Properties

Degree 14
Conductor $ 3^{7} \cdot 7^{14} \cdot 41^{7} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 7·3-s + 5·4-s − 5-s − 28·6-s + 28·9-s − 4·10-s + 11·11-s − 35·12-s + 7·13-s + 7·15-s − 6·16-s − 11·17-s + 112·18-s − 4·19-s − 5·20-s + 44·22-s + 7·23-s − 16·25-s + 28·26-s − 84·27-s + 4·29-s + 28·30-s + 7·31-s − 10·32-s − 77·33-s − 44·34-s + ⋯
L(s)  = 1  + 2.82·2-s − 4.04·3-s + 5/2·4-s − 0.447·5-s − 11.4·6-s + 28/3·9-s − 1.26·10-s + 3.31·11-s − 10.1·12-s + 1.94·13-s + 1.80·15-s − 3/2·16-s − 2.66·17-s + 26.3·18-s − 0.917·19-s − 1.11·20-s + 9.38·22-s + 1.45·23-s − 3.19·25-s + 5.49·26-s − 16.1·27-s + 0.742·29-s + 5.11·30-s + 1.25·31-s − 1.76·32-s − 13.4·33-s − 7.54·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{14} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{7} \cdot 7^{14} \cdot 41^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(3^{7} \cdot 7^{14} \cdot 41^{7}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(14,\ 3^{7} \cdot 7^{14} \cdot 41^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )$
$L(1)$  $\approx$  $3.787298968$
$L(\frac12)$  $\approx$  $3.787298968$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p(T)\) is a polynomial of degree 14. If $p \in \{3,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 13.
$p$$F_p(T)$
bad3 \( ( 1 + T )^{7} \)
7 \( 1 \)
41 \( ( 1 + T )^{7} \)
good2 \( 1 - p^{2} T + 11 T^{2} - 3 p^{3} T^{3} + 47 T^{4} - 41 p T^{5} + 133 T^{6} - 49 p^{2} T^{7} + 133 p T^{8} - 41 p^{3} T^{9} + 47 p^{3} T^{10} - 3 p^{7} T^{11} + 11 p^{5} T^{12} - p^{8} T^{13} + p^{7} T^{14} \)
5 \( 1 + T + 17 T^{2} + 12 T^{3} + 144 T^{4} + 72 T^{5} + 874 T^{6} + 366 T^{7} + 874 p T^{8} + 72 p^{2} T^{9} + 144 p^{3} T^{10} + 12 p^{4} T^{11} + 17 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
11 \( 1 - p T + 93 T^{2} - 50 p T^{3} + 2829 T^{4} - 12013 T^{5} + 46873 T^{6} - 160404 T^{7} + 46873 p T^{8} - 12013 p^{2} T^{9} + 2829 p^{3} T^{10} - 50 p^{5} T^{11} + 93 p^{5} T^{12} - p^{7} T^{13} + p^{7} T^{14} \)
13 \( 1 - 7 T + 77 T^{2} - 432 T^{3} + 2556 T^{4} - 12040 T^{5} + 50292 T^{6} - 197854 T^{7} + 50292 p T^{8} - 12040 p^{2} T^{9} + 2556 p^{3} T^{10} - 432 p^{4} T^{11} + 77 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
17 \( 1 + 11 T + 113 T^{2} + 778 T^{3} + 295 p T^{4} + 26849 T^{5} + 133003 T^{6} + 573436 T^{7} + 133003 p T^{8} + 26849 p^{2} T^{9} + 295 p^{4} T^{10} + 778 p^{4} T^{11} + 113 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \)
19 \( 1 + 4 T + 55 T^{2} + 144 T^{3} + 1355 T^{4} + 2800 T^{5} + 30893 T^{6} + 58744 T^{7} + 30893 p T^{8} + 2800 p^{2} T^{9} + 1355 p^{3} T^{10} + 144 p^{4} T^{11} + 55 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14} \)
23 \( 1 - 7 T + 91 T^{2} - 416 T^{3} + 3674 T^{4} - 12912 T^{5} + 98352 T^{6} - 293570 T^{7} + 98352 p T^{8} - 12912 p^{2} T^{9} + 3674 p^{3} T^{10} - 416 p^{4} T^{11} + 91 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
29 \( 1 - 4 T + 150 T^{2} - 604 T^{3} + 10745 T^{4} - 40100 T^{5} + 472193 T^{6} - 1504974 T^{7} + 472193 p T^{8} - 40100 p^{2} T^{9} + 10745 p^{3} T^{10} - 604 p^{4} T^{11} + 150 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \)
31 \( 1 - 7 T + 177 T^{2} - 1204 T^{3} + 14369 T^{4} - 88973 T^{5} + 693757 T^{6} - 3619024 T^{7} + 693757 p T^{8} - 88973 p^{2} T^{9} + 14369 p^{3} T^{10} - 1204 p^{4} T^{11} + 177 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
37 \( 1 + 118 T^{2} + 22 T^{3} + 7423 T^{4} - 1260 T^{5} + 346623 T^{6} - 120738 T^{7} + 346623 p T^{8} - 1260 p^{2} T^{9} + 7423 p^{3} T^{10} + 22 p^{4} T^{11} + 118 p^{5} T^{12} + p^{7} T^{14} \)
43 \( 1 - T + 199 T^{2} + 74 T^{3} + 19163 T^{4} + 15837 T^{5} + 1210385 T^{6} + 921684 T^{7} + 1210385 p T^{8} + 15837 p^{2} T^{9} + 19163 p^{3} T^{10} + 74 p^{4} T^{11} + 199 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \)
47 \( 1 + 14 T + 230 T^{2} + 2300 T^{3} + 22871 T^{4} + 175728 T^{5} + 1388785 T^{6} + 9267708 T^{7} + 1388785 p T^{8} + 175728 p^{2} T^{9} + 22871 p^{3} T^{10} + 2300 p^{4} T^{11} + 230 p^{5} T^{12} + 14 p^{6} T^{13} + p^{7} T^{14} \)
