# 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 of factors

## 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