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Properties

Label 1-151-151.80-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.498 - 0.867i$
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  + (0.913 − 0.406i)2-s + (−0.425 − 0.904i)3-s + (0.669 − 0.743i)4-s + (−0.348 − 0.937i)5-s + (−0.756 − 0.653i)6-s + (0.832 + 0.553i)7-s + (0.309 − 0.951i)8-s + (−0.637 + 0.770i)9-s + (−0.699 − 0.714i)10-s + (−0.570 − 0.821i)11-s + (−0.957 − 0.289i)12-s + (0.604 + 0.796i)13-s + (0.985 + 0.166i)14-s + (−0.699 + 0.714i)15-s + (−0.104 − 0.994i)16-s + (−0.895 − 0.444i)17-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.425 − 0.904i)3-s + (0.669 − 0.743i)4-s + (−0.348 − 0.937i)5-s + (−0.756 − 0.653i)6-s + (0.832 + 0.553i)7-s + (0.309 − 0.951i)8-s + (−0.637 + 0.770i)9-s + (−0.699 − 0.714i)10-s + (−0.570 − 0.821i)11-s + (−0.957 − 0.289i)12-s + (0.604 + 0.796i)13-s + (0.985 + 0.166i)14-s + (−0.699 + 0.714i)15-s + (−0.104 − 0.994i)16-s + (−0.895 − 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.498 - 0.867i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.498 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7910314724 - 1.366894108i\)
\(L(\frac12)\) \(\approx\) \(0.7910314724 - 1.366894108i\)
\(L(1)\) \(\approx\) \(1.144106366 - 0.9335926691i\)
\(L(1)\) \(\approx\) \(1.144106366 - 0.9335926691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (-0.425 - 0.904i)T \)
5 \( 1 + (-0.348 - 0.937i)T \)
7 \( 1 + (0.832 + 0.553i)T \)
11 \( 1 + (-0.570 - 0.821i)T \)
13 \( 1 + (0.604 + 0.796i)T \)
17 \( 1 + (-0.895 - 0.444i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.876 - 0.481i)T \)
31 \( 1 + (0.463 + 0.886i)T \)
37 \( 1 + (0.387 - 0.921i)T \)
41 \( 1 + (-0.187 + 0.982i)T \)
43 \( 1 + (0.832 - 0.553i)T \)
47 \( 1 + (0.944 - 0.328i)T \)
53 \( 1 + (0.728 + 0.684i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.996 - 0.0836i)T \)
67 \( 1 + (-0.637 - 0.770i)T \)
71 \( 1 + (-0.895 + 0.444i)T \)
73 \( 1 + (0.0627 - 0.998i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (-0.929 + 0.368i)T \)
89 \( 1 + (0.783 + 0.621i)T \)
97 \( 1 + (0.985 - 0.166i)T \)
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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

−28.31493263811241825446870570655, −27.25085967105334584339967785683, −26.27049315502670155715477339941, −25.72650784571435334395935376782, −24.0464030383624641517788889807, −23.37112093022915788435849500239, −22.4970889847432118129564449007, −21.80661489187295280480372582005, −20.656976470214748654266391960460, −20.01736443692131008410859792187, −17.87572245755684921467904320312, −17.45678227546771366392830362677, −15.83446841531360052623124164793, −15.37054948099602632313239427807, −14.48250438048889903686573657819, −13.37861508128838768595947837686, −11.87551286997987945922825894758, −10.99727785758805903914492523720, −10.29425485118529550844638551418, −8.30229759297389603527864612458, −7.20636073774590066347019678988, −6.01899697754178746531111538124, −4.75026847105643804861575256437, −3.93719417279311835559252728122, −2.61384135123085856232605408858, 1.20225019825266014713129157750, 2.39001185772569199682686477591, 4.24833882689609249661237275532, 5.349197868655449931394389041, 6.21042505170033839723111386303, 7.76611801167186421060141880668, 8.79643068203722262720698052560, 10.76171313728601452010318758478, 11.73976887305231337041333411921, 12.24487399894290535289606589567, 13.483762631262874718082177134299, 14.10402714431481152153096624748, 15.73413679272323635975968759252, 16.45487670440371710218904722377, 17.99788266998639976457961323412, 18.87920067194125151938397462461, 19.920865163870554327825647125626, 20.947515197593393401600159952452, 21.75693337009177977961272102007, 23.09187039839085646424322945355, 23.83385472780073644066664016345, 24.48214685620304323089901310283, 25.12805859068241983708195635327, 27.01693377346644272591306281666, 28.41365710917103474850016936844

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