Properties

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

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 8·11-s − 12·17-s + 8·19-s − 6·25-s − 4·27-s + 16·33-s + 4·41-s − 8·43-s + 2·49-s + 24·51-s − 16·57-s + 8·59-s − 8·67-s − 12·73-s + 12·75-s + 5·81-s − 24·83-s + 20·89-s − 28·97-s − 24·99-s + 8·107-s + 4·113-s + 26·121-s − 8·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 2.41·11-s − 2.91·17-s + 1.83·19-s − 6/5·25-s − 0.769·27-s + 2.78·33-s + 0.624·41-s − 1.21·43-s + 2/7·49-s + 3.36·51-s − 2.11·57-s + 1.04·59-s − 0.977·67-s − 1.40·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 2.11·89-s − 2.84·97-s − 2.41·99-s + 0.773·107-s + 0.376·113-s + 2.36·121-s − 0.721·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18432 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(18432\)    =    \(2^{11} \cdot 3^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18432} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 18432,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−10.84849442743057013529606410604, −10.20586918892848667331432436971, −9.826637660192491869357018654726, −9.133901360741508328144538474134, −8.391046533691483698004382219288, −7.83182394757322498939219938639, −7.17212008872630953090710242182, −6.80946897903754876349072202165, −5.77092473843782707280190480124, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.31158789218116792910899233548, −3.00571163091752467289605693825, −2.09803110754033007053716721218, 0, 2.09803110754033007053716721218, 3.00571163091752467289605693825, 4.31158789218116792910899233548, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 5.77092473843782707280190480124, 6.80946897903754876349072202165, 7.17212008872630953090710242182, 7.83182394757322498939219938639, 8.391046533691483698004382219288, 9.133901360741508328144538474134, 9.826637660192491869357018654726, 10.20586918892848667331432436971, 10.84849442743057013529606410604

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