Properties

Degree $2$
Conductor $32$
Sign $1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more about

Dirichlet series

$L(s,f)$  = 1  − 0.900·3-s + 0.720·5-s + 1.02·7-s − 0.188·9-s + 0.429·11-s − 0.330·13-s − 0.648·15-s + 1.14·17-s − 1.45·19-s − 0.927·21-s + 0.277·23-s − 0.480·25-s + 1.07·27-s + 1.12·29-s + 0.334·31-s − 0.386·33-s + 0.742·35-s + 0.360·37-s + 0.297·39-s − 0.716·41-s − 0.963·43-s − 0.136·45-s − 1.03·47-s + 0.0606·49-s − 1.02·51-s + 0.527·53-s + 0.309·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s+1.58i) \, \Gamma_{\R}(s-1.58i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 32,\ (1.58298402762i, -1.58298402762i:\ ),\ 1)\)

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