Dirichlet series
| $L(s,f)$ = 1 | − 0.700·2-s + 0.577·3-s − 0.508·4-s + 0.447·5-s − 0.404·6-s + 1.11·7-s + 1.05·8-s + 0.333·9-s − 0.313·10-s − 0.102·11-s − 0.293·12-s + 1.36·13-s − 0.778·14-s + 0.258·15-s − 0.232·16-s − 0.996·17-s − 0.233·18-s + 1.58·19-s − 0.227·20-s + 0.641·21-s + 0.0715·22-s − 0.394·23-s + 0.610·24-s + 0.199·25-s − 0.955·26-s + 0.192·27-s − 0.565·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+4.61i) \, \Gamma_{\R}(s-4.61i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(15\) = \(3 \cdot 5\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 15,\ (4.6113780114i, -4.6113780114i:\ ),\ 1)\) |
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