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Properties

Label 1-131-131.5-r0-0-0
Degree $1$
Conductor $131$
Sign $-0.487 + 0.873i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
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  + (−0.607 + 0.794i)2-s + (−0.989 + 0.144i)3-s + (−0.262 − 0.964i)4-s + (−0.168 + 0.985i)5-s + (0.485 − 0.873i)6-s + (0.981 − 0.192i)7-s + (0.926 + 0.377i)8-s + (0.958 − 0.285i)9-s + (−0.681 − 0.732i)10-s + (0.399 − 0.916i)11-s + (0.399 + 0.916i)12-s + (−0.681 + 0.732i)13-s + (−0.443 + 0.896i)14-s + (0.0241 − 0.999i)15-s + (−0.861 + 0.506i)16-s + (0.215 + 0.976i)17-s + ⋯
L(s)  = 1  + (−0.607 + 0.794i)2-s + (−0.989 + 0.144i)3-s + (−0.262 − 0.964i)4-s + (−0.168 + 0.985i)5-s + (0.485 − 0.873i)6-s + (0.981 − 0.192i)7-s + (0.926 + 0.377i)8-s + (0.958 − 0.285i)9-s + (−0.681 − 0.732i)10-s + (0.399 − 0.916i)11-s + (0.399 + 0.916i)12-s + (−0.681 + 0.732i)13-s + (−0.443 + 0.896i)14-s + (0.0241 − 0.999i)15-s + (−0.861 + 0.506i)16-s + (0.215 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $-0.487 + 0.873i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ -0.487 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2844987206 + 0.4847666735i\)
\(L(\frac12)\) \(\approx\) \(0.2844987206 + 0.4847666735i\)
\(L(1)\) \(\approx\) \(0.5089394733 + 0.3458462640i\)
\(L(1)\) \(\approx\) \(0.5089394733 + 0.3458462640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.607 + 0.794i)T \)
3 \( 1 + (-0.989 + 0.144i)T \)
5 \( 1 + (-0.168 + 0.985i)T \)
7 \( 1 + (0.981 - 0.192i)T \)
11 \( 1 + (0.399 - 0.916i)T \)
13 \( 1 + (-0.681 + 0.732i)T \)
17 \( 1 + (0.215 + 0.976i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (0.644 + 0.764i)T \)
29 \( 1 + (0.958 + 0.285i)T \)
31 \( 1 + (-0.0724 + 0.997i)T \)
37 \( 1 + (-0.998 - 0.0483i)T \)
41 \( 1 + (-0.861 - 0.506i)T \)
43 \( 1 + (0.715 - 0.698i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.995 + 0.0965i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.715 + 0.698i)T \)
71 \( 1 + (-0.970 + 0.239i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (0.779 - 0.626i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.906 - 0.421i)T \)
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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

−28.17697368794400764795293847673, −27.688377660188568896708468394545, −26.99363651778000397456046226616, −25.19516812269134122343608718238, −24.477220035265430361461525504794, −23.19275184873173871524439174261, −22.23392481808132858147864849556, −21.09900146023364107911272414097, −20.385262052462857368459208648110, −19.22468301764746753378439287599, −17.91253462198810612664624031024, −17.39082378967591555384273946411, −16.541655829854793652683569883273, −15.186360637210643818280930930662, −13.29874556030816532700731007089, −12.26744799311473051773972534553, −11.76116208015610073278578999321, −10.56351154581738525980261652622, −9.431332485188726984754388762, −8.18499059584718473282049508183, −7.09716267362700792221051272501, −5.049897039041425060634303067341, −4.46070876945290632983527209271, −2.15921376713303943372973283549, −0.78575363927296020507008336692, 1.5420675254898413136287603012, 4.07845015021296861310949365957, 5.4249707659703354134706469155, 6.50434994350707649241110050038, 7.39432205573381050026667510825, 8.71251976976797328386663485344, 10.31964622153908914669422730116, 10.882636008891110378676962891768, 11.97369875685540062137041789192, 13.97311804841065174167955934608, 14.74372785175875930222962000630, 15.803625863874077909734094538230, 17.054668395516344049504616706786, 17.49221051195567055591548819248, 18.741032359430116331178355618922, 19.32382848568015277585101790017, 21.311343327957334200572801594599, 22.106553516305836280280887039766, 23.4336114185722728600021342687, 23.808463273197755232677402659815, 24.9719521781869394418312844570, 26.31186443341466941101940004318, 27.22226027375543266041023697182, 27.50137965545007764218742545873, 28.9536955801876360014923633296

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