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 $0$

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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

Graph of the $Z$-function along the critical line