Properties

Degree 2
Conductor $ 1 $
Sign $1$
Primitive yes
Self-dual yes

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Dirichlet series

$L(s,f)$  = 1  + 1.54·2-s + 0.246·3-s + 1.40·4-s + 0.737·5-s + 0.382·6-s − 0.261·7-s + 0.620·8-s − 0.939·9-s + 1.14·10-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut &\Gamma_{\R}(s+13.7i) \, \Gamma_{\R}(s-13.7i) \, L(s,f)\cr =\mathstrut & \,\Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)    =    \(1\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 1,\ (13.7797513519i, -13.7797513519i:\ ),\ 1)$

Euler product

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

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.