Dirichlet series
| $L(s,f)$ = 1 | + 0.639·2-s − 1.02·3-s − 0.591·4-s + 0.274·5-s − 0.652·6-s − 0.609·7-s − 1.01·8-s + 0.0425·9-s + 0.175·10-s + 0.501·11-s + 0.603·12-s + 0.277·13-s − 0.389·14-s − 0.279·15-s − 0.0592·16-s − 1.81·17-s + 0.0272·18-s + 0.964·19-s − 0.162·20-s + 0.622·21-s + 0.320·22-s − 0.669·23-s + 1.03·24-s − 0.924·25-s + 0.177·26-s + 0.977·27-s + 0.360·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+4.23i) \, \Gamma_{\R}(s-4.23i) \, L(s,f)\cr =\mathstrut & -\, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(13\) |
| Sign: | $-1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 13,\ (4.23019956539i, -4.23019956539i:\ ),\ -1)\) |
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