Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 61 $
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·4-s + 2i·11-s − 16-s + 2i·19-s + 2·29-s − 2·44-s + i·49-s + 2·59-s − 61-s i·64-s − 2i·71-s − 2·76-s − 2·89-s − 2i·101-s + 2i·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13725 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(13725\)    =    \(3^{2} \cdot 5^{2} \cdot 61\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 13725,\ (0, 1:\ ),\ 0)$

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.