Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{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  + 2·3-s + 3·9-s − 2·11-s − 4·17-s − 6·25-s + 4·27-s − 4·33-s − 4·41-s + 2·49-s − 8·51-s + 8·59-s + 8·67-s − 28·73-s − 12·75-s + 5·81-s − 24·83-s − 12·89-s + 4·97-s − 6·99-s + 24·107-s − 12·113-s + 3·121-s − 8·123-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 0.603·11-s − 0.970·17-s − 6/5·25-s + 0.769·27-s − 0.696·33-s − 0.624·41-s + 2/7·49-s − 1.12·51-s + 1.04·59-s + 0.977·67-s − 3.27·73-s − 1.38·75-s + 5/9·81-s − 2.63·83-s − 1.27·89-s + 0.406·97-s − 0.603·99-s + 2.32·107-s − 1.12·113-s + 3/11·121-s − 0.721·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1115136\)    =    \(2^{10} \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1115136} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 1115136,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.75720782023225106841673661541, −7.57766902213530444274389318028, −6.91342543518502244816613622753, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −5.79508255106269640718109448990, −5.08911955765620457569841389288, −4.35461089374668409702872860477, −4.32526881755629167390593741274, −3.53074178558790394542885743604, −3.10863384707893856832893049605, −2.45250023112523203194083612695, −2.07552052405763697992139591576, −1.33148151575350928602632304721, 0, 1.33148151575350928602632304721, 2.07552052405763697992139591576, 2.45250023112523203194083612695, 3.10863384707893856832893049605, 3.53074178558790394542885743604, 4.32526881755629167390593741274, 4.35461089374668409702872860477, 5.08911955765620457569841389288, 5.79508255106269640718109448990, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 6.91342543518502244816613622753, 7.57766902213530444274389318028, 7.75720782023225106841673661541

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