Dirichlet series
| $L(s,f)$ = 1 | + 1.59·2-s − 0.462·3-s + 1.55·4-s + 0.268·5-s − 0.739·6-s − 0.915·7-s + 0.884·8-s − 0.785·9-s + 0.429·10-s + 1.44·11-s − 0.718·12-s + 0.971·13-s − 1.46·14-s − 0.124·15-s − 0.140·16-s − 0.719·17-s − 1.25·18-s + 1.76·19-s + 0.417·20-s + 0.423·21-s + 2.30·22-s − 0.291·23-s − 0.409·24-s − 0.927·25-s + 1.55·26-s + 0.826·27-s − 1.42·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.96i)) \, \Gamma_{\R}(s+(1 - 1.96i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(31\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 31,\ (1 + 1.96542621139i, 1 - 1.96542621139i:\ ),\ 1)\) |
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