Dirichlet series
| L(s) = 1 | + 7·3-s − 4·5-s + 6·7-s − 2·8-s + 28·9-s − 4·11-s + 4·13-s − 28·15-s − 16-s + 12·17-s + 42·21-s + 18·23-s − 14·24-s − 11·25-s + 84·27-s + 18·31-s − 2·32-s − 28·33-s − 24·35-s + 3·37-s + 28·39-s + 8·40-s − 11·41-s + 26·43-s − 112·45-s + 17·47-s − 7·48-s + ⋯ |
| L(s) = 1 | + 4.04·3-s − 1.78·5-s + 2.26·7-s − 0.707·8-s + 28/3·9-s − 1.20·11-s + 1.10·13-s − 7.22·15-s − 1/4·16-s + 2.91·17-s + 9.16·21-s + 3.75·23-s − 2.85·24-s − 2.19·25-s + 16.1·27-s + 3.23·31-s − 0.353·32-s − 4.87·33-s − 4.05·35-s + 0.493·37-s + 4.48·39-s + 1.26·40-s − 1.71·41-s + 3.96·43-s − 16.6·45-s + 2.47·47-s − 1.01·48-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 139^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 139^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 139^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4538.28\) |
| Root analytic conductor: | \(1.82476\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 139^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(25.64791642\) |
| \(L(\frac12)\) | \(\approx\) | \(25.64791642\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 139 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + p T^{3} + T^{4} + p T^{5} + 3 p T^{6} + 3 p^{2} T^{8} + p^{3} T^{9} + p^{3} T^{10} + p^{5} T^{11} + p^{7} T^{14} \) |
| 5 | \( 1 + 4 T + 27 T^{2} + 89 T^{3} + 66 p T^{4} + 921 T^{5} + 2474 T^{6} + 5764 T^{7} + 2474 p T^{8} + 921 p^{2} T^{9} + 66 p^{4} T^{10} + 89 p^{4} T^{11} + 27 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 - 6 T + 25 T^{2} - 87 T^{3} + 310 T^{4} - 983 T^{5} + 438 p T^{6} - 1224 p T^{7} + 438 p^{2} T^{8} - 983 p^{2} T^{9} + 310 p^{3} T^{10} - 87 p^{4} T^{11} + 25 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 4 T + 36 T^{2} + 129 T^{3} + 841 T^{4} + 2710 T^{5} + 13015 T^{6} + 34143 T^{7} + 13015 p T^{8} + 2710 p^{2} T^{9} + 841 p^{3} T^{10} + 129 p^{4} T^{11} + 36 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 4 T + 50 T^{2} - 165 T^{3} + 1103 T^{4} - 2962 T^{5} + 15671 T^{6} - 38711 T^{7} + 15671 p T^{8} - 2962 p^{2} T^{9} + 1103 p^{3} T^{10} - 165 p^{4} T^{11} + 50 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 12 T + 160 T^{2} - 1234 T^{3} + 9430 T^{4} - 52636 T^{5} + 284185 T^{6} - 1194316 T^{7} + 284185 p T^{8} - 52636 p^{2} T^{9} + 9430 p^{3} T^{10} - 1234 p^{4} T^{11} + 160 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 50 T^{2} + 154 T^{3} + 1400 T^{4} + 6824 T^{5} + 33291 T^{6} + 163644 T^{7} + 33291 p T^{8} + 6824 p^{2} T^{9} + 1400 p^{3} T^{10} + 154 p^{4} T^{11} + 50 p^{5} T^{12} + p^{7} T^{14} \) | |
| 23 | \( 1 - 18 T + 217 T^{2} - 1940 T^{3} + 14557 T^{4} - 93582 T^{5} + 531765 T^{6} - 2688472 T^{7} + 531765 p T^{8} - 93582 p^{2} T^{9} + 14557 p^{3} T^{10} - 1940 p^{4} T^{11} + 217 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 133 T^{2} - 113 T^{3} + 8326 T^{4} - 11553 T^{5} + 335868 T^{6} - 480328 T^{7} + 335868 p T^{8} - 11553 p^{2} T^{9} + 8326 p^{3} T^{10} - 113 p^{4} T^{11} + 133 p^{5} T^{12} + p^{7} T^{14} \) | |
| 31 | \( 1 - 18 T + 317 T^{2} - 3425 T^{3} + 34718 T^{4} - 266373 T^{5} + 1913062 T^{6} - 11006524 T^{7} + 1913062 p T^{8} - 266373 p^{2} T^{9} + 34718 p^{3} T^{10} - 3425 p^{4} T^{11} + 317 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 3 T + 89 T^{2} - 64 T^{3} + 3732 T^{4} - 5208 T^{5} + 174858 T^{6} - 445370 T^{7} + 174858 p T^{8} - 5208 p^{2} T^{9} + 3732 p^{3} T^{10} - 64 p^{4} T^{11} + 89 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 11 T + 182 T^{2} + 1039 T^{3} + 9072 T^{4} + 10177 T^{5} + 90433 T^{6} - 1161910 T^{7} + 90433 p T^{8} + 10177 p^{2} T^{9} + 9072 p^{3} T^{10} + 1039 p^{4} T^{11} + 182 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 26 T + 514 T^{2} - 7028 T^{3} + 80628 T^{4} - 754934 T^{5} + 6166655 T^{6} - 43044312 T^{7} + 6166655 p T^{8} - 754934 p^{2} T^{9} + 80628 p^{3} T^{10} - 7028 p^{4} T^{11} + 514 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 17 T + 207 T^{2} - 1528 T^{3} + 10624 T^{4} - 67432 T^{5} + 254 p^{2} T^{6} - 3941438 T^{7} + 254 p^{3} T^{8} - 67432 p^{2} T^{9} + 10624 p^{3} T^{10} - 1528 p^{4} T^{11} + 207 