↑

Properties

Label 1-151-151.118-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.683 - 0.729i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.683 - 0.729i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.683 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3603534097 - 0.8311815292i\)
\(L(\frac12)\) \(\approx\) \(0.3603534097 - 0.8311815292i\)
\(L(1)\) \(\approx\) \(0.7097555238 - 0.5796281141i\)
\(L(1)\) \(\approx\) \(0.7097555238 - 0.5796281141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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

−28.06239842310728826246952158804, −27.158008894349928953028532681786, −26.26141824964534737262316295185, −25.618319505238227340410506225633, −24.86107843546373516840297122401, −23.742737117345345138966854984185, −22.637791085748192255764511235346, −21.75167613800448062312200076603, −19.96997613464764262262775074005, −19.48449663298128957246417987257, −18.36631771667652061426383225554, −17.78071269957557299692998713648, −15.868839932851977916465257507226, −15.38249975981640931244986945316, −14.70782874648371068527246214006, −13.51773797079636215659186257519, −12.28099146683873870854082178402, −10.40578058920077061364411014322, −9.702475796963852732728364997833, −8.44001042327082351224207289138, −7.5821259803293068797883029733, −6.652343383489647449375122222213, −5.14664297018373821305381096444, −3.496774788523583023896421957054, −2.177363411196833982471932950212, 0.84588185844928242084553443379, 2.5314933269547026994407412243, 3.741303505589332802190387992478, 4.653467642563402901460818659874, 7.1724954033879539665965172759, 8.077933175390361524486618844196, 9.101716989922729900313431839018, 9.87741450631528856301434644358, 11.22195617342694956026706073658, 12.42528009095840160539504533575, 13.40014663800537902815408654341, 14.095274558130438180604510336, 16.10222771330670608906367743936, 16.4237331748114289833003212204, 18.02247156534338598212169560524, 19.168207486827317881771164354292, 19.78844107980744294757977185434, 20.54674091490891990157278007744, 21.3234075314120498742795476057, 22.59584580126526276214339519164, 23.96593631037536119682909785648, 24.80456008919292052729574741324, 26.36159445472723950968878323776, 26.56182475432901155670149832416, 27.555080525848490010720136569135

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