Properties

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

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s + 1.78·3-s + 0.5·4-s − 0.447·5-s − 1.26·6-s + 0.572·7-s − 0.353·8-s + 2.19·9-s + 0.316·10-s − 1.52·11-s + 0.893·12-s + 1.28·13-s − 0.404·14-s − 0.799·15-s + 0.250·16-s + 1.09·17-s − 1.55·18-s + 1.31·19-s − 0.223·20-s + 1.02·21-s + 1.08·22-s + 0.895·23-s − 0.631·24-s + 0.199·25-s − 0.911·26-s + 2.13·27-s + 0.286·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+28.0i) \, \Gamma_{\R}(s-28.0i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (28.0277820443i, -28.0277820443i:\ ),\ 1)\)

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