Properties

 Degree 1 Conductor 101 Sign $unknown$ Motivic weight 0 Primitive yes Self-dual no

Related objects

(not yet available)

Dirichlet series

 $L(s,\rho)$  = 1 + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + 0.999·6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + 0.999·14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + 17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ not computed}) \end{aligned}

Invariants

 $$d$$ = $$1$$ $$N$$ = $$101$$ $$\varepsilon$$ = $unknown$ primitive : yes self-dual : no Selberg data = $$(1,\ 101,\ (0:\ ),\ 0)$$

Euler product

\begin{aligned}L(s,\rho) = \prod_p \ (1 - \alpha_{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.