Properties

Label 1-151-151.99-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: $\chi_{151} (99, \cdot )$
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

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

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