Properties

Degree 2
Conductor 751
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  − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(751\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 751,\ (0, 1:\ ),\ 0)$

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.