# Properties

 Label 4-76e2-1.1-c4e2-0-0 Degree $4$ Conductor $5776$ Sign $1$ Analytic cond. $61.7185$ Root an. cond. $2.80287$ Motivic weight $4$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 31·5-s + 73·7-s + 162·9-s + 233·11-s + 353·17-s + 722·19-s − 316·23-s + 625·25-s − 2.26e3·35-s − 3.52e3·43-s − 5.02e3·45-s − 1.20e3·47-s + 2.40e3·49-s − 7.22e3·55-s − 3.16e3·61-s + 1.18e4·63-s + 1.00e4·73-s + 1.70e4·77-s + 1.96e4·81-s − 1.13e4·83-s − 1.09e4·85-s − 2.23e4·95-s + 3.77e4·99-s − 1.99e4·101-s + 9.79e3·115-s + 2.57e4·119-s + 1.46e4·121-s + ⋯
 L(s)  = 1 − 1.23·5-s + 1.48·7-s + 2·9-s + 1.92·11-s + 1.22·17-s + 2·19-s − 0.597·23-s + 25-s − 1.84·35-s − 1.90·43-s − 2.47·45-s − 0.546·47-s + 49-s − 2.38·55-s − 0.851·61-s + 2.97·63-s + 1.88·73-s + 2.86·77-s + 3·81-s − 1.64·83-s − 1.51·85-s − 2.47·95-s + 3.85·99-s − 1.96·101-s + 0.740·115-s + 1.81·119-s + 121-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$5776$$    =    $$2^{4} \cdot 19^{2}$$ Sign: $1$ Analytic conductor: $$61.7185$$ Root analytic conductor: $$2.80287$$ Motivic weight: $$4$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{76} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 5776,\ (\ :2, 2),\ 1)$$

## Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$3.157129682$$ $$L(\frac12)$$ $$\approx$$ $$3.157129682$$ $$L(3)$$ 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 $$1$$
19$C_1$ $$( 1 - p^{2} T )^{2}$$
good3$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
5$C_2^2$ $$1 + 31 T + 336 T^{2} + 31 p^{4} T^{3} + p^{8} T^{4}$$
7$C_2^2$ $$1 - 73 T + 2928 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4}$$
11$C_2^2$ $$1 - 233 T + 39648 T^{2} - 233 p^{4} T^{3} + p^{8} T^{4}$$
13$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
17$C_2^2$ $$1 - 353 T + 41088 T^{2} - 353 p^{4} T^{3} + p^{8} T^{4}$$
23$C_2$ $$( 1 + 158 T + p^{4} T^{2} )^{2}$$
29$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
31$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
37$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
41$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
43$C_2^2$ $$1 + 3527 T + 9020928 T^{2} + 3527 p^{4} T^{3} + p^{8} T^{4}$$
47$C_2^2$ $$1 + 1207 T - 3422832 T^{2} + 1207 p^{4} T^{3} + p^{8} T^{4}$$
53$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
59$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
61$C_2^2$ $$1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4}$$
67$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
71$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
73$C_2^2$ $$1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4}$$
79$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
83$C_2$ $$( 1 + 5678 T + p^{4} T^{2} )^{2}$$
89$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$