Dirichlet series
| L(s) = 1 | − 2-s + 7·3-s − 4·4-s − 7·6-s + 7·7-s + 7·8-s + 28·9-s − 28·12-s + 5·13-s − 7·14-s + 5·16-s + 2·17-s − 28·18-s − 13·19-s + 49·21-s + 6·23-s + 49·24-s − 5·26-s + 84·27-s − 28·28-s + 9·29-s + 5·31-s − 17·32-s − 2·34-s − 112·36-s − 5·37-s + 13·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4.04·3-s − 2·4-s − 2.85·6-s + 2.64·7-s + 2.47·8-s + 28/3·9-s − 8.08·12-s + 1.38·13-s − 1.87·14-s + 5/4·16-s + 0.485·17-s − 6.59·18-s − 2.98·19-s + 10.6·21-s + 1.25·23-s + 10.0·24-s − 0.980·26-s + 16.1·27-s − 5.29·28-s + 1.67·29-s + 0.898·31-s − 3.00·32-s − 0.342·34-s − 18.6·36-s − 0.821·37-s + 2.10·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 5^{14} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 5^{14} \cdot 47^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.39976\times 10^{10}\) |
| Root analytic conductor: | \(5.30539\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 3^{7} \cdot 5^{14} \cdot 47^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(155.5399514\) |
| \(L(\frac12)\) | \(\approx\) | \(155.5399514\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 5 | \( 1 \) | |
| 47 | \( ( 1 - T )^{7} \) | |
| good | 2 | \( 1 + T + 5 T^{2} + p T^{3} + 5 p T^{4} - 5 T^{5} + 5 p T^{6} - 25 T^{7} + 5 p^{2} T^{8} - 5 p^{2} T^{9} + 5 p^{4} T^{10} + p^{5} T^{11} + 5 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 7 | \( 1 - p T + 6 p T^{2} - 23 p T^{3} + 603 T^{4} - 1832 T^{5} + 5752 T^{6} - 15161 T^{7} + 5752 p T^{8} - 1832 p^{2} T^{9} + 603 p^{3} T^{10} - 23 p^{5} T^{11} + 6 p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 54 T^{2} + 4 T^{3} + 1409 T^{4} + 124 T^{5} + 23103 T^{6} + 1765 T^{7} + 23103 p T^{8} + 124 p^{2} T^{9} + 1409 p^{3} T^{10} + 4 p^{4} T^{11} + 54 p^{5} T^{12} + p^{7} T^{14} \) | |
| 13 | \( 1 - 5 T + 64 T^{2} - 177 T^{3} + 1442 T^{4} - 1791 T^{5} + 18331 T^{6} - 9075 T^{7} + 18331 p T^{8} - 1791 p^{2} T^{9} + 1442 p^{3} T^{10} - 177 p^{4} T^{11} + 64 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 2 T + 77 T^{2} - 181 T^{3} + 2935 T^{4} - 6920 T^{5} + 72517 T^{6} - 150257 T^{7} + 72517 p T^{8} - 6920 p^{2} T^{9} + 2935 p^{3} T^{10} - 181 p^{4} T^{11} + 77 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 13 T + 125 T^{2} + 895 T^{3} + 5427 T^{4} + 27598 T^{5} + 131775 T^{6} + 580649 T^{7} + 131775 p T^{8} + 27598 p^{2} T^{9} + 5427 p^{3} T^{10} + 895 p^{4} T^{11} + 125 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 6 T + 6 p T^{2} - 29 p T^{3} + 8507 T^{4} - 33659 T^{5} + 307403 T^{6} - 987723 T^{7} + 307403 p T^{8} - 33659 p^{2} T^{9} + 8507 p^{3} T^{10} - 29 p^{5} T^{11} + 6 p^{6} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 9 T + 3 p T^{2} - 546 T^{3} + 4067 T^{4} - 21487 T^{5} + 144799 T^{6} - 731747 T^{7} + 144799 p T^{8} - 21487 p^{2} T^{9} + 4067 p^{3} T^{10} - 546 p^{4} T^{11} + 3 p^{6} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 5 T + 132 T^{2} - 716 T^{3} + 9613 T^{4} - 45652 T^{5} + 444995 T^{6} - 1783011 T^{7} + 444995 p T^{8} - 45652 p^{2} T^{9} + 9613 p^{3} T^{10} - 716 p^{4} T^{11} + 132 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 5 T + 179 T^{2} + 646 T^{3} + 15358 T^{4} + 44901 T^{5} + 838681 T^{6} + 2027479 T^{7} + 838681 p T^{8} + 44901 p^{2} T^{9} + 15358 p^{3} T^{10} + 646 p^{4} T^{11} + 179 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 - 18 T + 318 T^{2} - 3497 T^{3} + 36413 T^{4} - 295444 T^{5} + 2302094 T^{6} - 14974305 T^{7} + 2302094 p T^{8} - 295444 p^{2} T^{9} + 36413 p^{3} T^{10} - 3497 p^{4} T^{11} + 318 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 14 T + 179 T^{2} - 1684 T^{3} + 17674 T^{4} - 141264 T^{5} + 1101154 T^{6} - 7030513 T^{7} + 1101154 p T^{8} - 141264 p^{2} T^{9} + 17674 p^{3} T^{10} - 1684 p^{4} T^{11} + 179 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 20 T + 287 T^{2} - 2551 T^{3} + 20329 T^{4} - 109232 T^{5} + 625093 T^{6} - 2907329 T^{7} + 625093 p T^{8} - 109232 p^{2} T^{9} + 20329 