Properties

Degree 2
Conductor 1
Sign $1$
Self-dual yes
Motivic weight 11

Origins

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Normalization:  

(not yet available)

Dirichlet series

$L(s,f)$  = 1  − 0.530·2-s + 0.598·3-s − 0.718·4-s + 0.691·5-s − 0.317·6-s − 0.376·7-s + 0.911·8-s − 0.641·9-s − 0.366·10-s + 1.000·11-s − 0.430·12-s − 0.431·13-s + 0.199·14-s + 0.413·15-s + 0.235·16-s − 1.179·17-s + 0.340·18-s + 0.987·19-s − 0.496·20-s − 0.225·21-s − 0.530·22-s + 0.603·23-s + 0.545·24-s − 0.522·25-s + 0.228·26-s − 0.982·27-s + 0.270·28-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut &\Gamma_{\C}(s+5.5) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 1,\ (\ :11/2),\ 1)$

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

\[L(1/2,f) \approx 0.7921228386\] \[L(1,f) \approx 0.839345512\]

Imaginary part of the first few zeros on the critical line

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