Properties

Label 3-1-1.1-r0e3-p0.22p30.80m31.02-0
Degree $3$
Conductor $1$
Sign $1$
Analytic cond. $0.216977$
Root an. cond. $0.600904$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (0.222 + 0.568i)2-s + (0.641 + 0.429i)3-s + (−0.496 + 0.821i)4-s + (−0.955 − 0.556i)5-s + (−0.101 + 0.460i)6-s + (0.492 − 0.587i)7-s + (0.0489 − 0.0995i)8-s + (−0.414 + 0.981i)9-s + (0.104 − 0.667i)10-s + (−0.634 − 1.09i)11-s + (−0.671 + 0.314i)12-s + (−0.348 − 0.568i)13-s + (0.443 + 0.149i)14-s + (−0.374 − 0.767i)15-s + (−0.0670 + 0.109i)16-s + (−0.251 + 0.414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s+30.8i) \, \Gamma_{\R}(s+0.220i) \, \Gamma_{\R}(s-31.0i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(0.216977\)
Root analytic conductor: \(0.600904\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 1,\ (30.8035371i, 0.2201810276i, -31.02371812i:\ ),\ 1)\)

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

−23.91931, −22.66232, −20.43442, −19.29597, −18.13866, −15.19389, −14.29029, −12.41668, −10.92357, −9.00284, −7.21544, −4.65163, −2.50640, 3.55707, 4.90288, 7.92097, 8.31284, 10.86487, 12.95891, 14.29503, 15.99689, 16.99508, 19.40759, 20.82755, 22.59395, 24.27488

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