Properties

Degree 1
Conductor 131
Sign $0.194 - 0.980i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.981 − 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (0.0241 − 0.999i)6-s + (0.958 + 0.285i)7-s + (0.836 − 0.548i)8-s + (−0.906 − 0.421i)9-s + (−0.943 − 0.331i)10-s + (−0.168 + 0.985i)11-s + (−0.168 − 0.985i)12-s + (−0.943 + 0.331i)13-s + (0.995 + 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (−0.443 − 0.896i)17-s + ⋯
L(s,χ)  = 1  + (0.981 − 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (0.0241 − 0.999i)6-s + (0.958 + 0.285i)7-s + (0.836 − 0.548i)8-s + (−0.906 − 0.421i)9-s + (−0.943 − 0.331i)10-s + (−0.168 + 0.985i)11-s + (−0.168 − 0.985i)12-s + (−0.943 + 0.331i)13-s + (0.995 + 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (−0.443 − 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.194 - 0.980i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.194 - 0.980i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $0.194 - 0.980i$
motivic weight  =  \(0\)
character  :  $\chi_{131} (41, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 131,\ (0:\ ),\ 0.194 - 0.980i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.427737055 - 1.172311202i$
$L(\frac12,\chi)$  $\approx$  $1.427737055 - 1.172311202i$
$L(\chi,1)$  $\approx$  1.541327977 - 0.7771967224i
$L(1,\chi)$  $\approx$  1.541327977 - 0.7771967224i

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

−29.04461249606561555588956027071, −27.68439723080575209611600250033, −26.73552785792584858599659177559, −26.193308161988686738538097187813, −24.53687065170446573778979322388, −23.968853681345633044855204550319, −22.59828220366031096385706195634, −22.10020650553907427027289728754, −20.94310427783434014587499564058, −20.2015606213784046087407725280, −19.10304738783183097590789366564, −17.2791111222065394261906786207, −16.29712051480356600175848169625, −15.26755102994563160688567571533, −14.63594205863176207845086031963, −13.750708208181711951644939085472, −12.05770017038048560272631099024, −11.14332392126406556126237898422, −10.39420034199352720611890289665, −8.34907318874653860117177445637, −7.58765149182022818549959516982, −5.886990216647363982621424341277, −4.655101438281449025495412393242, −3.785167951462365184023836170489, −2.609977725950688205345889924784, 1.527324227638635191763199535408, 2.7711938211190377674332145795, 4.47095358628278599907537941344, 5.36688104126253716299325944249, 7.207484124913736400422130389825, 7.6304693775219150252471769280, 9.28589679072677077290011637226, 11.31192608710935942429394254316, 11.92186857612428475552958681494, 12.723871655872693387262738608092, 13.92678976365637393518791522221, 14.85006780551582824219289909163, 15.79783196652579580491582598855, 17.30263012550997429922904200932, 18.44664216598007183743640264532, 19.84568777582931441037146100628, 20.13583868807976034382676969670, 21.38262829682481871341009082552, 22.7707573482081543026183941741, 23.49953839998020740053800794696, 24.60458444410686096562110788422, 24.68471510999946737290540880393, 26.321211181182738972867921784284, 27.81446554240286227356067296721, 28.64830037953726049760681763043

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