Properties

Degree 4
Conductor $ 3^{3} \cdot 7^{2} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s − 4-s − 2·6-s + 8·8-s + 9-s + 4·11-s − 12-s − 7·16-s − 12·17-s − 2·18-s − 8·22-s + 8·24-s − 6·25-s + 27-s − 4·29-s − 14·32-s + 4·33-s + 24·34-s − 36-s + 12·37-s + 4·41-s − 4·44-s − 7·48-s + 49-s + 12·50-s − 12·51-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.816·6-s + 2.82·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 7/4·16-s − 2.91·17-s − 0.471·18-s − 1.70·22-s + 1.63·24-s − 6/5·25-s + 0.192·27-s − 0.742·29-s − 2.47·32-s + 0.696·33-s + 4.11·34-s − 1/6·36-s + 1.97·37-s + 0.624·41-s − 0.603·44-s − 1.01·48-s + 1/7·49-s + 1.69·50-s − 1.68·51-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(160083\)    =    \(3^{3} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{160083} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 160083,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5668110775$
$L(\frac12)$  $\approx$  $0.5668110775$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.056096586309822555042245112360, −8.783200717465470995950515779184, −8.672365196683144779961098572501, −7.976525779912433683772333851675, −7.39398850413165927730875614743, −7.25047783802838427330028450541, −6.34518093653275179095390844834, −6.01581061696917710851209529028, −4.92403553365775964167265796866, −4.46792640657225865455095535714, −4.13559084050773741974089362056, −3.62088779784825026995011416777, −2.28059162307602342837183603744, −1.77611296934943774735876447029, −0.63334663369662260021570161928, 0.63334663369662260021570161928, 1.77611296934943774735876447029, 2.28059162307602342837183603744, 3.62088779784825026995011416777, 4.13559084050773741974089362056, 4.46792640657225865455095535714, 4.92403553365775964167265796866, 6.01581061696917710851209529028, 6.34518093653275179095390844834, 7.25047783802838427330028450541, 7.39398850413165927730875614743, 7.976525779912433683772333851675, 8.672365196683144779961098572501, 8.783200717465470995950515779184, 9.056096586309822555042245112360

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