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  + 32·13-s + 16·25-s − 80·37-s − 4·49-s − 16·61-s − 32·73-s − 32·97-s − 56·109-s − 64·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 472·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 8.87·13-s + 16/5·25-s − 13.1·37-s − 4/7·49-s − 2.04·61-s − 3.74·73-s − 3.24·97-s − 5.36·109-s − 5.81·121-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 + 36.3·169-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  =  $(16,\ 2^{40} \cdot 3^{24} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )$
$L(1)$  $\approx$  $1.003004528$
$L(\frac12)$  $\approx$  $1.003004528$
$L(\frac{3}{2})$   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 \( ( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 32 T^{2} + 471 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 4 T + p T^{2} )^{8} \)
17 \( ( 1 - 16 T^{2} + 450 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 + 40 T^{2} + 783 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 52 T^{2} + 1590 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 20 T + 171 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 16 T^{2} + 3279 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 76 T^{2} + 5430 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 220 T^{2} + 19014 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 140 T^{2} + 10806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 248 T^{2} + 25215 T^{4} + 248 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 260 T^{2} + 28614 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 124 T^{2} + 14550 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 64 T^{2} + 15999 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−3.32609406058859284291320912630, −3.28695563499783577684956514838, −3.13926490914665287324036868599, −3.09763536587661317739924617664, −3.06421778394097865855517736340, −2.81883859585133722905702375271, −2.67713297597380962947425978184, −2.55650877570963606628726976703, −2.46372626430393017834904340794, −2.27228068855398728491646747521, −1.94634508671348713731728004051, −1.80167273418313417034843478182, −1.78054439928037071553291603344, −1.61883407155371568127029743921, −1.56428453440115140464913586536, −1.41372709481705912490498029011, −1.40236039253164226727185466096, −1.36855607950418212611061209078, −1.28024240140347935555932253455, −1.24025422444832665408212657608, −1.03671576512556810368373951686, −0.69069654014593368674084607377, −0.41146556034045590459938346929, −0.30943520918012587124233452503, −0.05901414511772646209255841341, 0.05901414511772646209255841341, 0.30943520918012587124233452503, 0.41146556034045590459938346929, 0.69069654014593368674084607377, 1.03671576512556810368373951686, 1.24025422444832665408212657608, 1.28024240140347935555932253455, 1.36855607950418212611061209078, 1.40236039253164226727185466096, 1.41372709481705912490498029011, 1.56428453440115140464913586536, 1.61883407155371568127029743921, 1.78054439928037071553291603344, 1.80167273418313417034843478182, 1.94634508671348713731728004051, 2.27228068855398728491646747521, 2.46372626430393017834904340794, 2.55650877570963606628726976703, 2.67713297597380962947425978184, 2.81883859585133722905702375271, 3.06421778394097865855517736340, 3.09763536587661317739924617664, 3.13926490914665287324036868599, 3.28695563499783577684956514838, 3.32609406058859284291320912630

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

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