# Properties

 Label 4-37e2-1.1-c1e2-0-0 Degree $4$ Conductor $1369$ Sign $1$ Analytic cond. $0.0872886$ Root an. cond. $0.543549$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·3-s + 6·7-s − 3·9-s − 6·11-s − 4·16-s − 12·21-s + 6·25-s + 14·27-s + 12·33-s − 2·37-s − 6·41-s + 6·47-s + 8·48-s + 13·49-s + 18·53-s − 18·63-s − 24·67-s − 6·71-s + 18·73-s − 12·75-s − 36·77-s − 4·81-s + 18·83-s + 18·99-s − 6·101-s − 24·107-s + 4·111-s + ⋯
 L(s)  = 1 − 1.15·3-s + 2.26·7-s − 9-s − 1.80·11-s − 16-s − 2.61·21-s + 6/5·25-s + 2.69·27-s + 2.08·33-s − 0.328·37-s − 0.937·41-s + 0.875·47-s + 1.15·48-s + 13/7·49-s + 2.47·53-s − 2.26·63-s − 2.93·67-s − 0.712·71-s + 2.10·73-s − 1.38·75-s − 4.10·77-s − 4/9·81-s + 1.97·83-s + 1.80·99-s − 0.597·101-s − 2.32·107-s + 0.379·111-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1369$$    =    $$37^{2}$$ Sign: $1$ Analytic conductor: $$0.0872886$$ Root analytic conductor: $$0.543549$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1369,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4468046130$$ $$L(\frac12)$$ $$\approx$$ $$0.4468046130$$ $$L(\frac{3}{2})$$ 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)$
bad37$C_2$ $$1 + 2 T + p T^{2}$$
good2$C_2$ $$( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )$$
3$C_2$ $$( 1 + T + p T^{2} )^{2}$$
5$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
11$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
19$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
31$C_2$ $$( 1 - p T^{2} )^{2}$$
41$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
43$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
47$C_2$ $$( 1 - 3 T + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
59$C_2^2$ $$1 - 102 T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - p T^{2} )^{2}$$
67$C_2$ $$( 1 + 12 T + p T^{2} )^{2}$$
71$C_2$ $$( 1 + 3 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - 122 T^{2} + p^{2} T^{4}$$
83$C_2$ $$( 1 - 9 T + p T^{2} )^{2}$$
89$C_2^2$ $$1 + 18 T^{2} + p^{2} T^{4}$$
97$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$