Dirichlet series
| L(s) = 1 | + 2·2-s + 7·3-s + 4-s − 2·5-s + 14·6-s + 4·7-s − 2·8-s + 28·9-s − 4·10-s + 2·11-s + 7·12-s + 6·13-s + 8·14-s − 14·15-s − 7·16-s − 8·17-s + 56·18-s + 4·19-s − 2·20-s + 28·21-s + 4·22-s + 8·23-s − 14·24-s − 6·25-s + 12·26-s + 84·27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4.04·3-s + 1/2·4-s − 0.894·5-s + 5.71·6-s + 1.51·7-s − 0.707·8-s + 28/3·9-s − 1.26·10-s + 0.603·11-s + 2.02·12-s + 1.66·13-s + 2.13·14-s − 3.61·15-s − 7/4·16-s − 1.94·17-s + 13.1·18-s + 0.917·19-s − 0.447·20-s + 6.11·21-s + 0.852·22-s + 1.66·23-s − 2.85·24-s − 6/5·25-s + 2.35·26-s + 16.1·27-s + 0.755·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 79^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 79^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 79^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(86.9316\) |
| Root analytic conductor: | \(1.37566\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 79^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.61422370\) |
| \(L(\frac12)\) | \(\approx\) | \(23.61422370\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 79 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - p T + 3 T^{2} - p T^{3} + p^{2} T^{4} - 9 T^{5} + 9 p T^{6} - 29 T^{7} + 9 p^{2} T^{8} - 9 p^{2} T^{9} + p^{5} T^{10} - p^{5} T^{11} + 3 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 2 T + 2 p T^{2} + 28 T^{3} + 91 T^{4} + 212 T^{5} + 574 T^{6} + 1252 T^{7} + 574 p T^{8} + 212 p^{2} T^{9} + 91 p^{3} T^{10} + 28 p^{4} T^{11} + 2 p^{6} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 4 T + 26 T^{2} - 10 p T^{3} + 236 T^{4} - 460 T^{5} + 1035 T^{6} - 2196 T^{7} + 1035 p T^{8} - 460 p^{2} T^{9} + 236 p^{3} T^{10} - 10 p^{5} T^{11} + 26 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 2 T + 35 T^{2} - 92 T^{3} + 647 T^{4} - 1922 T^{5} + 8882 T^{6} - 25228 T^{7} + 8882 p T^{8} - 1922 p^{2} T^{9} + 647 p^{3} T^{10} - 92 p^{4} T^{11} + 35 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 6 T + 75 T^{2} - 274 T^{3} + 1981 T^{4} - 4506 T^{5} + 28948 T^{6} - 50850 T^{7} + 28948 p T^{8} - 4506 p^{2} T^{9} + 1981 p^{3} T^{10} - 274 p^{4} T^{11} + 75 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 8 T + 58 T^{2} + 274 T^{3} + 1828 T^{4} + 8632 T^{5} + 43073 T^{6} + 159548 T^{7} + 43073 p T^{8} + 8632 p^{2} T^{9} + 1828 p^{3} T^{10} + 274 p^{4} T^{11} + 58 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 4 T + 62 T^{2} - 312 T^{3} + 2089 T^{4} - 11492 T^{5} + 53852 T^{6} - 266384 T^{7} + 53852 p T^{8} - 11492 p^{2} T^{9} + 2089 p^{3} T^{10} - 312 p^{4} T^{11} + 62 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 8 T + 137 T^{2} - 772 T^{3} + 337 p T^{4} - 34138 T^{5} + 261448 T^{6} - 950172 T^{7} + 261448 p T^{8} - 34138 p^{2} T^{9} + 337 p^{4} T^{10} - 772 p^{4} T^{11} + 137 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 10 T + 114 T^{2} + 840 T^{3} + 7444 T^{4} + 47574 T^{5} + 311525 T^{6} + 1600752 T^{7} + 311525 p T^{8} + 47574 p^{2} T^{9} + 7444 p^{3} T^{10} + 840 p^{4} T^{11} + 114 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 4 T + 138 T^{2} - 348 T^{3} + 9049 T^{4} - 16880 T^{5} + 397000 T^{6} - 613472 T^{7} + 397000 p T^{8} - 16880 p^{2} T^{9} + 9049 p^{3} T^{10} - 348 p^{4} T^{11} + 138 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 10 T + 163 T^{2} - 36 p T^{3} + 13421 T^{4} - 2606 p T^{5} + 717047 T^{6} - 4367384 T^{7} + 717047 p T^{8} - 2606 p^{3} T^{9} + 13421 p^{3} T^{10} - 36 p^{5} T^{11} + 163 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 20 T + 303 T^{2} + 3464 T^{3} + 34933 T^{4} + 302412 T^{5} + 2301099 T^{6} + 15564336 T^{7} + 2301099 p T^{8} + 302412 p^{2} T^{9} + 34933 p^{3} T^{10} + 3464 p^{4} T^{11} + 303 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 22 T + 362 T^{2} - 4096 T^{3} + 41340 T^{4} - 352314 T^{5} + 2798799 T^{6} - 19169920 T^{7} + 2798799 p T^{8} - 352314 p^{2} T^{9} + 41340 p^{3} T^{10} - 4096 p^{4} T^{11} + 362 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 10 T + 225 T^{2} + 1668 T^{3} + 20573 T^{4} + 117238 T^{5} + 1143773 T^{6} + 5726584 T^{7} + 1143773 p T^{8} + 117238 p^{2} T^{9} + 20573 p^{3} T^{10} + 1668 p^{4} T^{11} + 225 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 135 T^{2} - 96 T^{3} + 11281 T^{4} - 31072 T^{5} + 642143 T^{6} - 2090496 T^{7} + 642143 p T^{8} - 31072 p^{2} T^{9} + 11281 p^{3} T^{10} - 96 p^{4} T^{11} + 135 p^{5} T^{12} + p^{7} T^{14} \) | |
