Properties

Degree 16
Conductor $ 2^{40} \cdot 3^{24} \cdot 7^{8} $
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  + 8·11-s − 8·13-s + 8·23-s + 16·25-s + 40·37-s − 16·47-s − 4·49-s + 64·59-s − 16·61-s − 72·71-s + 24·73-s − 32·83-s + 48·97-s + 48·107-s − 16·121-s + 127-s + 131-s + 137-s + 139-s − 64·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯
L(s)  = 1  + 2.41·11-s − 2.21·13-s + 1.66·23-s + 16/5·25-s + 6.57·37-s − 2.33·47-s − 4/7·49-s + 8.33·59-s − 2.04·61-s − 8.54·71-s + 2.80·73-s − 3.51·83-s + 4.87·97-s + 4.64·107-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.35·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(16\)
\( N \)  =  \(2^{40} \cdot 3^{24} \cdot 7^{8}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $32.44891025$
$L(\frac12)$  $\approx$  $32.44891025$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 16. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 15.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{4} \)
good5 \( 1 - 16 T^{2} + 174 T^{4} - 256 p T^{6} + 7379 T^{8} - 256 p^{3} T^{10} + 174 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 - 4 T + 32 T^{2} - 136 T^{3} + 463 T^{4} - 136 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 4 T + 36 T^{2} + 116 T^{3} + 602 T^{4} + 116 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 56 T^{2} + 1330 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 26 T^{2} + 795 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 4 T + 32 T^{2} - 40 T^{3} + 655 T^{4} - 40 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( 1 - 64 T^{2} + 4452 T^{4} - 163520 T^{6} + 6050150 T^{8} - 163520 p^{2} T^{10} + 4452 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \)
31 \( 1 - 116 T^{2} + 7098 T^{4} - 315760 T^{6} + 11036483 T^{8} - 315760 p^{2} T^{10} + 7098 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 - 10 T + 91 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( 1 - 288 T^{2} + 37342 T^{4} - 2869632 T^{6} + 143812995 T^{8} - 2869632 p^{2} T^{10} + 37342 p^{4} T^{12} - 288 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 40 T^{2} + 5668 T^{4} - 165496 T^{6} + 14065894 T^{8} - 165496 p^{2} T^{10} + 5668 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 4 T + 96 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
53 \( 1 - 136 T^{2} + 6300 T^{4} + 118408 T^{6} - 20822362 T^{8} + 118408 p^{2} T^{10} + 6300 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 - 32 T + 584 T^{2} - 7136 T^{3} + 63778 T^{4} - 7136 p T^{5} + 584 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( 1 - 200 T^{2} + 27492 T^{4} - 2608600 T^{6} + 201429158 T^{8} - 2608600 p^{2} T^{10} + 27492 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 + 36 T + 760 T^{2} + 10392 T^{3} + 103479 T^{4} + 10392 p T^{5} + 760 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 12 T + 228 T^{2} - 1644 T^{3} + 20234 T^{4} - 1644 p T^{5} + 228 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( 1 - 424 T^{2} + 86788 T^{4} - 11475640 T^{6} + 1070068678 T^{8} - 11475640 p^{2} T^{10} + 86788 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 16 T + 88 T^{2} + 560 T^{3} + 8322 T^{4} + 560 p T^{5} + 88 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 - 496 T^{2} + 119790 T^{4} - 18269696 T^{6} + 1929877235 T^{8} - 18269696 p^{2} T^{10} + 119790 p^{4} T^{12} - 496 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 - 24 T + 444 T^{2} - 5928 T^{3} + 63110 T^{4} - 5928 p T^{5} + 444 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.18911033557827748992604948350, −3.17651300696022795769662876076, −2.99719430008900515706932125422, −2.91376992838494018089215133427, −2.85499499830057701871566955108, −2.75621253919212765250398720042, −2.67445716043996174366511224728, −2.57153215485631563335810402541, −2.38564921050900223056319735961, −2.26396220984973643585930574238, −2.19645680772829071272224434621, −2.08198856930287828568411354558, −2.01869548569584547670410123845, −1.88791899265750238759320607617, −1.47123304723352687171768168702, −1.33369040534124816482429740362, −1.33188331343395734967348885463, −1.32092748651853921460799368334, −1.21302322791734309803042531012, −0.886379589141649775447588503392, −0.874405994031680972316733739956, −0.77759432400983154397563839760, −0.49804155593539625233567121315, −0.34256551091459152926923513888, −0.33316073626517574743890114605, 0.33316073626517574743890114605, 0.34256551091459152926923513888, 0.49804155593539625233567121315, 0.77759432400983154397563839760, 0.874405994031680972316733739956, 0.886379589141649775447588503392, 1.21302322791734309803042531012, 1.32092748651853921460799368334, 1.33188331343395734967348885463, 1.33369040534124816482429740362, 1.47123304723352687171768168702, 1.88791899265750238759320607617, 2.01869548569584547670410123845, 2.08198856930287828568411354558, 2.19645680772829071272224434621, 2.26396220984973643585930574238, 2.38564921050900223056319735961, 2.57153215485631563335810402541, 2.67445716043996174366511224728, 2.75621253919212765250398720042, 2.85499499830057701871566955108, 2.91376992838494018089215133427, 2.99719430008900515706932125422, 3.17651300696022795769662876076, 3.18911033557827748992604948350

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

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