Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{3} $
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-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 3·9-s + 2·12-s − 14-s − 16-s − 3·18-s − 8·19-s − 2·21-s − 6·24-s − 6·25-s − 4·27-s − 28-s − 4·29-s − 5·32-s − 3·36-s + 12·37-s + 8·38-s + 2·42-s + 2·48-s + 49-s + 6·50-s + 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s + 0.577·12-s − 0.267·14-s − 1/4·16-s − 0.707·18-s − 1.83·19-s − 0.436·21-s − 1.22·24-s − 6/5·25-s − 0.769·27-s − 0.188·28-s − 0.742·29-s − 0.883·32-s − 1/2·36-s + 1.97·37-s + 1.29·38-s + 0.308·42-s + 0.288·48-s + 1/7·49-s + 0.848·50-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(49392\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{49392} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 49392,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.863214598051417092763914791206, −9.198895146278336601365818133221, −9.163045893798789171919999202805, −8.068098981215555715874277637796, −7.977686193886306753143677629185, −7.38325958034671995701748313853, −6.53484921474790582970192984243, −6.19643457342138342025418009798, −5.48510765590071063887274763802, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −4.01671863219499060051377207353, −2.39234840548789475685588947322, −1.40786771277750203137263322761, 0, 1.40786771277750203137263322761, 2.39234840548789475685588947322, 4.01671863219499060051377207353, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.48510765590071063887274763802, 6.19643457342138342025418009798, 6.53484921474790582970192984243, 7.38325958034671995701748313853, 7.977686193886306753143677629185, 8.068098981215555715874277637796, 9.163045893798789171919999202805, 9.198895146278336601365818133221, 9.863214598051417092763914791206

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