Properties

Degree 2
Conductor $ 3 \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  + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.363i)7-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)12-s + (−0.499 + 1.53i)13-s + (0.309 − 0.951i)16-s + (−0.500 − 1.53i)19-s + (0.190 + 0.587i)21-s + 25-s + (−0.809 + 0.587i)27-s + (0.190 − 0.587i)28-s + (0.309 − 0.951i)31-s + 0.999·36-s + 0.618·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\R}(s+1) \, L(s,\rho)\cr =\mathstrut & \epsilon \cdot \overline{\Lambda(1-\overline{s})} \quad (\text{with }\epsilon \text{ unknown}) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(93\)    =    \(3 \cdot 31\)
\( \varepsilon \)  =  $unknown$
primitive  :  yes
self-dual  :  no
Selberg data  =  \((2,\ 93,\ (0, 1:\ ),\ 0)\)

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.