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Properties

Label 1-131-131.105-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: $\chi_{131} (105, \cdot )$
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

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

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