Dirichlet series
| L(s) = 1 | − 3·2-s + 7·3-s − 3·5-s − 21·6-s − 5·7-s + 9·8-s + 28·9-s + 9·10-s − 4·11-s − 18·13-s + 15·14-s − 21·15-s − 11·16-s − 3·17-s − 84·18-s − 4·19-s − 35·21-s + 12·22-s + 7·23-s + 63·24-s − 17·25-s + 54·26-s + 84·27-s − 7·29-s + 63·30-s − 22·31-s + 4·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 4.04·3-s − 1.34·5-s − 8.57·6-s − 1.88·7-s + 3.18·8-s + 28/3·9-s + 2.84·10-s − 1.20·11-s − 4.99·13-s + 4.00·14-s − 5.42·15-s − 2.75·16-s − 0.727·17-s − 19.7·18-s − 0.917·19-s − 7.63·21-s + 2.55·22-s + 1.45·23-s + 12.8·24-s − 3.39·25-s + 10.5·26-s + 16.1·27-s − 1.29·29-s + 11.5·30-s − 3.95·31-s + 0.707·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 23^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{7} \cdot 23^{7} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(3^{7} \cdot 23^{7} \cdot 29^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.65870\times 10^{8}\) |
| Root analytic conductor: | \(3.99725\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 3^{7} \cdot 23^{7} \cdot 29^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T )^{7} \) |
| 23 | \( ( 1 - T )^{7} \) | |
| 29 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 + 3 T + 9 T^{2} + 9 p T^{3} + 19 p T^{4} + 31 p T^{5} + 105 T^{6} + 9 p^{4} T^{7} + 105 p T^{8} + 31 p^{3} T^{9} + 19 p^{4} T^{10} + 9 p^{5} T^{11} + 9 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) |
| 5 | \( 1 + 3 T + 26 T^{2} + 12 p T^{3} + 59 p T^{4} + 554 T^{5} + 2056 T^{6} + 3282 T^{7} + 2056 p T^{8} + 554 p^{2} T^{9} + 59 p^{4} T^{10} + 12 p^{5} T^{11} + 26 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 5 T + 41 T^{2} + 135 T^{3} + 656 T^{4} + 234 p T^{5} + 6288 T^{6} + 13196 T^{7} + 6288 p T^{8} + 234 p^{3} T^{9} + 656 p^{3} T^{10} + 135 p^{4} T^{11} + 41 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 4 T + 43 T^{2} + 177 T^{3} + 1011 T^{4} + 3683 T^{5} + 15431 T^{6} + 49616 T^{7} + 15431 p T^{8} + 3683 p^{2} T^{9} + 1011 p^{3} T^{10} + 177 p^{4} T^{11} + 43 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 18 T + 210 T^{2} + 1721 T^{3} + 11354 T^{4} + 61060 T^{5} + 21516 p T^{6} + 83504 p T^{7} + 21516 p^{2} T^{8} + 61060 p^{2} T^{9} + 11354 p^{3} T^{10} + 1721 p^{4} T^{11} + 210 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 3 T + 80 T^{2} + 188 T^{3} + 3142 T^{4} + 6147 T^{5} + 77998 T^{6} + 126724 T^{7} + 77998 p T^{8} + 6147 p^{2} T^{9} + 3142 p^{3} T^{10} + 188 p^{4} T^{11} + 80 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 4 T + 96 T^{2} + 250 T^{3} + 3793 T^{4} + 5980 T^{5} + 90955 T^{6} + 101614 T^{7} + 90955 p T^{8} + 5980 p^{2} T^{9} + 3793 p^{3} T^{10} + 250 p^{4} T^{11} + 96 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 22 T + 385 T^{2} + 4655 T^{3} + 47023 T^{4} + 387247 T^{5} + 2731617 T^{6} + 16373192 T^{7} + 2731617 p T^{8} + 387247 p^{2} T^{9} + 47023 p^{3} T^{10} + 4655 p^{4} T^{11} + 385 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 25 T + 383 T^{2} + 4139 T^{3} + 35920 T^{4} + 263242 T^{5} + 1744343 T^{6} + 10831352 T^{7} + 1744343 p T^{8} + 263242 p^{2} T^{9} + 35920 p^{3} T^{10} + 4139 p^{4} T^{11} + 383 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 13 T + 266 T^{2} + 2638 T^{3} + 30449 T^{4} + 241362 T^{5} + 2003812 T^{6} + 12720406 T^{7} + 2003812 p T^{8} + 241362 p^{2} T^{9} + 30449 p^{3} T^{10} + 2638 p^{4} T^{11} + 266 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 2 T + 167 T^{2} + 179 T^{3} + 12436 T^{4} + 420 T^{5} + 603809 T^{6} - 305536 T^{7} + 603809 p T^{8} + 420 p^{2} T^{9} + 12436 p^{3} T^{10} + 179 p^{4} T^{11} + 167 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 25 T + 432 T^{2} + 4881 T^{3} + 44174 T^{4} + 6566 p T^{5} + 1991113 T^{6} + 266592 p T^{7} + 1991113 p T^{8} + 6566 p^{3} T^{9} + 44174 p^{3} T^{10} + 4881 p^{4} T^{11} + 432 p^{5} T^{12} + 25 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 + 5 T + 162 T^{2} + 932 T^{3} + 12979 T^{4} + 102946 T^{5} + 809118 T^{6} + 7139802 T^{7} + 809118 p T^{8} + 