# Properties

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

# Related objects

(not yet available)

## Dirichlet series

 $L(s,\rho)$  = 1 + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)10-s + (−0.500 − 1.53i)13-s + (0.309 + 0.951i)16-s + (−0.5 + 0.363i)17-s + 0.999·18-s + 0.999·20-s + (0.309 − 0.951i)25-s − 1.61·26-s + (−0.5 − 0.363i)29-s + 0.999·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{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

 $$d$$ = $$2$$ $$N$$ = $$100$$    =    $$2^{2} \cdot 5^{2}$$ $$\varepsilon$$ = $unknown$ primitive : yes self-dual : no Selberg data = $$(2,\ 100,\ (0, 1:\ ),\ 0)$$

## 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.