Dirichlet series
| L(s) = 1 | + 2-s + 3-s − 3·4-s + 7·5-s + 6-s − 3·7-s − 3·8-s − 5·9-s + 7·10-s + 11-s − 3·12-s + 2·13-s − 3·14-s + 7·15-s + 16-s + 12·17-s − 5·18-s + 3·19-s − 21·20-s − 3·21-s + 22-s + 23-s − 3·24-s + 28·25-s + 2·26-s − 6·27-s + 9·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 3/2·4-s + 3.13·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s − 5/3·9-s + 2.21·10-s + 0.301·11-s − 0.866·12-s + 0.554·13-s − 0.801·14-s + 1.80·15-s + 1/4·16-s + 2.91·17-s − 1.17·18-s + 0.688·19-s − 4.69·20-s − 0.654·21-s + 0.213·22-s + 0.208·23-s − 0.612·24-s + 28/5·25-s + 0.392·26-s − 1.15·27-s + 1.70·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{7} \cdot 47^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(5^{7} \cdot 47^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(81.9246\) |
| Root analytic conductor: | \(1.36984\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 5^{7} \cdot 47^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(4.640044848\) |
| \(L(\frac12)\) | \(\approx\) | \(4.640044848\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - T )^{7} \) |
| 47 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 3 p^{2} T^{4} - 13 T^{5} + 29 T^{6} - 17 p T^{7} + 29 p T^{8} - 13 p^{2} T^{9} + 3 p^{5} T^{10} - p^{6} T^{11} + p^{7} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - T + 2 p T^{2} - 5 T^{3} + 7 p T^{4} - 16 T^{5} + 22 p T^{6} - 68 T^{7} + 22 p^{2} T^{8} - 16 p^{2} T^{9} + 7 p^{4} T^{10} - 5 p^{4} T^{11} + 2 p^{6} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + 3 T + 26 T^{2} + 73 T^{3} + 45 p T^{4} + 750 T^{5} + 2580 T^{6} + 5420 T^{7} + 2580 p T^{8} + 750 p^{2} T^{9} + 45 p^{4} T^{10} + 73 p^{4} T^{11} + 26 p^{5} T^{12} + 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - T + 31 T^{2} - 26 T^{3} + 523 T^{4} - 135 T^{5} + 53 p^{2} T^{6} + 404 T^{7} + 53 p^{3} T^{8} - 135 p^{2} T^{9} + 523 p^{3} T^{10} - 26 p^{4} T^{11} + 31 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 - 2 T + 56 T^{2} - 120 T^{3} + 1402 T^{4} - 3294 T^{5} + 22641 T^{6} - 53840 T^{7} + 22641 p T^{8} - 3294 p^{2} T^{9} + 1402 p^{3} T^{10} - 120 p^{4} T^{11} + 56 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 - 12 T + 134 T^{2} - 942 T^{3} + 6311 T^{4} - 32780 T^{5} + 9778 p T^{6} - 691380 T^{7} + 9778 p^{2} T^{8} - 32780 p^{2} T^{9} + 6311 p^{3} T^{10} - 942 p^{4} T^{11} + 134 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 - 3 T + 33 T^{2} + 42 T^{3} + 385 T^{4} + 1915 T^{5} + 16025 T^{6} + 5644 T^{7} + 16025 p T^{8} + 1915 p^{2} T^{9} + 385 p^{3} T^{10} + 42 p^{4} T^{11} + 33 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - T + 31 T^{2} - 142 T^{3} + 1767 T^{4} - 2559 T^{5} + 44969 T^{6} - 144388 T^{7} + 44969 p T^{8} - 2559 p^{2} T^{9} + 1767 p^{3} T^{10} - 142 p^{4} T^{11} + 31 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 26 T + 451 T^{2} - 5548 T^{3} + 55093 T^{4} - 445974 T^{5} + 3064863 T^{6} - 17801960 T^{7} + 3064863 p T^{8} - 445974 p^{2} T^{9} + 55093 p^{3} T^{10} - 5548 p^{4} T^{11} + 451 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 5 T + 143 T^{2} + 298 T^{3} + 7151 T^{4} - 7221 T^{5} + 187521 T^{6} - 721524 T^{7} + 187521 p T^{8} - 7221 p^{2} T^{9} + 7151 p^{3} T^{10} + 298 p^{4} T^{11} + 143 p^{5} T^{12} + 5 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 + 58 T^{2} - 2 p T^{3} + 3519 T^{4} + 816 T^{5} + 156862 T^{6} - 134988 T^{7} + 156862 p T^{8} + 816 p^{2} T^{9} + 3519 p^{3} T^{10} - 2 p^{5} T^{11} + 58 p^{5} T^{12} + p^{7} T^{14} \) | |
