Properties

Degree 2
Conductor 6
Sign $1$
Self-dual yes
Motivic weight 3

Origins

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Normalization:  

(not yet available)

Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.536·5-s + 0.408·6-s − 0.863·7-s − 0.353·8-s + 0.333·9-s − 0.379·10-s + 0.328·11-s − 0.288·12-s + 0.810·13-s + 0.610·14-s − 0.309·15-s + 0.250·16-s − 1.797·17-s − 0.235·18-s + 0.241·19-s + 0.268·20-s + 0.498·21-s − 0.232·22-s + 1.523·23-s + 0.204·24-s − 0.712·25-s − 0.573·26-s − 0.192·27-s − 0.431·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 6 ^{s/2} \Gamma_{\C}(s+1.5) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6\)    =    \(2 \cdot 3\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 6,\ (\ :3/2),\ 1)$

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

\[L(1/2,f) \approx 0.5097104234\] \[L(1,f) \approx 0.6311371888\]

Imaginary part of the first few zeros on the critical line

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