# Properties

 Degree 4 Conductor $13^{2} \cdot 37^{2}$ Sign $-1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 1

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s − 4·4-s − 3·9-s − 8·12-s − 4·13-s + 12·16-s + 12·17-s + 12·23-s − 10·25-s − 14·27-s − 12·29-s + 12·36-s − 8·39-s + 16·43-s + 24·48-s − 13·49-s + 24·51-s + 16·52-s − 6·53-s + 16·61-s − 32·64-s − 48·68-s + 24·69-s − 20·75-s − 20·79-s − 4·81-s − 24·87-s + ⋯
 L(s)  = 1 + 1.15·3-s − 2·4-s − 9-s − 2.30·12-s − 1.10·13-s + 3·16-s + 2.91·17-s + 2.50·23-s − 2·25-s − 2.69·27-s − 2.22·29-s + 2·36-s − 1.28·39-s + 2.43·43-s + 3.46·48-s − 1.85·49-s + 3.36·51-s + 2.21·52-s − 0.824·53-s + 2.04·61-s − 4·64-s − 5.82·68-s + 2.88·69-s − 2.30·75-s − 2.25·79-s − 4/9·81-s − 2.57·87-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 231361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$231361$$    =    $$13^{2} \cdot 37^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{231361} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : no self-dual : yes analytic rank = $$1$$ Selberg data = $$(4,\ 231361,\ (\ :1/2, 1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{13,\;37\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ $$1 + 4 T + p T^{2}$$
37$C_1$$\times$$C_1$ $$( 1 - T )( 1 + T )$$
good2$C_2$ $$( 1 + p T^{2} )^{2}$$
3$C_2$ $$( 1 - T + p T^{2} )^{2}$$
5$C_2$ $$( 1 + p T^{2} )^{2}$$
7$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
11$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
17$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2$ $$( 1 - 6 T + p T^{2} )^{2}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
31$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
41$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
43$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
47$C_2$ $$( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )$$
53$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
59$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )$$
73$C_2$ $$( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
79$C_2$ $$( 1 + 10 T + p T^{2} )^{2}$$
83$C_2$ $$( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}