Dirichlet series
| $L(s,f)$ = 1 | + 1.13·2-s + 0.393·3-s + 0.289·4-s + 1.48·5-s + 0.447·6-s + 1.70·7-s − 0.807·8-s − 0.844·9-s + 1.69·10-s − 0.730·11-s + 0.113·12-s + 0.277·13-s + 1.93·14-s + 0.586·15-s − 1.20·16-s − 0.778·17-s − 0.959·18-s − 0.225·19-s + 0.430·20-s + 0.671·21-s − 0.829·22-s + 0.869·23-s − 0.317·24-s + 1.21·25-s + 0.314·26-s − 0.726·27-s + 0.493·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+(1 + 4.96i)) \, \Gamma_{\R}(s+(1 - 4.96i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(13\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 13,\ (1 + 4.96964302847i, 1 - 4.96964302847i:\ ),\ 1)\) |
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