Dirichlet series
| $L(s,f)$ = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s + 0.916·5-s − 0.967·6-s − 0.377·7-s + 0.353·8-s + 0.870·9-s + 0.647·10-s + 0.266·11-s − 0.683·12-s − 0.595·13-s − 0.267·14-s − 1.25·15-s + 0.250·16-s − 0.519·17-s + 0.615·18-s + 1.10·19-s + 0.458·20-s + 0.516·21-s + 0.188·22-s − 0.131·23-s − 0.483·24-s − 0.160·25-s − 0.420·26-s + 0.176·27-s − 0.188·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 14 ^{s/2} \, \Gamma_{\R}(s+(1 + 1.79i)) \, \Gamma_{\R}(s+(1 - 1.79i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(14\) = \(2 \cdot 7\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 14,\ (1 + 1.79416536046i, 1 - 1.79416536046i:\ ),\ 1)\) |
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