Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 3·9-s + 4·13-s + 5·16-s − 8·19-s + 25-s − 16·31-s + 9·36-s − 4·37-s + 8·43-s − 14·49-s − 12·52-s − 20·61-s − 3·64-s − 32·67-s + 28·73-s + 24·76-s + 16·79-s + 9·81-s + 20·97-s − 3·100-s − 8·103-s − 36·109-s − 12·117-s + 121-s + 48·124-s + 127-s + ⋯
L(s)  = 1  − 3/2·4-s − 9-s + 1.10·13-s + 5/4·16-s − 1.83·19-s + 1/5·25-s − 2.87·31-s + 3/2·36-s − 0.657·37-s + 1.21·43-s − 2·49-s − 1.66·52-s − 2.56·61-s − 3/8·64-s − 3.90·67-s + 3.27·73-s + 2.75·76-s + 1.80·79-s + 81-s + 2.03·97-s − 0.299·100-s − 0.788·103-s − 3.44·109-s − 1.10·117-s + 1/11·121-s + 4.31·124-s + 0.0887·127-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  :  no
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$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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

−10.63828874077951155959665793654, −9.365267708430525132762162914792, −9.301659499748329006626376523886, −8.877176431244694382657420951838, −8.248642768475844318802620430882, −7.963920428420679646091651044326, −7.06776652987294055072227501878, −6.16821463973303244938343944630, −5.89992194836808220350577312387, −5.09766059609940450216711100862, −4.52659700416500112028212971426, −3.79000489498233733500749844862, −3.25423179226253980082861279896, −1.86919846930486895513918273600, 0, 1.86919846930486895513918273600, 3.25423179226253980082861279896, 3.79000489498233733500749844862, 4.52659700416500112028212971426, 5.09766059609940450216711100862, 5.89992194836808220350577312387, 6.16821463973303244938343944630, 7.06776652987294055072227501878, 7.963920428420679646091651044326, 8.248642768475844318802620430882, 8.877176431244694382657420951838, 9.301659499748329006626376523886, 9.365267708430525132762162914792, 10.63828874077951155959665793654

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