Properties

Degree $2$
Conductor $4$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 1.83·3-s − 0.634·5-s + 1.41·7-s + 2.37·9-s − 0.275·11-s + 0.264·13-s − 1.16·15-s − 0.906·17-s + 1.56·19-s + 2.60·21-s + 1.10·23-s − 0.597·25-s + 2.53·27-s − 1.08·29-s + 0.353·31-s − 0.506·33-s − 0.899·35-s + 1.28·37-s + 0.486·39-s + 0.780·41-s − 0.247·43-s − 1.52·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 4 ^{s/2} \, \Gamma_{\R}(s+(1 + 26.0i)) \, \Gamma_{\R}(s+(1 - 26.0i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 4,\ (1 + 26.0378899623i, 1 - 26.0378899623i:\ ),\ 1)\)

Euler product

\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)

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.