53 \( 1 - 23 T + 403 T^{2} - 5284 T^{3} + 60362 T^{4} - 576532 T^{5} + 5009514 T^{6} - 38172514 T^{7} + 5009514 p T^{8} - 576532 p^{2} T^{9} + 60362 p^{3} T^{10} - 5284 p^{4} T^{11} + 403 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \)
59 \( 1 + 8 T + 309 T^{2} + 2188 T^{3} + 42973 T^{4} + 268692 T^{5} + 3667649 T^{6} + 19759184 T^{7} + 3667649 p T^{8} + 268692 p^{2} T^{9} + 42973 p^{3} T^{10} + 2188 p^{4} T^{11} + 309 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \)
61 \( 1 - 3 T + 253 T^{2} - 1292 T^{3} + 32651 T^{4} - 177077 T^{5} + 2922699 T^{6} - 13311088 T^{7} + 2922699 p T^{8} - 177077 p^{2} T^{9} + 32651 p^{3} T^{10} - 1292 p^{4} T^{11} + 253 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
67 \( 1 - 3 T + 195 T^{2} - 754 T^{3} + 20178 T^{4} - 83910 T^{5} + 1549642 T^{6} - 6051514 T^{7} + 1549642 p T^{8} - 83910 p^{2} T^{9} + 20178 p^{3} T^{10} - 754 p^{4} T^{11} + 195 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \)
71 \( 1 - 7 T + 323 T^{2} - 2160 T^{3} + 53359 T^{4} - 311321 T^{5} + 5588401 T^{6} - 27474912 T^{7} + 5588401 p T^{8} - 311321 p^{2} T^{9} + 53359 p^{3} T^{10} - 2160 p^{4} T^{11} + 323 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \)
73 \( 1 - 11 T + 433 T^{2} - 3456 T^{3} + 78235 T^{4} - 476001 T^{5} + 8321367 T^{6} - 41221920 T^{7} + 8321367 p T^{8} - 476001 p^{2} T^{9} + 78235 p^{3} T^{10} - 3456 p^{4} T^{11} + 433 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} \)
79 \( 1 + T + 331 T^{2} + 382 T^{3} + 52054 T^{4} + 82270 T^{5} + 5419866 T^{6} + 9127174 T^{7} + 5419866 p T^{8} + 82270 p^{2} T^{9} + 52054 p^{3} T^{10} + 382 p^{4} T^{11} + 331 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \)
83 \( 1 + 193 T^{2} + 1296 T^{3} + 19473 T^{4} + 190016 T^{5} + 2510665 T^{6} + 13606752 T^{7} + 2510665 p T^{8} + 190016 p^{2} T^{9} + 19473 p^{3} T^{10} + 1296 p^{4} T^{11} + 193 p^{5} T^{12} + p^{7} T^{14} \)
89 \( 1 + 32 T + 847 T^{2} + 14824 T^{3} + 230117 T^{4} + 2864080 T^{5} + 32862267 T^{6} + 320741968 T^{7} + 32862267 p T^{8} + 2864080 p^{2} T^{9} + 230117 p^{3} T^{10} + 14824 p^{4} T^{11} + 847 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} \)
97 \( 1 - 25 T + 665 T^{2} - 11462 T^{3} + 182540 T^{4} - 2417768 T^{5} + 28408058 T^{6} - 298204818 T^{7} + 28408058 p T^{8} - 2417768 p^{2} T^{9} + 182540 p^{3} T^{10} - 11462 p^{4} T^{11} + 665 p^{5} T^{12} - 25 p^{6} T^{13} + p^{7} T^{14} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.82912168522317938107904624549, −3.76599130895314934051712405594, −3.75244070943888777985450118221, −3.71124954793088503663117972923, −3.33317781049080442061146920682, −3.08890126228408164816503625611, −3.03504558777575910235490997944, −3.02208945120627413066884011123, −2.78536874216101350051918263477, −2.51819846827693960544078551454, −2.37016070939859386023541339324, −2.23677562015477554710393427098, −2.05756731137298783008072924090, −1.88270931436110004607930638424, −1.78921727718960751016433582635, −1.61273240860405541332901706840, −1.49214575989798564887943244507, −1.42209026368258556837820450263, −1.40396202301518336003023647582, −0.888067159259169045486882482614, −0.846621230616195102577836592232, −0.73366716070138387665287298533, −0.46976463337647541018749586675, −0.46886837091238763160572202084, −0.14864401419505364942586242449, 0.14864401419505364942586242449, 0.46886837091238763160572202084, 0.46976463337647541018749586675, 0.73366716070138387665287298533, 0.846621230616195102577836592232, 0.888067159259169045486882482614, 1.40396202301518336003023647582, 1.42209026368258556837820450263, 1.49214575989798564887943244507, 1.61273240860405541332901706840, 1.78921727718960751016433582635, 1.88270931436110004607930638424, 2.05756731137298783008072924090, 2.23677562015477554710393427098, 2.37016070939859386023541339324, 2.51819846827693960544078551454, 2.78536874216101350051918263477, 3.02208945120627413066884011123, 3.03504558777575910235490997944, 3.08890126228408164816503625611, 3.33317781049080442061146920682, 3.71124954793088503663117972923, 3.75244070943888777985450118221, 3.76599130895314934051712405594, 3.82912168522317938107904624549

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.