Properties

Label 1-311-311.171-r1-0-0
Degree $1$
Conductor $311$
Sign $0.983 + 0.183i$
Analytic cond. $33.4215$
Root an. cond. $33.4215$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(311\)
Sign: $0.983 + 0.183i$
Analytic conductor: \(33.4215\)
Root analytic conductor: \(33.4215\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{311} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 311,\ (1:\ ),\ 0.983 + 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.607527770 + 0.5179849215i\)
\(L(\frac12)\) \(\approx\) \(5.607527770 + 0.5179849215i\)
\(L(1)\) \(\approx\) \(2.788798567 + 0.09500446186i\)
\(L(1)\) \(\approx\) \(2.788798567 + 0.09500446186i\)

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

−24.9530730012816838124434485957, −24.20591755550069151849325149675, −23.415745348937179824426755570858, −22.11706596794014106619869849474, −21.54955531620616348952149016044, −20.386464454708427057101932761523, −19.94607426721238108537616062912, −19.0852085931239408886073499819, −17.63768082691550464060080186360, −16.419318589844120770115130412891, −15.695214928414823438978739015590, −14.94738465258773459192639857887, −13.87255619141807892904457509719, −12.97092853223776666126856425924, −12.567348683248152638351762124150, −11.46368429337648428135300878496, −9.85559112781702545406328693824, −8.763611197880579391321580282347, −8.13118423606830803640155256143, −6.675981972648792308265054786509, −5.92832114126941447820811826010, −4.44909347573031722112597005457, −3.67732989863936338208579570380, −2.51118822549346829581428977808, −1.28057492687278021114310209425, 1.44982077899366882962578156185, 2.77857434973957359112409375404, 3.637529606444848900306395171932, 4.20652447265654468823044180445, 6.077777046115065929350346103458, 6.84410598415771426047602457589, 7.74916167006874797892626384774, 9.37187576578425045570828136685, 10.196330994990650060212008975, 11.136637707805663722942885340331, 12.226664856844395263883855532936, 13.51400204445971113742512694829, 14.03708668860305890787586452406, 14.60884207989977962127153424730, 15.88509971483886576933998397330, 16.2641871812411074040489967361, 18.072347371411511616441299493356, 19.31840110513395775618599115612, 19.57942800710578368821431185262, 20.6832277967293992862651165414, 21.46065808569504216909069278773, 22.48744420960844845996608144801, 22.96505467474005126227078488999, 24.068144372561431720262900562283, 25.29530646892760118450507084470

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