Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s − 2·9-s + 8·13-s − 4·16-s + 4·21-s − 9·25-s + 5·27-s + 14·31-s + 6·37-s − 8·39-s − 12·43-s + 4·48-s − 2·49-s + 24·61-s + 8·63-s − 14·67-s + 8·73-s + 9·75-s − 20·79-s + 81-s − 32·91-s − 14·93-s − 14·97-s − 32·103-s + 20·109-s − 6·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s − 2/3·9-s + 2.21·13-s − 16-s + 0.872·21-s − 9/5·25-s + 0.962·27-s + 2.51·31-s + 0.986·37-s − 1.28·39-s − 1.82·43-s + 0.577·48-s − 2/7·49-s + 3.07·61-s + 1.00·63-s − 1.71·67-s + 0.936·73-s + 1.03·75-s − 2.25·79-s + 1/9·81-s − 3.35·91-s − 1.45·93-s − 1.42·97-s − 3.15·103-s + 1.91·109-s − 0.569·111-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1089\)    =    \(3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1089} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1089,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4275956837$
$L(\frac12)$  $\approx$  $0.4275956837$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 + T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−13.68040274047176229306789941909, −13.56863905712999451342489435843, −13.05532878430427737685314941779, −12.07025537765623530596257929043, −11.45125861034521058353374083886, −11.14136051051457299622132768480, −10.03550909718107888433464868208, −9.701226498296212058900989539920, −8.603539619290756001226038948684, −8.228854358433494441874279371750, −6.66503987304844816686807720890, −6.36261389471308870138602900888, −5.66407660024727617127120629681, −4.19085115174357929506261869806, −3.09000916592887094247460325417, 3.09000916592887094247460325417, 4.19085115174357929506261869806, 5.66407660024727617127120629681, 6.36261389471308870138602900888, 6.66503987304844816686807720890, 8.228854358433494441874279371750, 8.603539619290756001226038948684, 9.701226498296212058900989539920, 10.03550909718107888433464868208, 11.14136051051457299622132768480, 11.45125861034521058353374083886, 12.07025537765623530596257929043, 13.05532878430427737685314941779, 13.56863905712999451342489435843, 13.68040274047176229306789941909

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