Properties

Degree 2
Conductor 3
Sign $0.555 + 0.831i$
Self-dual no
Motivic weight 8

Origins

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Normalization:  

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Dirichlet series

$L(s,f)$  = 1  + 1.403i ·2-s + (0.555 − 0.831i) 3-s − 0.968·4-s − 0.359i·5-s + (1.166 + 0.779i) 6-s − 0.728·7-s + 0.043i ·8-s + (−0.382 − 0.923i) 9-s + 0.504·10-s + 0.475i ·11-s + (−0.538 + 0.805i) 12-s + 0.900·13-s − 1.022i·14-s + (−0.298 − 0.199i) 15-s − 1.030·16-s + 0.896i ·17-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 3 ^{s/2} \Gamma_{\C}(s+4) \cdot L(s,f)\cr =\mathstrut & (0.555 + 0.831i) \Lambda(1-s,\overline{f}) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $0.555 + 0.831i$
primitive  :  yes
self-dual  :  no
Selberg data  =  $(2,\ 3,\ (\ :4),\ 0.555 + 0.831i)$

Euler product

\[\begin{equation} L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1} \end{equation}\]

Particular Values

\[L(1/2,f) \approx 1.0371110676 + 0.5543591839i\] \[L(1,f) \approx 1.0319188436 + 0.4406820218i\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line