# Properties

 Label 3-1-1.1-r0e3-m3.50m32.99p36.49-0 Degree $3$ Conductor $1$ Sign $1$ Analytic cond. $16.9237$ Root an. cond. $2.56743$ Arithmetic no Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.141 − 0.853i)2-s + (0.0379 − 0.413i)3-s + (−0.850 − 1.09i)4-s + (−0.0982 + 0.668i)5-s + (−0.347 − 0.0910i)6-s + (−0.490 + 0.998i)7-s + (−0.804 + 0.570i)8-s + (−0.207 − 0.444i)9-s + (0.557 + 0.178i)10-s + (0.183 + 0.858i)11-s + (−0.485 + 0.310i)12-s + (1.62 + 0.191i)13-s + (0.783 + 0.559i)14-s + (0.272 + 0.0660i)15-s + (−0.299 + 0.795i)16-s + (−0.569 − 0.201i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-32.9i) \, \Gamma_{\R}(s-3.50i) \, \Gamma_{\R}(s+36.4i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}

## Invariants

 Degree: $$3$$ Conductor: $$1$$ Sign: $1$ Analytic conductor: $$16.9237$$ Root analytic conductor: $$2.56743$$ Rational: no Arithmetic: no Primitive: yes Self-dual: no Selberg data: $$(3,\ 1,\ (-32.987429682i, -3.5002128372i, 36.48764252i:\ ),\ 1)$$

## Euler product

$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{3} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−23.996557, −22.632144, −21.380980, −19.832215, −17.725395, −16.475513, −15.975519, −13.697314, −13.085643, −10.957415, −8.954849, −8.030527, −6.397202, −4.514330, −3.569462, −0.455495, 1.593869, 6.163996, 9.088982, 10.658646, 12.247686, 13.923596, 15.318678, 18.266187, 18.868431, 20.432673, 22.423698, 23.420472