Properties

Degree 1
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 17 $
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  + i·7-s − 11-s i·13-s + 19-s + i·23-s − 29-s + 31-s i·37-s + 41-s + i·43-s + i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ not computed}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(1,\ 1020,\ (1:\ ),\ 0)$

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.