102946 p^{2} T^{9} + 12979 p^{3} T^{10} + 932 p^{4} T^{11} + 162 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 11 T + 203 T^{2} - 1834 T^{3} + 22191 T^{4} - 181693 T^{5} + 1772309 T^{6} - 223684 p T^{7} + 1772309 p T^{8} - 181693 p^{2} T^{9} + 22191 p^{3} T^{10} - 1834 p^{4} T^{11} + 203 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 + 33 T + 797 T^{2} + 13241 T^{3} + 183526 T^{4} + 2058638 T^{5} + 20168438 T^{6} + 167441348 T^{7} + 20168438 p T^{8} + 2058638 p^{2} T^{9} + 183526 p^{3} T^{10} + 13241 p^{4} T^{11} + 797 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 8 T + 182 T^{2} - 2291 T^{3} + 24185 T^{4} - 260917 T^{5} + 2432692 T^{6} - 20277336 T^{7} + 2432692 p T^{8} - 260917 p^{2} T^{9} + 24185 p^{3} T^{10} - 2291 p^{4} T^{11} + 182 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 6 T + 302 T^{2} + 1347 T^{3} + 42856 T^{4} + 163270 T^{5} + 4135912 T^{6} + 13977690 T^{7} + 4135912 p T^{8} + 163270 p^{2} T^{9} + 42856 p^{3} T^{10} + 1347 p^{4} T^{11} + 302 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 - 15 T + 6 p T^{2} - 5215 T^{3} + 83728 T^{4} - 810862 T^{5} + 9403735 T^{6} - 74481820 T^{7} + 9403735 p T^{8} - 810862 p^{2} T^{9} + 83728 p^{3} T^{10} - 5215 p^{4} T^{11} + 6 p^{6} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 15 T + 565 T^{2} + 6921 T^{3} + 135750 T^{4} + 1345178 T^{5} + 17992607 T^{6} + 140941514 T^{7} + 17992607 p T^{8} + 1345178 p^{2} T^{9} + 135750 p^{3} T^{10} + 6921 p^{4} T^{11} + 565 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 21 T + 408 T^{2} - 4818 T^{3} + 59881 T^{4} - 573036 T^{5} + 6250008 T^{6} - 53935978 T^{7} + 6250008 p T^{8} - 573036 p^{2} T^{9} + 59881 p^{3} T^{10} - 4818 p^{4} T^{11} + 408 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 8 T + 432 T^{2} - 2603 T^{3} + 85202 T^{4} - 395462 T^{5} + 10586330 T^{6} - 40422326 T^{7} + 10586330 p T^{8} - 395462 p^{2} T^{9} + 85202 p^{3} T^{10} - 2603 p^{4} T^{11} + 432 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - 13 T + 555 T^{2} - 6021 T^{3} + 142804 T^{4} - 1285466 T^{5} + 21777034 T^{6} - 159349292 T^{7} + 21777034 p T^{8} - 1285466 p^{2} T^{9} + 142804 p^{3} T^{10} - 6021 p^{4} T^{11} + 555 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−4.58235497223594517057033821793, −4.55044108040538804185647944625, −4.47173938147046130968370335788, −4.45387690681618710874905565352, −3.91422748192339750027640790953, −3.81429446373271666524148456148, −3.66984452968375463229237287443, −3.66120216551366245954769805317, −3.54585864728428179566093507799, −3.50301651376845247222701889284, −3.50023417849304708803727768226, −3.27851684412075141378758473367, −3.06911542463815757260558197807, −2.97158089731959578853628216902, −2.78520282663666424411302311339, −2.42250102426845090066656055727, −2.38272842662696696594179343753, −2.29396793374380473688411339290, −2.21442877862716523103219016889, −2.15497642808473395639846435875, −1.80691191805346908965086357817, −1.75791002852859904551242507440, −1.50388471664017327210206339184, −1.40243800028650936821880551474, −1.24522085264184878085266314942, 0, 0, 0, 0, 0, 0, 0, 1.24522085264184878085266314942, 1.40243800028650936821880551474, 1.50388471664017327210206339184, 1.75791002852859904551242507440, 1.80691191805346908965086357817, 2.15497642808473395639846435875, 2.21442877862716523103219016889, 2.29396793374380473688411339290, 2.38272842662696696594179343753, 2.42250102426845090066656055727, 2.78520282663666424411302311339, 2.97158089731959578853628216902, 3.06911542463815757260558197807, 3.27851684412075141378758473367, 3.50023417849304708803727768226, 3.50301651376845247222701889284, 3.54585864728428179566093507799, 3.66120216551366245954769805317, 3.66984452968375463229237287443, 3.81429446373271666524148456148, 3.91422748192339750027640790953, 4.45387690681618710874905565352, 4.47173938147046130968370335788, 4.55044108040538804185647944625, 4.58235497223594517057033821793
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.