| 41 | \( 1 - 12 T + 251 T^{2} - 2064 T^{3} + 25761 T^{4} - 166900 T^{5} + 1590099 T^{6} - 8455520 T^{7} + 1590099 p T^{8} - 166900 p^{2} T^{9} + 25761 p^{3} T^{10} - 2064 p^{4} T^{11} + 251 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 33 T + 601 T^{2} + 7666 T^{3} + 76769 T^{4} + 641143 T^{5} + 4734545 T^{6} + 32102428 T^{7} + 4734545 p T^{8} + 641143 p^{2} T^{9} + 76769 p^{3} T^{10} + 7666 p^{4} T^{11} + 601 p^{5} T^{12} + 33 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 4 T + 240 T^{2} - 1446 T^{3} + 26813 T^{4} - 199674 T^{5} + 1927462 T^{6} - 14249280 T^{7} + 1927462 p T^{8} - 199674 p^{2} T^{9} + 26813 p^{3} T^{10} - 1446 p^{4} T^{11} + 240 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 13 T + 216 T^{2} - 1967 T^{3} + 21317 T^{4} - 188888 T^{5} + 1754228 T^{6} - 14126040 T^{7} + 1754228 p T^{8} - 188888 p^{2} T^{9} + 21317 p^{3} T^{10} - 1967 p^{4} T^{11} + 216 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 20 T + 409 T^{2} - 5988 T^{3} + 74937 T^{4} - 804614 T^{5} + 7664204 T^{6} - 62685670 T^{7} + 7664204 p T^{8} - 804614 p^{2} T^{9} + 74937 p^{3} T^{10} - 5988 p^{4} T^{11} + 409 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 + 32 T + 773 T^{2} + 13032 T^{3} + 186493 T^{4} + 2175424 T^{5} + 22307777 T^{6} + 193955056 T^{7} + 22307777 p T^{8} + 2175424 p^{2} T^{9} + 186493 p^{3} T^{10} + 13032 p^{4} T^{11} + 773 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 11 T + 270 T^{2} - 1961 T^{3} + 30913 T^{4} - 159400 T^{5} + 2348676 T^{6} - 10187352 T^{7} + 2348676 p T^{8} - 159400 p^{2} T^{9} + 30913 p^{3} T^{10} - 1961 p^{4} T^{11} + 270 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 8 T + 302 T^{2} + 2284 T^{3} + 43436 T^{4} + 291192 T^{5} + 4143933 T^{6} + 24427080 T^{7} + 4143933 p T^{8} + 291192 p^{2} T^{9} + 43436 p^{3} T^{10} + 2284 p^{4} T^{11} + 302 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 - 17 T + 396 T^{2} - 4859 T^{3} + 73293 T^{4} - 729084 T^{5} + 8457148 T^{6} - 69386608 T^{7} + 8457148 p T^{8} - 729084 p^{2} T^{9} + 73293 p^{3} T^{10} - 4859 p^{4} T^{11} + 396 p^{5} T^{12} - 17 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 19 T + 559 T^{2} - 8414 T^{3} + 136371 T^{4} - 1619909 T^{5} + 18697989 T^{6} - 174706756 T^{7} + 18697989 p T^{8} - 1619909 p^{2} T^{9} + 136371 p^{3} T^{10} - 8414 p^{4} T^{11} + 559 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 13 T + 436 T^{2} - 5793 T^{3} + 96441 T^{4} - 1118574 T^{5} + 13377890 T^{6} - 125648408 T^{7} + 13377890 p T^{8} - 1118574 p^{2} T^{9} + 96441 p^{3} T^{10} - 5793 p^{4} T^{11} + 436 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 + 12 T + 248 T^{2} + 1590 T^{3} + 20069 T^{4} + 85206 T^{5} + 1509214 T^{6} + 8987480 T^{7} + 1509214 p T^{8} + 85206 p^{2} T^{9} + 20069 p^{3} T^{10} + 1590 p^{4} T^{11} + 248 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} \) | |
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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
−6.09208765205785706565615752396, −5.77902578761272394418665934714, −5.71109758129224850851014960714, −5.48304453422927859576327657171, −5.33907830783883321036958024463, −5.13861663584814383660856735403, −4.94454218437335286507968404093, −4.89513685849246393401750191143, −4.82215384242277206107245670508, −4.73328193075728331067671079237, −4.39540090705871009886098191570, −3.94840425434104863500276766682, −3.83560618187670065595099401676, −3.49526343163694265116845512489, −3.36790271203972073352784712988, −3.24834168053119780744156365309, −3.02525494948015180747463392585, −3.00876301915524520835367965754, −2.58680613255640483266590097075, −2.55829407722626045973060035210, −2.05313512761130739950755647381, −1.83950775748515891086731247113, −1.47594235896325861679484254935, −1.05018778628061951391627360850, −0.78465220262632329084685083501, 0.78465220262632329084685083501, 1.05018778628061951391627360850, 1.47594235896325861679484254935, 1.83950775748515891086731247113, 2.05313512761130739950755647381, 2.55829407722626045973060035210, 2.58680613255640483266590097075, 3.00876301915524520835367965754, 3.02525494948015180747463392585, 3.24834168053119780744156365309, 3.36790271203972073352784712988, 3.49526343163694265116845512489, 3.83560618187670065595099401676, 3.94840425434104863500276766682, 4.39540090705871009886098191570, 4.73328193075728331067671079237, 4.82215384242277206107245670508, 4.89513685849246393401750191143, 4.94454218437335286507968404093, 5.13861663584814383660856735403, 5.33907830783883321036958024463, 5.48304453422927859576327657171, 5.71109758129224850851014960714, 5.77902578761272394418665934714, 6.09208765205785706565615752396
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.