Properties

Label 3-1-1.1-r0e3-m0.31m20.04p20.36-0
Degree $3$
Conductor $1$
Sign $1$
Analytic cond. $0.150168$
Root an. cond. $0.531528$
Arithmetic no
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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Dirichlet series

L(s)  = 1  + (−0.581 − 1.14i)2-s + (0.203 − 0.225i)3-s + (−0.393 + 0.185i)4-s + (−0.879 + 1.67i)5-s + (−0.377 − 0.101i)6-s + (0.245 − 0.645i)7-s + (−0.208 + 0.342i)8-s + (−0.213 − 0.317i)9-s + (2.43 + 0.0311i)10-s + (−0.498 − 0.359i)11-s + (−0.0380 + 0.126i)12-s + (−0.630 − 0.0785i)13-s + (−0.881 + 0.0937i)14-s + (0.200 + 0.540i)15-s + (−0.0832 − 0.547i)16-s + (0.853 − 0.0520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\R}(s-20.0i) \, \Gamma_{\R}(s-0.312i) \, \Gamma_{\R}(s+20.3i) \, L(s)\cr=\mathstrut & \,\overline{\Lambda}(1-s)\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(0.150168\)
Root analytic conductor: \(0.531528\)
Rational: no
Arithmetic: no
Primitive: yes
Self-dual: no
Selberg data: \((3,\ 1,\ (-20.0449461277998i, -0.3127111071210366i, 20.35765723492084i:\ ),\ 1)\)

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

−24.16351729735, −16.82752552414, −15.57424291881, −12.40123454361, −8.83190268322, −7.80794407460, −4.88203527638, 3.02160698877, 7.40020598873, 10.30439325147, 11.41876849284, 14.54844614571, 18.75696734645, 23.19141666783

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