Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $-1$
Primitive yes
Self-dual yes

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Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 1.66·3-s + 0.499·4-s − 0.447·5-s − 1.17·6-s − 1.12·7-s + 0.353·8-s + 1.76·9-s − 0.316·10-s − 0.438·11-s − 0.831·12-s + 0.224·13-s − 0.796·14-s + 0.743·15-s + 0.250·16-s − 1.12·17-s + 1.24·18-s − 0.340·19-s − 0.223·20-s + 1.87·21-s − 0.310·22-s − 0.839·23-s − 0.588·24-s + 0.200·25-s + 0.158·26-s − 1.27·27-s − 0.563·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+4.58i) \, \Gamma_{\R}(s-4.58i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(2,\ 10,\ (4.58236150313i, -4.58236150313i:\ ),\ -1)$

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

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