Properties

Degree 1
Conductor 131
Sign $-0.487 + 0.873i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $-0.487 + 0.873i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (5, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ -0.487 + 0.873i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2844987206 + 0.4847666735i$
$L(\frac12,\chi)$  $\approx$  $0.2844987206 + 0.4847666735i$
$L(\chi,1)$  $\approx$  0.5089394733 + 0.3458462640i
$L(1,\chi)$  $\approx$  0.5089394733 + 0.3458462640i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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