↑

Properties

Label 1-311-311.20-r0-0-0
Degree $1$
Conductor $311$
Sign $0.989 - 0.143i$
Analytic cond. $1.44427$
Root an. cond. $1.44427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.979 + 0.201i)2-s + (0.979 − 0.201i)3-s + (0.918 + 0.394i)4-s + (−0.0506 − 0.998i)5-s + 6-s + (−0.758 + 0.651i)7-s + (0.820 + 0.571i)8-s + (0.918 − 0.394i)9-s + (0.151 − 0.988i)10-s + (−0.994 − 0.101i)11-s + (0.979 + 0.201i)12-s + (0.918 − 0.394i)13-s + (−0.874 + 0.485i)14-s + (−0.250 − 0.968i)15-s + (0.688 + 0.724i)16-s + (0.151 − 0.988i)17-s + ⋯
L(s)  = 1  + (0.979 + 0.201i)2-s + (0.979 − 0.201i)3-s + (0.918 + 0.394i)4-s + (−0.0506 − 0.998i)5-s + 6-s + (−0.758 + 0.651i)7-s + (0.820 + 0.571i)8-s + (0.918 − 0.394i)9-s + (0.151 − 0.988i)10-s + (−0.994 − 0.101i)11-s + (0.979 + 0.201i)12-s + (0.918 − 0.394i)13-s + (−0.874 + 0.485i)14-s + (−0.250 − 0.968i)15-s + (0.688 + 0.724i)16-s + (0.151 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(1.44427\)
Root analytic conductor: \(1.44427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (0:\ ),\ 0.989 - 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.937280121 - 0.2118454662i\)
\(L(\frac12)\) \(\approx\) \(2.937280121 - 0.2118454662i\)
\(L(1)\) \(\approx\) \(2.294050002 - 0.06660678403i\)
\(L(1)\) \(\approx\) \(2.294050002 - 0.06660678403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad311 \( 1 \)
good2 \( 1 + (0.979 + 0.201i)T \)
3 \( 1 + (0.979 - 0.201i)T \)
5 \( 1 + (-0.0506 - 0.998i)T \)
7 \( 1 + (-0.758 + 0.651i)T \)
11 \( 1 + (-0.994 - 0.101i)T \)
13 \( 1 + (0.918 - 0.394i)T \)
17 \( 1 + (0.151 - 0.988i)T \)
19 \( 1 + (-0.0506 + 0.998i)T \)
23 \( 1 + (0.820 + 0.571i)T \)
29 \( 1 + (-0.874 - 0.485i)T \)
31 \( 1 + (0.151 + 0.988i)T \)
37 \( 1 + (-0.758 - 0.651i)T \)
41 \( 1 + (-0.440 - 0.897i)T \)
43 \( 1 + (-0.758 + 0.651i)T \)
47 \( 1 + (-0.874 - 0.485i)T \)
53 \( 1 + (-0.758 + 0.651i)T \)
59 \( 1 + (-0.758 + 0.651i)T \)
61 \( 1 + (-0.0506 + 0.998i)T \)
67 \( 1 + (-0.440 + 0.897i)T \)
71 \( 1 + (0.528 - 0.848i)T \)
73 \( 1 + (0.347 + 0.937i)T \)
79 \( 1 + (-0.440 - 0.897i)T \)
83 \( 1 + (-0.250 + 0.968i)T \)
89 \( 1 + (-0.758 - 0.651i)T \)
97 \( 1 + (0.528 - 0.848i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.54308227114446626534722094607, −24.11085120896547926718719973444, −23.44496731053732457080697351149, −22.52868156674974904599299916853, −21.673283082845232328628194486026, −20.85515765575434122626448696488, −20.051203018923874413512108627488, −19.11155086538452199490476858719, −18.56245943231293661666309407533, −16.78333696283692947322059496711, −15.65301891978469901655566405492, −15.17714064987137056629463989613, −14.19263714411554118464243611394, −13.322908315341796086009866003281, −12.893866375048779595512931648298, −11.14320296236234320004204392377, −10.54424994239559597274216677021, −9.64834504682903848564560028951, −8.085210458685513502828280239076, −7.03694712621002409693905729209, −6.32793157667026144065086769833, −4.73592267494300720319255568585, −3.56606839209004407692337100982, −3.06648742936511552248568280830, −1.87216023943704597824185536562, 1.6252515409805164496235949354, 2.916990325952675127827422257070, 3.67111078648702381565814598450, 5.05258304345933888921028573598, 5.88724070222555860589146663742, 7.24701392412852345298939840015, 8.19198838268514237370284544508, 9.04473242560140390986337448043, 10.2566496736047768971670117325, 11.78003164216301757610132356886, 12.74751215959030099475315436792, 13.19063230725443390521289658621, 14.01670476738896755089017538770, 15.357869822426199977762059913188, 15.78676297919312576405328784821, 16.58646218078182822712347856429, 18.172793994167023518539364144966, 19.15635296280620168105997633722, 20.12047055155315414578905685094, 20.96895153131571650592459759531, 21.25426218293302124008773113543, 22.75892667333739631739314620745, 23.44051513520963640915036775345, 24.4820090062560249580932006337, 25.16652025102174569607948005578

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