# Properties

 Degree $2$ Conductor $77$ Sign $unknown$ Motivic weight $0$ Primitive yes Self-dual no

# Related objects

(not yet available)

## Dirichlet series

 $L(s,\rho)$  = 1 + (−0.500 − 1.53i)2-s + (−1.30 + 0.951i)4-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (1.30 + 0.951i)14-s + (1.30 − 0.951i)18-s − 1.61·22-s − 1.61·23-s + (−0.809 − 0.587i)25-s + (0.499 − 1.53i)28-s + (−0.5 + 0.363i)29-s + 1.00·32-s + (−1.30 − 0.951i)36-s + (1.30 − 0.951i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{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

 Degree: $$2$$ Conductor: $$77$$    =    $$7 \cdot 11$$ Sign: $unknown$ Primitive: yes Self-dual: no Selberg data: $$(2,\ 77,\ (0, 1:\ ),\ 0)$$

## Particular Values

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

## Euler product

$$L(s,\rho) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## 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.