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  + 24·13-s + 8·25-s − 32·37-s − 4·49-s + 64·61-s − 32·73-s − 48·97-s − 64·109-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 244·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 6.65·13-s + 8/5·25-s − 5.26·37-s − 4/7·49-s + 8.19·61-s − 3.74·73-s − 4.87·97-s − 6.13·109-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 18.7·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-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  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $0.4159159858$
$L(\frac12)$  $\approx$  $0.4159159858$
$L(\frac{3}{2})$   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 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + 4 T^{2} - 138 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 58 T^{2} + 1395 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 82 T^{2} + 2715 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 70 T^{2} + 2763 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 - 142 T^{2} + 8523 T^{4} - 142 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 28 T^{2} - 1530 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 142 T^{2} + 9795 T^{4} - 142 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 70 T^{2} + 5787 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 254 T^{2} + 25083 T^{4} - 254 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 242 T^{2} + 24507 T^{4} + 242 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 4 T + p T^{2} )^{8} \)
79 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 - 58 T^{2} + 15507 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−3.31066993989103792816737376624, −3.14809806138303908064659059353, −3.08639485708398051626228762603, −3.03567225201574593517954454867, −3.01734288593445113006318765298, −2.72797726114151154067711036227, −2.62461150650587471012642969032, −2.55390699054728251058882411503, −2.45773615555051832536099576482, −2.23878206673743941876592801026, −2.05055950807791847734200226640, −2.00881860080318880006025449059, −1.88609004962703733541430868713, −1.72929704757702201866203713460, −1.56989622863231988247112996275, −1.37985690773701066659229118108, −1.36215721285144881376278606286, −1.22189054088878066310580453652, −1.21351447577162771640250664648, −1.02561594894084176741256905138, −0.965905186833715964811239405347, −0.938969031279005878031023495077, −0.40470572102400790697840739960, −0.34702982273408692854889535084, −0.03355091358480577318533867946, 0.03355091358480577318533867946, 0.34702982273408692854889535084, 0.40470572102400790697840739960, 0.938969031279005878031023495077, 0.965905186833715964811239405347, 1.02561594894084176741256905138, 1.21351447577162771640250664648, 1.22189054088878066310580453652, 1.36215721285144881376278606286, 1.37985690773701066659229118108, 1.56989622863231988247112996275, 1.72929704757702201866203713460, 1.88609004962703733541430868713, 2.00881860080318880006025449059, 2.05055950807791847734200226640, 2.23878206673743941876592801026, 2.45773615555051832536099576482, 2.55390699054728251058882411503, 2.62461150650587471012642969032, 2.72797726114151154067711036227, 3.01734288593445113006318765298, 3.03567225201574593517954454867, 3.08639485708398051626228762603, 3.14809806138303908064659059353, 3.31066993989103792816737376624

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

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