Properties

Degree $2$
Conductor $34$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  + 0.707·2-s + 0.368·3-s + 0.5·4-s − 1.09·5-s + 0.260·6-s + 0.650·7-s + 0.353·8-s − 0.863·9-s − 0.772·10-s − 0.953·11-s + 0.184·12-s + 1.43·13-s + 0.459·14-s − 0.402·15-s + 0.250·16-s − 0.242·17-s − 0.610·18-s + 1.52·19-s − 0.546·20-s + 0.239·21-s − 0.674·22-s + 0.810·23-s + 0.130·24-s + 0.192·25-s + 1.01·26-s − 0.687·27-s + 0.325·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 34 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.43i)) \, \Gamma_{\R}(s+(1 - 1.43i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34\)    =    \(2 \cdot 17\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 34,\ (1 + 1.43772610252i, 1 - 1.43772610252i:\ ),\ 1)\)

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