# Properties

 Degree 4 Conductor $2^{4} \cdot 3^{6}$ Sign $1$ Motivic weight 4 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 10·7-s − 68·13-s − 128·19-s + 1.16e3·25-s − 1.39e3·31-s − 1.49e3·37-s + 5.23e3·43-s − 4.72e3·49-s + 1.28e4·61-s − 1.04e4·67-s − 9.03e3·73-s + 1.50e4·79-s − 680·91-s + 2.11e4·97-s − 1.16e4·103-s − 1.00e4·109-s + 1.55e4·121-s + 127-s + 131-s − 1.28e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 + 0.204·7-s − 0.402·13-s − 0.354·19-s + 1.87·25-s − 1.45·31-s − 1.09·37-s + 2.83·43-s − 1.96·49-s + 3.44·61-s − 2.32·67-s − 1.69·73-s + 2.40·79-s − 0.0821·91-s + 2.24·97-s − 1.09·103-s − 0.845·109-s + 1.06·121-s + 6.20e−5·127-s + 5.82e−5·131-s − 0.0723·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 11664 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$11664$$    =    $$2^{4} \cdot 3^{6}$$ $$\varepsilon$$ = $1$ motivic weight = $$4$$ character : induced by $\chi_{108} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 11664,\ (\ :2, 2),\ 1)$$ $$L(\frac{5}{2})$$ $$\approx$$ $$2.01467$$ $$L(\frac12)$$ $$\approx$$ $$2.01467$$ $$L(3)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 4. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5$C_2^2$ $$1 - 1169 T^{2} + p^{8} T^{4}$$
7$C_2$ $$( 1 - 5 T + p^{4} T^{2} )^{2}$$
11$C_2^2$ $$1 - 15593 T^{2} + p^{8} T^{4}$$
13$C_2$ $$( 1 + 34 T + p^{4} T^{2} )^{2}$$
17$C_2^2$ $$1 + 35458 T^{2} + p^{8} T^{4}$$
19$C_2$ $$( 1 + 64 T + p^{4} T^{2} )^{2}$$
23$C_2^2$ $$1 - 185138 T^{2} + p^{8} T^{4}$$
29$C_2^2$ $$1 - 286718 T^{2} + p^{8} T^{4}$$
31$C_2$ $$( 1 + 697 T + p^{4} T^{2} )^{2}$$
37$C_2$ $$( 1 + 748 T + p^{4} T^{2} )^{2}$$
41$C_2^2$ $$1 - 5183666 T^{2} + p^{8} T^{4}$$
43$C_2$ $$( 1 - 2618 T + p^{4} T^{2} )^{2}$$
47$C_2^2$ $$1 - 2758046 T^{2} + p^{8} T^{4}$$
53$C_2^2$ $$1 - 14633921 T^{2} + p^{8} T^{4}$$
59$C_2^2$ $$1 + 9567874 T^{2} + p^{8} T^{4}$$
61$C_2$ $$( 1 - 6404 T + p^{4} T^{2} )^{2}$$
67$C_2$ $$( 1 + 5218 T + p^{4} T^{2} )^{2}$$
71$C_2^2$ $$1 - 7658462 T^{2} + p^{8} T^{4}$$
73$C_2$ $$( 1 + 4519 T + p^{4} T^{2} )^{2}$$
79$C_2$ $$( 1 - 7502 T + p^{4} T^{2} )^{2}$$
83$C_2^2$ $$1 - 64875281 T^{2} + p^{8} T^{4}$$
89$C_2^2$ $$1 - 46736606 T^{2} + p^{8} T^{4}$$
97$C_2$ $$( 1 - 10571 T + p^{4} T^{2} )^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}