p^{2} T^{9} + 32453 p^{3} T^{10} + 2426 p^{4} T^{11} + 282 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 2 T + 212 T^{2} - 366 T^{3} + 497 p T^{4} - 30878 T^{5} + 1346480 T^{6} - 1615092 T^{7} + 1346480 p T^{8} - 30878 p^{2} T^{9} + 497 p^{4} T^{10} - 366 p^{4} T^{11} + 212 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 8 T + 177 T^{2} - 912 T^{3} + 11693 T^{4} - 35256 T^{5} + 432605 T^{6} - 866144 T^{7} + 432605 p T^{8} - 35256 p^{2} T^{9} + 11693 p^{3} T^{10} - 912 p^{4} T^{11} + 177 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 2 T + 211 T^{2} - 604 T^{3} + 22621 T^{4} - 63566 T^{5} + 1684215 T^{6} - 3939464 T^{7} + 1684215 p T^{8} - 63566 p^{2} T^{9} + 22621 p^{3} T^{10} - 604 p^{4} T^{11} + 211 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 10 T + 68 T^{2} + 564 T^{3} + 7490 T^{4} + 69606 T^{5} + 453907 T^{6} + 1842328 T^{7} + 453907 p T^{8} + 69606 p^{2} T^{9} + 7490 p^{3} T^{10} + 564 p^{4} T^{11} + 68 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 14 T + 393 T^{2} - 4080 T^{3} + 66043 T^{4} - 544594 T^{5} + 105303 p T^{6} - 42469120 T^{7} + 105303 p^{2} T^{8} - 544594 p^{2} T^{9} + 66043 p^{3} T^{10} - 4080 p^{4} T^{11} + 393 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 8 T + 299 T^{2} - 1908 T^{3} + 43661 T^{4} - 221222 T^{5} + 4054526 T^{6} - 17312388 T^{7} + 4054526 p T^{8} - 221222 p^{2} T^{9} + 43661 p^{3} T^{10} - 1908 p^{4} T^{11} + 299 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 10 T + 363 T^{2} - 3316 T^{3} + 60569 T^{4} - 7042 p T^{5} + 6248102 T^{6} - 44683996 T^{7} + 6248102 p T^{8} - 7042 p^{3} T^{9} + 60569 p^{3} T^{10} - 3316 p^{4} T^{11} + 363 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 6 T + 291 T^{2} + 1036 T^{3} + 35145 T^{4} + 59322 T^{5} + 2766403 T^{6} + 2354856 T^{7} + 2766403 p T^{8} + 59322 p^{2} T^{9} + 35145 p^{3} T^{10} + 1036 p^{4} T^{11} + 291 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 52 T + 1523 T^{2} - 31800 T^{3} + 522203 T^{4} - 7037132 T^{5} + 79661705 T^{6} - 766418576 T^{7} + 79661705 p T^{8} - 7037132 p^{2} T^{9} + 522203 p^{3} T^{10} - 31800 p^{4} T^{11} + 1523 p^{5} T^{12} - 52 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 10 T + 362 T^{2} + 1618 T^{3} + 45511 T^{4} - 11482 T^{5} + 3276004 T^{6} - 12186140 T^{7} + 3276004 p T^{8} - 11482 p^{2} T^{9} + 45511 p^{3} T^{10} + 1618 p^{4} T^{11} + 362 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 222 T^{2} - 698 T^{3} + 38288 T^{4} - 87304 T^{5} + 4414265 T^{6} - 12681948 T^{7} + 4414265 p T^{8} - 87304 p^{2} T^{9} + 38288 p^{3} T^{10} - 698 p^{4} T^{11} + 222 p^{5} T^{12} + p^{7} T^{14} \) | |
| 97 | \( 1 + 24 T + 490 T^{2} + 7290 T^{3} + 114080 T^{4} + 1401624 T^{5} + 16484869 T^{6} + 161147084 T^{7} + 16484869 p T^{8} + 1401624 p^{2} T^{9} + 114080 p^{3} T^{10} + 7290 p^{4} T^{11} + 490 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.89546609616105394877368986097, −5.84624693367181628228119132762, −5.74970565770299766656927693801, −5.62973329720563020305381508200, −5.36806905607139323352477802686, −5.35883619559174249974714204608, −5.17842296779522439176261900762, −5.00755147600689311263117885575, −4.97883523731470481734744245001, −4.58390726684160789428592755383, −4.44474428230450902140583557009, −4.38511610735756297537590625805, −3.79459898502165447823002849810, −3.76566986960473015694358687349, −3.33603189733119050953858467186, −3.27911967032119883950331400134, −3.08140952855753023029935480399, −2.63444902040624750349263126739, −2.56246171539699370489019396293, −2.44642084002183667295276153334, −2.06675571676242139840767917581, −1.83487682027436614199988471346, −1.79382865408832722659243733553, −1.32305777517800720586171586944, −0.63699790344572476713681412121, 0.63699790344572476713681412121, 1.32305777517800720586171586944, 1.79382865408832722659243733553, 1.83487682027436614199988471346, 2.06675571676242139840767917581, 2.44642084002183667295276153334, 2.56246171539699370489019396293, 2.63444902040624750349263126739, 3.08140952855753023029935480399, 3.27911967032119883950331400134, 3.33603189733119050953858467186, 3.76566986960473015694358687349, 3.79459898502165447823002849810, 4.38511610735756297537590625805, 4.44474428230450902140583557009, 4.58390726684160789428592755383, 4.97883523731470481734744245001, 5.00755147600689311263117885575, 5.17842296779522439176261900762, 5.35883619559174249974714204608, 5.36806905607139323352477802686, 5.62973329720563020305381508200, 5.74970565770299766656927693801, 5.84624693367181628228119132762, 5.89546609616105394877368986097
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.