Properties

Degree 4
Conductor $ 2^{14} \cdot 3^{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 + 9-s + 4·11-s − 4·13-s + 8·23-s − 6·25-s + 4·27-s − 8·33-s − 20·37-s + 8·39-s − 16·47-s + 2·49-s − 28·59-s − 4·61-s − 16·69-s + 24·71-s + 28·73-s + 12·75-s − 11·81-s + 12·83-s − 4·97-s + 4·99-s + 4·107-s + 12·109-s + 40·111-s − 4·117-s − 10·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 1.66·23-s − 6/5·25-s + 0.769·27-s − 1.39·33-s − 3.28·37-s + 1.28·39-s − 2.33·47-s + 2/7·49-s − 3.64·59-s − 0.512·61-s − 1.92·69-s + 2.84·71-s + 3.27·73-s + 1.38·75-s − 1.22·81-s + 1.31·83-s − 0.406·97-s + 0.402·99-s + 0.386·107-s + 1.14·109-s + 3.79·111-s − 0.369·117-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(147456\)    =    \(2^{14} \cdot 3^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{147456} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 147456,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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

−9.174652574334721806401375408825, −8.635939205168057076823393046651, −8.075661729852922503369502159976, −7.44440089656946414681700247943, −6.97464122003923989664965494039, −6.40345159211036072533921924868, −6.33050260025483473718379487161, −5.36530815656846370209456927790, −4.88235750431299832390164653972, −4.84373431553687523132526670248, −3.62014189552160099547672624510, −3.36730269661969347724618284285, −2.20034159198920039710092677140, −1.34910975935301153966920068234, 0, 1.34910975935301153966920068234, 2.20034159198920039710092677140, 3.36730269661969347724618284285, 3.62014189552160099547672624510, 4.84373431553687523132526670248, 4.88235750431299832390164653972, 5.36530815656846370209456927790, 6.33050260025483473718379487161, 6.40345159211036072533921924868, 6.97464122003923989664965494039, 7.44440089656946414681700247943, 8.075661729852922503369502159976, 8.635939205168057076823393046651, 9.174652574334721806401375408825

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