# Properties

 Label 4-442e2-1.1-c1e2-0-35 Degree $4$ Conductor $195364$ Sign $-1$ Analytic cond. $12.4565$ Root an. cond. $1.87866$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s − 2·19-s − 6·25-s − 6·32-s + 6·36-s + 4·38-s − 18·43-s − 6·47-s + 6·49-s + 12·50-s − 12·59-s + 7·64-s + 16·67-s − 8·72-s − 6·76-s − 5·81-s + 6·83-s + 36·86-s − 12·89-s + 12·94-s − 12·98-s − 18·100-s + ⋯
 L(s)  = 1 − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s + 5/4·16-s − 0.942·18-s − 0.458·19-s − 6/5·25-s − 1.06·32-s + 36-s + 0.648·38-s − 2.74·43-s − 0.875·47-s + 6/7·49-s + 1.69·50-s − 1.56·59-s + 7/8·64-s + 1.95·67-s − 0.942·72-s − 0.688·76-s − 5/9·81-s + 0.658·83-s + 3.88·86-s − 1.27·89-s + 1.23·94-s − 1.21·98-s − 9/5·100-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$195364$$    =    $$2^{2} \cdot 13^{2} \cdot 17^{2}$$ Sign: $-1$ Analytic conductor: $$12.4565$$ Root analytic conductor: $$1.87866$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{195364} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 195364,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ $$( 1 + T )^{2}$$
13$C_2$ $$1 + p T^{2}$$
17$C_2$ $$1 + p T^{2}$$
good3$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
7$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
11$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
19$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 2 T + p T^{2} )$$
23$C_2^2$ $$1 - 10 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 4 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 18 T^{2} + p^{2} T^{4}$$
37$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
41$C_2^2$ $$1 + 24 T^{2} + p^{2} T^{4}$$
43$C_2$$\times$$C_2$ $$( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
47$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 6 T + p T^{2} )$$
53$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
59$C_2$$\times$$C_2$ $$( 1 + p T^{2} )( 1 + 12 T + p T^{2} )$$
61$C_2^2$ $$1 - 4 T^{2} + p^{2} T^{4}$$
67$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
71$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
73$C_2^2$ $$1 - 120 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
83$C_2$$\times$$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
89$C_2$$\times$$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} )$$
97$C_2^2$ $$1 + 48 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$