Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 9-s + 2·12-s − 3·16-s + 25-s + 4·27-s − 8·31-s − 36-s − 20·37-s + 6·48-s + 2·49-s + 7·64-s − 20·67-s − 2·75-s − 11·81-s + 16·93-s − 20·97-s − 100-s + 4·103-s − 4·108-s + 40·111-s − 11·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 1/3·9-s + 0.577·12-s − 3/4·16-s + 1/5·25-s + 0.769·27-s − 1.43·31-s − 1/6·36-s − 3.28·37-s + 0.866·48-s + 2/7·49-s + 7/8·64-s − 2.44·67-s − 0.230·75-s − 1.22·81-s + 1.65·93-s − 2.03·97-s − 0.0999·100-s + 0.394·103-s − 0.384·108-s + 3.79·111-s − 121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{27225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 27225,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
good2$V_4$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$V_4$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$V_4$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$V_4$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$V_4$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$V_4$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 - 134 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.53816257725248073130842019726, −9.955951025311467647832971998211, −9.107317506658110308566345614657, −8.923644105979883940049846677080, −8.320437631237746381160261512250, −7.44997762456031466582066888485, −6.93370463159796136190684082007, −6.48980745991730713651759347225, −5.60065879084227609478701806218, −5.33377947704344434601290840834, −4.66473467177095454165541913899, −3.94727908945697743769298534363, −3.05940369593580445690460715793, −1.74717117060943513123928878837, 0, 1.74717117060943513123928878837, 3.05940369593580445690460715793, 3.94727908945697743769298534363, 4.66473467177095454165541913899, 5.33377947704344434601290840834, 5.60065879084227609478701806218, 6.48980745991730713651759347225, 6.93370463159796136190684082007, 7.44997762456031466582066888485, 8.320437631237746381160261512250, 8.923644105979883940049846677080, 9.107317506658110308566345614657, 9.955951025311467647832971998211, 10.53816257725248073130842019726

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