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 14 T + 292 T^{2} + 2868 T^{3} + 32590 T^{4} + 247650 T^{5} + 2119737 T^{6} + 14306520 T^{7} + 2119737 p T^{8} + 247650 p^{2} T^{9} + 32590 p^{3} T^{10} + 2868 p^{4} T^{11} + 292 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 20 T + 417 T^{2} + 5440 T^{3} + 67321 T^{4} + 681196 T^{5} + 6289953 T^{6} + 51029568 T^{7} + 6289953 p T^{8} + 681196 p^{2} T^{9} + 67321 p^{3} T^{10} + 5440 p^{4} T^{11} + 417 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 28 T + 603 T^{2} - 8944 T^{3} + 114525 T^{4} - 1196996 T^{5} + 11324543 T^{6} - 92185248 T^{7} + 11324543 p T^{8} - 1196996 p^{2} T^{9} + 114525 p^{3} T^{10} - 8944 p^{4} T^{11} + 603 p^{5} T^{12} - 28 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 10 T + 245 T^{2} + 1831 T^{3} + 29670 T^{4} + 215047 T^{5} + 2692706 T^{6} + 17671176 T^{7} + 2692706 p T^{8} + 215047 p^{2} T^{9} + 29670 p^{3} T^{10} + 1831 p^{4} T^{11} + 245 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 18 T + 427 T^{2} + 4959 T^{3} + 73022 T^{4} + 665863 T^{5} + 7617072 T^{6} + 57350816 T^{7} + 7617072 p T^{8} + 665863 p^{2} T^{9} + 73022 p^{3} T^{10} + 4959 p^{4} T^{11} + 427 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 12 T + 419 T^{2} + 3696 T^{3} + 74937 T^{4} + 522004 T^{5} + 8031507 T^{6} + 46161120 T^{7} + 8031507 p T^{8} + 522004 p^{2} T^{9} + 74937 p^{3} T^{10} + 3696 p^{4} T^{11} + 419 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 22 T + 585 T^{2} - 8531 T^{3} + 136126 T^{4} - 1507987 T^{5} + 17635562 T^{6} - 153092920 T^{7} + 17635562 p T^{8} - 1507987 p^{2} T^{9} + 136126 p^{3} T^{10} - 8531 p^{4} T^{11} + 585 p^{5} T^{12} - 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 12 T + 165 T^{2} + 627 T^{3} + 11882 T^{4} + 63343 T^{5} + 1369262 T^{6} + 6581564 T^{7} + 1369262 p T^{8} + 63343 p^{2} T^{9} + 11882 p^{3} T^{10} + 627 p^{4} T^{11} + 165 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 16 T + 325 T^{2} + 3973 T^{3} + 53030 T^{4} + 519797 T^{5} + 5608352 T^{6} + 50025728 T^{7} + 5608352 p T^{8} + 519797 p^{2} T^{9} + 53030 p^{3} T^{10} + 3973 p^{4} T^{11} + 325 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 30 T + 659 T^{2} - 9812 T^{3} + 134577 T^{4} - 1498850 T^{5} + 16270971 T^{6} - 155872088 T^{7} + 16270971 p T^{8} - 1498850 p^{2} T^{9} + 134577 p^{3} T^{10} - 9812 p^{4} T^{11} + 659 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−5.30253177990692266633875363161, −5.18646250625511399341758963607, −5.15867873752200297138623069528, −4.83973121210301381371019204379, −4.62162340943164206217750810572, −4.53103992560110387742456939417, −4.38439765801079279080832255778, −4.10176646342259468160779017626, −4.06630069110904826224290596021, −4.04455229798956289321489009574, −3.75639292049642095315199411348, −3.40022983815217139027780086370, −3.38741049304550563467011909415, −3.09350225034980259688646674330, −3.04859729400594740900503530977, −3.02930439983317955208121900874, −2.86568615436856508692502433221, −2.43924455647890283113485549602, −2.24268924897835293060453479201, −2.15780199069187460067420175654, −1.95915175836001310290591050133, −1.34755703024373319568217144114, −1.34373517990069237372044718990, −1.01419269916534562517705924470, −0.892571915075470521168122346749, 0.892571915075470521168122346749, 1.01419269916534562517705924470, 1.34373517990069237372044718990, 1.34755703024373319568217144114, 1.95915175836001310290591050133, 2.15780199069187460067420175654, 2.24268924897835293060453479201, 2.43924455647890283113485549602, 2.86568615436856508692502433221, 3.02930439983317955208121900874, 3.04859729400594740900503530977, 3.09350225034980259688646674330, 3.38741049304550563467011909415, 3.40022983815217139027780086370, 3.75639292049642095315199411348, 4.04455229798956289321489009574, 4.06630069110904826224290596021, 4.10176646342259468160779017626, 4.38439765801079279080832255778, 4.53103992560110387742456939417, 4.62162340943164206217750810572, 4.83973121210301381371019204379, 5.15867873752200297138623069528, 5.18646250625511399341758963607, 5.30253177990692266633875363161
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.