| 59 | \( 1 + 2 T + 49 T^{2} + 716 T^{3} + 10537 T^{4} + 20366 T^{5} + 595329 T^{6} + 4889512 T^{7} + 595329 p T^{8} + 20366 p^{2} T^{9} + 10537 p^{3} T^{10} + 716 p^{4} T^{11} + 49 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 235 T^{2} - 936 T^{3} + 30301 T^{4} - 110204 T^{5} + 2627055 T^{6} - 8298736 T^{7} + 2627055 p T^{8} - 110204 p^{2} T^{9} + 30301 p^{3} T^{10} - 936 p^{4} T^{11} + 235 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 20 T + 378 T^{2} - 5132 T^{3} + 63293 T^{4} - 659568 T^{5} + 6462696 T^{6} - 54343904 T^{7} + 6462696 p T^{8} - 659568 p^{2} T^{9} + 63293 p^{3} T^{10} - 5132 p^{4} T^{11} + 378 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 20 T + 537 T^{2} + 7712 T^{3} + 119181 T^{4} + 1293004 T^{5} + 14341893 T^{6} + 120175744 T^{7} + 14341893 p T^{8} + 1293004 p^{2} T^{9} + 119181 p^{3} T^{10} + 7712 p^{4} T^{11} + 537 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 2 T + 291 T^{2} - 614 T^{3} + 42469 T^{4} - 99734 T^{5} + 4203292 T^{6} - 9581654 T^{7} + 4203292 p T^{8} - 99734 p^{2} T^{9} + 42469 p^{3} T^{10} - 614 p^{4} T^{11} + 291 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 10 T + 412 T^{2} - 4496 T^{3} + 82086 T^{4} - 866006 T^{5} + 10219863 T^{6} - 93230464 T^{7} + 10219863 p T^{8} - 866006 p^{2} T^{9} + 82086 p^{3} T^{10} - 4496 p^{4} T^{11} + 412 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 20 T + 644 T^{2} + 9066 T^{3} + 168861 T^{4} + 1831726 T^{5} + 24665898 T^{6} + 210567376 T^{7} + 24665898 p T^{8} + 1831726 p^{2} T^{9} + 168861 p^{3} T^{10} + 9066 p^{4} T^{11} + 644 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 14 T + 543 T^{2} - 6190 T^{3} + 136345 T^{4} - 1283466 T^{5} + 20459672 T^{6} - 157691574 T^{7} + 20459672 p T^{8} - 1283466 p^{2} T^{9} + 136345 p^{3} T^{10} - 6190 p^{4} T^{11} + 543 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.99482387937810214160069417855, −5.93609307476963158654883738660, −5.66160398423825396266209447617, −5.43989910002199156790514378497, −5.10479903849106104373775766040, −5.00480141014289318035556610625, −4.67933522798956126060697433360, −4.67624992891089875121224839261, −4.47223802851529662361577193390, −4.42567554350007724024169125963, −4.27324335213620953734285309419, −3.96604259789939796092710004717, −3.70957157996684710971935390707, −3.58680098265045896559355793346, −3.54987256424385452228420250014, −3.35252959383502785930892999889, −3.28216020525515659439133933516, −2.78563646340832148365007268663, −2.60754894257669714651177621388, −2.50359637215986647218948176381, −2.22624874355911802231248594150, −1.98757174124795095559608188200, −1.55443013035193247633393789638, −1.49989269695725376023983058192, −1.10219870714847639320295312530, 1.10219870714847639320295312530, 1.49989269695725376023983058192, 1.55443013035193247633393789638, 1.98757174124795095559608188200, 2.22624874355911802231248594150, 2.50359637215986647218948176381, 2.60754894257669714651177621388, 2.78563646340832148365007268663, 3.28216020525515659439133933516, 3.35252959383502785930892999889, 3.54987256424385452228420250014, 3.58680098265045896559355793346, 3.70957157996684710971935390707, 3.96604259789939796092710004717, 4.27324335213620953734285309419, 4.42567554350007724024169125963, 4.47223802851529662361577193390, 4.67624992891089875121224839261, 4.67933522798956126060697433360, 5.00480141014289318035556610625, 5.10479903849106104373775766040, 5.43989910002199156790514378497, 5.66160398423825396266209447617, 5.93609307476963158654883738660, 5.99482387937810214160069417855
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.