Properties

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

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s − 0.941·3-s + 0.5·4-s + 0.447·5-s − 0.665·6-s + 1.38·7-s + 0.353·8-s − 0.114·9-s + 0.316·10-s + 1.09·11-s − 0.470·12-s + 1.39·13-s + 0.979·14-s − 0.420·15-s + 0.250·16-s + 1.66·17-s − 0.0807·18-s + 1.35·19-s + 0.223·20-s − 1.30·21-s + 0.772·22-s − 0.109·23-s − 0.332·24-s + 0.199·25-s + 0.988·26-s + 1.04·27-s + 0.692·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+29.2i) \, \Gamma_{\R}(s-29.2i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((2,\ 10,\ (29.2867591963i, -29.2867591963i:\ ),\ 1)\)

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