Properties

Degree $2$
Conductor $26$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  − 0.707·2-s + 1.46·3-s + 0.5·4-s + 1.11·5-s − 1.03·6-s − 1.32·7-s − 0.353·8-s + 1.13·9-s − 0.789·10-s − 0.848·11-s + 0.730·12-s + 0.277·13-s + 0.940·14-s + 1.63·15-s + 0.250·16-s + 0.0591·17-s − 0.800·18-s − 1.14·19-s + 0.558·20-s − 1.94·21-s + 0.600·22-s + 0.708·23-s − 0.516·24-s + 0.248·25-s − 0.196·26-s + 0.192·27-s − 0.664·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 26 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.97i)) \, \Gamma_{\R}(s+(1 - 1.97i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 26,\ (1 + 1.9732685967i, 1 - 1.9732685967i:\ ),\ 1)\)

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

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