Properties

Degree 2
Conductor 7
Sign $1$
Self-dual yes
Motivic weight 2

Origins

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Normalization:  

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Dirichlet series

$L(s,f)$  = 1  − 1.5·2-s + 1.250·4-s − 7-s − 0.375·8-s + 9-s − 0.545·11-s + 1.5·14-s − 0.687·16-s − 1.5·18-s + 0.818·22-s + 0.782·23-s + 25-s − 1.250·28-s − 1.862·29-s + 1.406·32-s + 1.250·36-s − 1.027·37-s + 1.348·43-s − 0.681·44-s − 1.173·46-s + 49-s − 1.5·50-s − 0.113·53-s + 0.375·56-s + 2.793·58-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,f)=\mathstrut & 7 ^{s/2} \Gamma_{\C}(s+1) \cdot L(s,f)\cr =\mathstrut & \Lambda(1-s,f) \end{align} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 7,\ (\ :1),\ 1)$

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

\[L(1/2,f) \approx 0.3329817715\] \[L(1,f) \approx 0.4672117689\]

Imaginary part of the first few zeros on the critical line

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