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Properties

Label 1-151-151.90-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.877 - 0.480i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.0627 − 0.998i)3-s + (0.913 + 0.406i)4-s + (0.604 − 0.796i)5-s + (−0.268 + 0.963i)6-s + (−0.570 − 0.821i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.756 + 0.653i)10-s + (−0.895 − 0.444i)11-s + (0.463 − 0.886i)12-s + (0.985 + 0.166i)13-s + (0.387 + 0.921i)14-s + (−0.756 − 0.653i)15-s + (0.669 + 0.743i)16-s + (0.996 + 0.0836i)17-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.0627 − 0.998i)3-s + (0.913 + 0.406i)4-s + (0.604 − 0.796i)5-s + (−0.268 + 0.963i)6-s + (−0.570 − 0.821i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.756 + 0.653i)10-s + (−0.895 − 0.444i)11-s + (0.463 − 0.886i)12-s + (0.985 + 0.166i)13-s + (0.387 + 0.921i)14-s + (−0.756 − 0.653i)15-s + (0.669 + 0.743i)16-s + (0.996 + 0.0836i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.877 - 0.480i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1601641177 - 0.6262599836i\)
\(L(\frac12)\) \(\approx\) \(0.1601641177 - 0.6262599836i\)
\(L(1)\) \(\approx\) \(0.5200350268 - 0.4509690214i\)
\(L(1)\) \(\approx\) \(0.5200350268 - 0.4509690214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (0.0627 - 0.998i)T \)
5 \( 1 + (0.604 - 0.796i)T \)
7 \( 1 + (-0.570 - 0.821i)T \)
11 \( 1 + (-0.895 - 0.444i)T \)
13 \( 1 + (0.985 + 0.166i)T \)
17 \( 1 + (0.996 + 0.0836i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.929 + 0.368i)T \)
31 \( 1 + (0.228 + 0.973i)T \)
37 \( 1 + (-0.348 - 0.937i)T \)
41 \( 1 + (0.968 - 0.248i)T \)
43 \( 1 + (-0.570 + 0.821i)T \)
47 \( 1 + (-0.699 - 0.714i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.832 - 0.553i)T \)
67 \( 1 + (-0.992 + 0.125i)T \)
71 \( 1 + (0.996 - 0.0836i)T \)
73 \( 1 + (-0.425 + 0.904i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (0.876 + 0.481i)T \)
89 \( 1 + (-0.0209 - 0.999i)T \)
97 \( 1 + (0.387 - 0.921i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.172362755508869632586999604003, −27.64564784314787890904395025816, −26.18674492330148277627625083319, −25.86683644087071835280642697723, −25.23861042730836586675216543290, −23.51554790757248485135760598390, −22.482448128527067575566623796652, −21.301409360704853595935014107043, −20.75263273010763059126998890988, −19.30417957702048198825531318860, −18.48456379291589517484893369078, −17.540911716133178611477873062812, −16.421322488587139545665012401204, −15.38049022846406434785420343682, −14.966128945652872834338738050160, −13.392328473486277049669074794863, −11.6395732095557723060724466784, −10.61697667385758217037898679994, −9.85033014637822069180557981151, −9.02777216112906155619766377524, −7.755348314783062852214737114835, −6.20123735194264427613312123741, −5.47969845042974107152304980508, −3.28653718631433711880735112508, −2.27908185222311739431344967781, 0.73688332783669305806512421029, 1.940691603369772421133982417992, 3.438799803012225384853792083631, 5.73668391356184486252562218680, 6.69793592249447826603481528672, 7.9870949758319947557146839491, 8.70341199174198924079966644421, 10.0432404934995689208579894132, 11.01707930400610190786010430038, 12.54122117519204290874226642904, 13.04085076018673720460717536900, 14.261715980399790747908368266422, 16.240186662823476292516784984689, 16.68037339711229106930195824355, 17.8175055014167973229060791405, 18.6542509336927980263408047093, 19.54175600177260191548816859063, 20.60026256731244211025585664223, 21.18989549158080239379412349883, 23.08770719085131644093599925492, 23.92915768113494232543699410935, 24.94080170170125965491316634233, 25.76822227632367160483404555015, 26.40321249865366823212578189764, 27.97318940632795571096999215939

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