# Properties

 Degree 4 Conductor $5^{2} \cdot 61$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes

# Related objects

(not yet available)

## Dirichlet series

 $L(s,f)$  = 1 − 1.61·4-s + 1.27·5-s − 0.460·9-s − 1.16·11-s + 1.61·16-s − 1.71·19-s − 2.06·20-s + 0.629·25-s + 2.10·29-s + 0.924·31-s + 0.745·36-s − 0.119·41-s + 1.88·44-s − 0.587·45-s + 0.638·49-s − 1.48·55-s − 0.111·59-s − 0.128·61-s − 1.00·64-s + 0.0453·71-s + 2.77·76-s − 0.364·79-s + 2.06·80-s − 0.787·81-s − 0.0964·89-s − 2.18·95-s + 0.535·99-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,f)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1525$$    =    $$5^{2} \cdot 61$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(4,\ 1525,\ (\ :1/2, 1/2),\ 1)$$

## Euler product

\begin{aligned}L(s,f) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}

## Particular Values

$L(1/2,f) \approx 0.5065018928$ $L(1,f) \approx 0.7423837223$