Properties

Degree $2$
Conductor $51$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

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Dirichlet series

$L(s,f)$  = 1  + 0.988·2-s + 0.577·3-s − 0.0233·4-s − 1.31·5-s + 0.570·6-s + 0.311·7-s − 1.01·8-s + 0.333·9-s − 1.30·10-s + 1.77·11-s − 0.0134·12-s − 0.179·13-s + 0.307·14-s − 0.759·15-s − 0.976·16-s − 0.242·17-s + 0.329·18-s − 0.471·19-s + 0.0307·20-s + 0.179·21-s + 1.75·22-s + 0.316·23-s − 0.583·24-s + 0.731·25-s − 0.177·26-s + 0.192·27-s − 0.00726·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+(1 + 0.806i)) \, \Gamma_{\R}(s+(1 - 0.806i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 51,\ (1 + 0.8065348943180i, 1 - 0.8065348943180i:\ ),\ 1)\)

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