Properties

Degree 1
Conductor 23
Sign $unknown$
Motivic weight 0
Primitive yes
Self-dual no

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)11-s + (−0.959 − 0.281i)12-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ not computed}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(1,\ 23,\ (0:\ ),\ 0)$

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.