Properties

Label 3-1-1.1-r0e3-m4.04m16.76p20.80-0
Degree $3$
Conductor $1$
Sign $1$
Analytic cond. $5.66046$
Root an. cond. $1.78217$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (0.434 + 0.644i)2-s + (0.678 + 0.270i)3-s + (−0.660 + 1.20i)4-s + (0.316 − 0.423i)5-s + (0.120 + 0.554i)6-s + (0.0105 − 0.112i)7-s + (−0.666 + 0.0981i)8-s + (−0.291 + 0.636i)9-s + (0.410 + 0.0197i)10-s + (0.0742 + 0.486i)11-s + (−0.773 + 0.638i)12-s + (1.76 − 0.0105i)13-s + (0.0769 − 0.0420i)14-s + (0.328 − 0.201i)15-s + (−0.406 − 0.691i)16-s + (0.538 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-16.7i) \, \Gamma_{\R}(s-4.03i) \, \Gamma_{\R}(s+20.7i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(5.66046\)
Root analytic conductor: \(1.78217\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 1,\ (-16.759573097542123i, -4.039412751323375i, 20.798985848865495i:\ ),\ 1)\)

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

−23.3605810701196, −18.7333508945641, −14.6200398230811, −13.3744862075461, −10.8083661984539, −8.8984577210562, −5.9146796144559, −3.4754690144629, −1.3812470524849, 8.4689964375154, 13.4921360470266, 20.9912819295316, 22.9431745527250

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