Properties

Degree 2
Conductor $ 2^{2} \cdot 31 $
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·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.499 + 0.866i)5-s + (−0.500 − 0.866i)6-s + (0.866 − 0.5i)7-s i·8-s + (−0.866 − 0.499i)10-s + (0.866 + 0.499i)11-s + (0.866 − 0.500i)12-s + (0.499 − 0.866i)13-s + (0.500 + 0.866i)14-s − 0.999i·15-s + 16-s + (−0.500 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(124\)    =    \(2^{2} \cdot 31\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 124,\ (0, 1:\ ),\ 0)$

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.