p^{3} T^{10} - 2551 p^{4} T^{11} + 287 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 10 T + 166 T^{2} - 1097 T^{3} + 17355 T^{4} - 117565 T^{5} + 1281147 T^{6} - 6776881 T^{7} + 1281147 p T^{8} - 117565 p^{2} T^{9} + 17355 p^{3} T^{10} - 1097 p^{4} T^{11} + 166 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 8 T + 265 T^{2} + 1526 T^{3} + 34236 T^{4} + 165566 T^{5} + 3007292 T^{6} + 12482867 T^{7} + 3007292 p T^{8} + 165566 p^{2} T^{9} + 34236 p^{3} T^{10} + 1526 p^{4} T^{11} + 265 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 4 T + 251 T^{2} - 985 T^{3} + 32760 T^{4} - 115695 T^{5} + 2930050 T^{6} - 132061 p T^{7} + 2930050 p T^{8} - 115695 p^{2} T^{9} + 32760 p^{3} T^{10} - 985 p^{4} T^{11} + 251 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 12 T + 434 T^{2} - 4275 T^{3} + 84215 T^{4} - 683133 T^{5} + 9505965 T^{6} - 62470019 T^{7} + 9505965 p T^{8} - 683133 p^{2} T^{9} + 84215 p^{3} T^{10} - 4275 p^{4} T^{11} + 434 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 4 T + 299 T^{2} - 1184 T^{3} + 46132 T^{4} - 184983 T^{5} + 4723393 T^{6} - 17142053 T^{7} + 4723393 p T^{8} - 184983 p^{2} T^{9} + 46132 p^{3} T^{10} - 1184 p^{4} T^{11} + 299 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 5 T + 266 T^{2} + 91 T^{3} + 28420 T^{4} - 137768 T^{5} + 1963224 T^{6} - 18510599 T^{7} + 1963224 p T^{8} - 137768 p^{2} T^{9} + 28420 p^{3} T^{10} + 91 p^{4} T^{11} + 266 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 52 T + 1504 T^{2} - 30382 T^{3} + 476198 T^{4} - 6128199 T^{5} + 813446 p T^{6} - 653149999 T^{7} + 813446 p^{2} T^{8} - 6128199 p^{2} T^{9} + 476198 p^{3} T^{10} - 30382 p^{4} T^{11} + 1504 p^{5} T^{12} - 52 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 32 T + 761 T^{2} - 12489 T^{3} + 183768 T^{4} - 2254809 T^{5} + 25873656 T^{6} - 253822965 T^{7} + 25873656 p T^{8} - 2254809 p^{2} T^{9} + 183768 p^{3} T^{10} - 12489 p^{4} T^{11} + 761 p^{5} T^{12} - 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 12 T + 243 T^{2} + 2123 T^{3} + 32622 T^{4} + 332331 T^{5} + 4391644 T^{6} + 40355907 T^{7} + 4391644 p T^{8} + 332331 p^{2} T^{9} + 32622 p^{3} T^{10} + 2123 p^{4} T^{11} + 243 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−3.85904108490778994155982641587, −3.84337438105916651215961604834, −3.77041060326273406848131003273, −3.74395070569621403364418427928, −3.25560240357172134127908838714, −3.19851272807741442759130033955, −3.13705728270804080559026661632, −3.12771547947744154768919068020, −3.00436044815363675992006022676, −2.67359395928405364226804543524, −2.42232150909866044722657428713, −2.35918330657532430141531196647, −2.34748681444449350679937741860, −2.15306394191690340093790851674, −2.02875866431245093551470347744, −1.99670205628018385412299311690, −1.71571504787310880373659206835, −1.63698758252246552760881354628, −1.62248563600705281396708964161, −1.01830426420344204026882271983, −1.01096476985566781445041411946, −0.921892272226796321330013845035, −0.72263452177008680505011061010, −0.69419178016807065697144644436, −0.58251674099735174222063472715, 0.58251674099735174222063472715, 0.69419178016807065697144644436, 0.72263452177008680505011061010, 0.921892272226796321330013845035, 1.01096476985566781445041411946, 1.01830426420344204026882271983, 1.62248563600705281396708964161, 1.63698758252246552760881354628, 1.71571504787310880373659206835, 1.99670205628018385412299311690, 2.02875866431245093551470347744, 2.15306394191690340093790851674, 2.34748681444449350679937741860, 2.35918330657532430141531196647, 2.42232150909866044722657428713, 2.67359395928405364226804543524, 3.00436044815363675992006022676, 3.12771547947744154768919068020, 3.13705728270804080559026661632, 3.19851272807741442759130033955, 3.25560240357172134127908838714, 3.74395070569621403364418427928, 3.77041060326273406848131003273, 3.84337438105916651215961604834, 3.85904108490778994155982641587
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.