Dirichlet series
| L(s) = 1 | + 7·3-s − 3·4-s + 4·5-s + 9·7-s − 8-s + 28·9-s − 3·11-s − 21·12-s + 11·13-s + 28·15-s + 16-s − 5·17-s + 6·19-s − 12·20-s + 63·21-s − 10·23-s − 7·24-s − 3·25-s + 84·27-s − 27·28-s + 3·29-s + 9·31-s + 3·32-s − 21·33-s + 36·35-s − 84·36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 4.04·3-s − 3/2·4-s + 1.78·5-s + 3.40·7-s − 0.353·8-s + 28/3·9-s − 0.904·11-s − 6.06·12-s + 3.05·13-s + 7.22·15-s + 1/4·16-s − 1.21·17-s + 1.37·19-s − 2.68·20-s + 13.7·21-s − 2.08·23-s − 1.42·24-s − 3/5·25-s + 16.1·27-s − 5.10·28-s + 0.557·29-s + 1.61·31-s + 0.530·32-s − 3.65·33-s + 6.08·35-s − 14·36-s + 0.657·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 97^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 97^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 97^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(365.755\) |
| Root analytic conductor: | \(1.52435\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 97^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.89196767\) |
| \(L(\frac12)\) | \(\approx\) | \(23.89196767\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 97 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + 3 T^{2} + T^{3} + p^{3} T^{4} + 3 T^{5} + 5 p^{2} T^{6} + 5 p^{3} T^{8} + 3 p^{2} T^{9} + p^{6} T^{10} + p^{4} T^{11} + 3 p^{5} T^{12} + p^{7} T^{14} \) |
| 5 | \( 1 - 4 T + 19 T^{2} - 68 T^{3} + 236 T^{4} - 628 T^{5} + 1704 T^{6} - 3944 T^{7} + 1704 p T^{8} - 628 p^{2} T^{9} + 236 p^{3} T^{10} - 68 p^{4} T^{11} + 19 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 9 T + 62 T^{2} - 316 T^{3} + 1360 T^{4} - 4975 T^{5} + 15963 T^{6} - 44888 T^{7} + 15963 p T^{8} - 4975 p^{2} T^{9} + 1360 p^{3} T^{10} - 316 p^{4} T^{11} + 62 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 3 T + 38 T^{2} + 10 p T^{3} + 724 T^{4} + 2245 T^{5} + 9835 T^{6} + 30244 T^{7} + 9835 p T^{8} + 2245 p^{2} T^{9} + 724 p^{3} T^{10} + 10 p^{5} T^{11} + 38 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 11 T + 116 T^{2} - 776 T^{3} + 4898 T^{4} - 23821 T^{5} + 109021 T^{6} - 404944 T^{7} + 109021 p T^{8} - 23821 p^{2} T^{9} + 4898 p^{3} T^{10} - 776 p^{4} T^{11} + 116 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 5 T + 78 T^{2} + 278 T^{3} + 2792 T^{4} + 8395 T^{5} + 68169 T^{6} + 175924 T^{7} + 68169 p T^{8} + 8395 p^{2} T^{9} + 2792 p^{3} T^{10} + 278 p^{4} T^{11} + 78 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 6 T + 104 T^{2} - 514 T^{3} + 5062 T^{4} - 20834 T^{5} + 148187 T^{6} - 500236 T^{7} + 148187 p T^{8} - 20834 p^{2} T^{9} + 5062 p^{3} T^{10} - 514 p^{4} T^{11} + 104 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 10 T + 134 T^{2} + 1056 T^{3} + 8668 T^{4} + 52318 T^{5} + 320207 T^{6} + 1538896 T^{7} + 320207 p T^{8} + 52318 p^{2} T^{9} + 8668 p^{3} T^{10} + 1056 p^{4} T^{11} + 134 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 3 T + 5 p T^{2} - 357 T^{3} + 9296 T^{4} - 19097 T^{5} + 370054 T^{6} - 653550 T^{7} + 370054 p T^{8} - 19097 p^{2} T^{9} + 9296 p^{3} T^{10} - 357 p^{4} T^{11} + 5 p^{6} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 9 T + 205 T^{2} - 1427 T^{3} + 18206 T^{4} - 101027 T^{5} + 917438 T^{6} - 4053122 T^{7} + 917438 p T^{8} - 101027 p^{2} T^{9} + 18206 p^{3} T^{10} - 1427 p^{4} T^{11} + 205 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 4 T + 92 T^{2} - 678 T^{3} + 6330 T^{4} - 41916 T^{5} + 316421 T^{6} - 1891540 T^{7} + 316421 p T^{8} - 41916 p^{2} T^{9} + 6330 p^{3} T^{10} - 678 p^{4} T^{11} + 92 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 232 T^{2} + 40 T^{3} + 24922 T^{4} + 5408 T^{5} + 1593541 T^{6} + 316144 T^{7} + 1593541 p T^{8} + 5408 p^{2} T^{9} + 24922 p^{3} T^{10} + 40 p^{4} T^{11} + 232 p^{5} T^{12} + p^{7} T^{14} \) | |
