# Properties

 Degree 2 Conductor $3^{2} \cdot 5^{2} \cdot 61$ Sign $unknown$ Motivic weight 0 Primitive yes Self-dual no

# Related objects

(not yet available)

## Dirichlet series

 $L(s,\rho)$  = 1 + i·4-s + 2i·11-s − 16-s + 2i·19-s + 2·29-s − 2·44-s + i·49-s + 2·59-s − 61-s − i·64-s − 2i·71-s − 2·76-s − 2·89-s − 2i·101-s + 2i·109-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 13725 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$13725$$    =    $$3^{2} \cdot 5^{2} \cdot 61$$ $$\varepsilon$$ = $unknown$ primitive : yes self-dual : no Selberg data = $(2,\ 13725,\ (0, 1:\ ),\ 0)$

## Euler product

\begin{aligned} L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Particular Values

Not enough information (Dirichlet series coefficients/sign of the functional equation) to compute special values.

## Imaginary part of the first few zeros on the critical line

Zeros not available.

## Graph of the $Z$-function along the critical line

Plot not available.