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  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.5705761482$
$L(\frac12)$  $\approx$  $0.5705761482$
$L(\frac{3}{2})$   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

−3.12639352942696345035140030263, −3.00508279960514779968296908214, −3.00087245902669297926527335831, −2.94021879970568770922825661705, −2.84064677557649676571379900505, −2.63405569111159572147835280757, −2.58790426217531541825395658355, −2.48998799819155797493753611556, −2.41009083856948803308834011849, −2.37292424376077574886755864900, −2.28316888101350791962345837764, −2.21407394102098134448469724813, −2.04233273438328819867364554648, −1.80791014689722698321022390770, −1.66746313885331905129269781127, −1.53969361647884991701933662989, −1.33685097044150679318386684295, −1.25279091536722655624917797072, −1.14398684132876561380804447281, −0.849805310832086401744539908103, −0.70143804855573540336420300465, −0.59000010012690945533322683630, −0.55284156046337761778354619187, −0.42933589368998279855420924295, −0.04667426789876708475996127399, 0.04667426789876708475996127399, 0.42933589368998279855420924295, 0.55284156046337761778354619187, 0.59000010012690945533322683630, 0.70143804855573540336420300465, 0.849805310832086401744539908103, 1.14398684132876561380804447281, 1.25279091536722655624917797072, 1.33685097044150679318386684295, 1.53969361647884991701933662989, 1.66746313885331905129269781127, 1.80791014689722698321022390770, 2.04233273438328819867364554648, 2.21407394102098134448469724813, 2.28316888101350791962345837764, 2.37292424376077574886755864900, 2.41009083856948803308834011849, 2.48998799819155797493753611556, 2.58790426217531541825395658355, 2.63405569111159572147835280757, 2.84064677557649676571379900505, 2.94021879970568770922825661705, 3.00087245902669297926527335831, 3.00508279960514779968296908214, 3.12639352942696345035140030263

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

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