Properties

Degree 4
Conductor $ 53^{3} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 8·7-s + 3·9-s − 6·13-s + 5·16-s − 6·17-s − 10·25-s + 24·28-s − 14·29-s − 9·36-s + 10·37-s − 4·43-s − 4·47-s + 34·49-s + 18·52-s − 53-s − 4·59-s − 24·63-s − 3·64-s + 18·68-s − 28·89-s + 48·91-s + 2·97-s + 30·100-s + 12·107-s − 40·112-s + 30·113-s + ⋯
L(s)  = 1  − 3/2·4-s − 3.02·7-s + 9-s − 1.66·13-s + 5/4·16-s − 1.45·17-s − 2·25-s + 4.53·28-s − 2.59·29-s − 3/2·36-s + 1.64·37-s − 0.609·43-s − 0.583·47-s + 34/7·49-s + 2.49·52-s − 0.137·53-s − 0.520·59-s − 3.02·63-s − 3/8·64-s + 2.18·68-s − 2.96·89-s + 5.03·91-s + 0.203·97-s + 3·100-s + 1.16·107-s − 3.77·112-s + 2.82·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 148877 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 148877 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(148877\)    =    \(53^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{148877} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 148877,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 53$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 53$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad53$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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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

−9.264238826193943461669666199356, −8.483283280441272019249165232631, −7.73762398215887034121172697318, −7.20736947999270954805514319970, −7.00110496145002859293965527187, −6.09508866331367665451434742179, −6.04347889941024738587947019148, −5.18496964971722877756379342667, −4.50862835068296220400083952922, −3.97664069548018217930915160992, −3.69121393392699379806557761029, −2.87754256168513552194439396359, −2.07674380795307242484164104235, 0, 0, 2.07674380795307242484164104235, 2.87754256168513552194439396359, 3.69121393392699379806557761029, 3.97664069548018217930915160992, 4.50862835068296220400083952922, 5.18496964971722877756379342667, 6.04347889941024738587947019148, 6.09508866331367665451434742179, 7.00110496145002859293965527187, 7.20736947999270954805514319970, 7.73762398215887034121172697318, 8.483283280441272019249165232631, 9.264238826193943461669666199356

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