Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $unknown$
Motivic weight 0
Primitive yes
Self-dual no

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s,\rho)$  = 1  + (0.499 − 0.866i)2-s + (0.499 − 0.866i)3-s + (−0.500 − 0.866i)4-s + (−0.500 − 0.866i)6-s − 7-s − 0.999·8-s + (−0.500 − 0.866i)9-s + 11-s − 0.999·12-s + (0.500 + 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.499 + 0.866i)16-s + (0.499 − 0.866i)17-s − 0.999·18-s + (−0.499 + 0.866i)21-s + (0.499 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ not computed}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  \((1,\ 95,\ (0:\ ),\ 0)\)

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.