| 43 | \( 1 - 5 T + 93 T^{2} - 279 T^{3} + 4338 T^{4} - 15607 T^{5} + 316670 T^{6} - 1117754 T^{7} + 316670 p T^{8} - 15607 p^{2} T^{9} + 4338 p^{3} T^{10} - 279 p^{4} T^{11} + 93 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 2 T + 86 T^{2} - 688 T^{3} + 3576 T^{4} - 47954 T^{5} + 451795 T^{6} - 1383296 T^{7} + 451795 p T^{8} - 47954 p^{2} T^{9} + 3576 p^{3} T^{10} - 688 p^{4} T^{11} + 86 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 12 T + 184 T^{2} + 1466 T^{3} + 20754 T^{4} + 158028 T^{5} + 1534913 T^{6} + 8917884 T^{7} + 1534913 p T^{8} + 158028 p^{2} T^{9} + 20754 p^{3} T^{10} + 1466 p^{4} T^{11} + 184 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 12 T + 201 T^{2} + 2204 T^{3} + 24680 T^{4} + 238788 T^{5} + 2054420 T^{6} + 16410096 T^{7} + 2054420 p T^{8} + 238788 p^{2} T^{9} + 24680 p^{3} T^{10} + 2204 p^{4} T^{11} + 201 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 12 T + 352 T^{2} - 2926 T^{3} + 51290 T^{4} - 328828 T^{5} + 4513417 T^{6} - 23881652 T^{7} + 4513417 p T^{8} - 328828 p^{2} T^{9} + 51290 p^{3} T^{10} - 2926 p^{4} T^{11} + 352 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 5 T + 248 T^{2} + 548 T^{3} + 29878 T^{4} + 46395 T^{5} + 2820299 T^{6} + 4543880 T^{7} + 2820299 p T^{8} + 46395 p^{2} T^{9} + 29878 p^{3} T^{10} + 548 p^{4} T^{11} + 248 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 24 T + 554 T^{2} + 8256 T^{3} + 113536 T^{4} + 1236048 T^{5} + 12722783 T^{6} + 109470224 T^{7} + 12722783 p T^{8} + 1236048 p^{2} T^{9} + 113536 p^{3} T^{10} + 8256 p^{4} T^{11} + 554 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 14 T + 221 T^{2} - 2188 T^{3} + 31296 T^{4} - 296660 T^{5} + 3027258 T^{6} - 22952068 T^{7} + 3027258 p T^{8} - 296660 p^{2} T^{9} + 31296 p^{3} T^{10} - 2188 p^{4} T^{11} + 221 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 16 T + 314 T^{2} - 2178 T^{3} + 30092 T^{4} - 196600 T^{5} + 3689411 T^{6} - 25694060 T^{7} + 3689411 p T^{8} - 196600 p^{2} T^{9} + 30092 p^{3} T^{10} - 2178 p^{4} T^{11} + 314 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 16 T + 426 T^{2} + 6024 T^{3} + 88060 T^{4} + 1069624 T^{5} + 11115271 T^{6} + 112486944 T^{7} + 11115271 p T^{8} + 1069624 p^{2} T^{9} + 88060 p^{3} T^{10} + 6024 p^{4} T^{11} + 426 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + T + 332 T^{2} - 1000 T^{3} + 54330 T^{4} - 282449 T^{5} + 6416769 T^{6} - 33754992 T^{7} + 6416769 p T^{8} - 282449 p^{2} T^{9} + 54330 p^{3} T^{10} - 1000 p^{4} T^{11} + 332 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.70499917694634813858331316360, −5.67274341064318617025222141471, −5.58352720198086906952080408212, −5.25578005259739536688006626219, −4.86211460293723244255756860377, −4.72104759036527617502369202200, −4.68183450611373443810672685336, −4.50436271667587628046580375614, −4.40386142409479485294837508159, −4.39977360149628608971024156455, −4.23819609001603146265951454947, −3.72641826214746676623464795614, −3.56003165572438437765544388416, −3.47298187752638806643707296029, −3.32656690640988220111635832077, −3.26880271112861081151020080827, −2.81494054093281100547513340977, −2.40985550958983670632234138699, −2.34333176843061137602373996877, −2.30917685267195249132425368136, −1.97368464800869018185279543489, −1.69037935420477276579346989491, −1.58111400919361134463121861148, −1.29221345436682679157142758312, −1.17862033194694814890911773121, 1.17862033194694814890911773121, 1.29221345436682679157142758312, 1.58111400919361134463121861148, 1.69037935420477276579346989491, 1.97368464800869018185279543489, 2.30917685267195249132425368136, 2.34333176843061137602373996877, 2.40985550958983670632234138699, 2.81494054093281100547513340977, 3.26880271112861081151020080827, 3.32656690640988220111635832077, 3.47298187752638806643707296029, 3.56003165572438437765544388416, 3.72641826214746676623464795614, 4.23819609001603146265951454947, 4.39977360149628608971024156455, 4.40386142409479485294837508159, 4.50436271667587628046580375614, 4.68183450611373443810672685336, 4.72104759036527617502369202200, 4.86211460293723244255756860377, 5.25578005259739536688006626219, 5.58352720198086906952080408212, 5.67274341064318617025222141471, 5.70499917694634813858331316360
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.