Properties

 Degree 2 Conductor $2^{3} \cdot 11$ 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 + (−0.5 + 0.363i)3-s + (−0.809 − 0.587i)4-s + (0.190 + 0.587i)6-s + (−0.809 + 0.587i)8-s + (−0.190 + 0.587i)9-s + (−0.809 + 0.587i)11-s + 0.618·12-s + (0.309 + 0.951i)16-s + (−0.500 − 1.53i)17-s + (0.499 + 0.363i)18-s + (1.30 − 0.951i)19-s + (0.309 + 0.951i)22-s + (0.190 − 0.587i)24-s + (−0.809 + 0.587i)25-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{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

 $$d$$ = $$2$$ $$N$$ = $$88$$    =    $$2^{3} \cdot 11$$ $$\varepsilon$$ = $unknown$ primitive : yes self-dual : no Selberg data = $$(2,\ 88,\ (0, 1:\ ),\ 0)$$

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.