# Properties

 Label 4-7360e2-1.1-c1e2-0-6 Degree $4$ Conductor $54169600$ Sign $1$ Analytic cond. $3453.90$ Root an. cond. $7.66615$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s − 2·5-s + 7-s − 4·9-s − 11-s + 3·13-s + 2·15-s + 17-s + 3·19-s − 21-s + 2·23-s + 3·25-s + 6·27-s + 14·29-s + 7·31-s + 33-s − 2·35-s − 4·37-s − 3·39-s − 9·41-s + 8·45-s + 6·47-s − 12·49-s − 51-s + 8·53-s + 2·55-s − 3·57-s + ⋯
 L(s)  = 1 − 0.577·3-s − 0.894·5-s + 0.377·7-s − 4/3·9-s − 0.301·11-s + 0.832·13-s + 0.516·15-s + 0.242·17-s + 0.688·19-s − 0.218·21-s + 0.417·23-s + 3/5·25-s + 1.15·27-s + 2.59·29-s + 1.25·31-s + 0.174·33-s − 0.338·35-s − 0.657·37-s − 0.480·39-s − 1.40·41-s + 1.19·45-s + 0.875·47-s − 1.71·49-s − 0.140·51-s + 1.09·53-s + 0.269·55-s − 0.397·57-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$54169600$$    =    $$2^{12} \cdot 5^{2} \cdot 23^{2}$$ Sign: $1$ Analytic conductor: $$3453.90$$ Root analytic conductor: $$7.66615$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 54169600,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.028978720$$ $$L(\frac12)$$ $$\approx$$ $$2.028978720$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_1$ $$( 1 + T )^{2}$$
23$C_1$ $$( 1 - T )^{2}$$
good3$D_{4}$ $$1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4}$$
7$D_{4}$ $$1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4}$$
11$D_{4}$ $$1 + T + p T^{2} + p T^{3} + p^{2} T^{4}$$
13$D_{4}$ $$1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
17$D_{4}$ $$1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4}$$
19$D_{4}$ $$1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
29$D_{4}$ $$1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4}$$
31$D_{4}$ $$1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4}$$
37$D_{4}$ $$1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
41$D_{4}$ $$1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4}$$
43$C_2$ $$( 1 + p T^{2} )^{2}$$
47$D_{4}$ $$1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
53$D_{4}$ $$1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
59$D_{4}$ $$1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
61$D_{4}$ $$1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
67$D_{4}$ $$1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4}$$
71$D_{4}$ $$1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4}$$
73$D_{4}$ $$1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
79$D_{4}$ $$1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
83$D_{4}$ $$1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4}$$
89$C_4$ $$1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
97$D_{4}$ $$1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$