Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 7^{3} $
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  − 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  =  0
Selberg data  =  $(4,\ 49392,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.230683220$
$L(\frac12)$  $\approx$  $1.230683220$
$L(\frac{3}{2})$   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

−9.762171554631668381167652624967, −9.724661210726173973340366981146, −9.090946623633278801475280262159, −8.672365196683144779961098572501, −8.050951011396219930904197989085, −7.70869243915887411656672354688, −7.25047783802838427330028450541, −6.69032425884932976329953320783, −5.58120289551734980246344429859, −5.36367026412859974981569612163, −4.13559084050773741974089362056, −3.99570051430579360707309966688, −3.06814457319249350690123364058, −2.26612547708831611492323080497, −1.10536343915846851689160557608, 1.10536343915846851689160557608, 2.26612547708831611492323080497, 3.06814457319249350690123364058, 3.99570051430579360707309966688, 4.13559084050773741974089362056, 5.36367026412859974981569612163, 5.58120289551734980246344429859, 6.69032425884932976329953320783, 7.25047783802838427330028450541, 7.70869243915887411656672354688, 8.050951011396219930904197989085, 8.672365196683144779961098572501, 9.090946623633278801475280262159, 9.724661210726173973340366981146, 9.762171554631668381167652624967

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