Properties

Degree $2$
Conductor $68$
Sign $-1$
Arithmetic no
Primitive yes
Self-dual yes

Related objects

Learn more

Dirichlet series

$L(s,f)$  = 1  − 0.882·3-s − 0.366·5-s − 0.181·7-s − 0.220·9-s + 0.852·11-s + 1.60·13-s + 0.323·15-s − 0.242·17-s + 1.48·19-s + 0.160·21-s − 1.53·23-s − 0.865·25-s + 1.07·27-s − 0.366·29-s + 0.0768·31-s − 0.752·33-s + 0.0667·35-s + 1.12·37-s − 1.41·39-s + 0.100·41-s − 1.74·43-s + 0.0810·45-s − 1.77·47-s − 0.966·49-s + 0.214·51-s − 0.742·53-s − 0.312·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,f)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+5.33i) \, \Gamma_{\R}(s-5.33i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-1$
Arithmetic: no
Primitive: yes
Self-dual: yes
Selberg data: \((2,\ 68,\ (5.3302064817i, -5.3302064817i:\ ),\ -1)\)

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