Properties

Degree 1
Conductor 71
Sign $0.854 + 0.519i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.393 + 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 − 0.722i)4-s + (0.309 − 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (0.936 − 0.351i)8-s + (0.936 + 0.351i)9-s + (0.753 + 0.657i)10-s + (−0.963 + 0.266i)11-s + (−0.550 − 0.834i)12-s + (−0.963 − 0.266i)13-s + (0.309 + 0.951i)14-s + (0.473 − 0.880i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s,χ)  = 1  + (−0.393 + 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 − 0.722i)4-s + (0.309 − 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (0.936 − 0.351i)8-s + (0.936 + 0.351i)9-s + (0.753 + 0.657i)10-s + (−0.963 + 0.266i)11-s + (−0.550 − 0.834i)12-s + (−0.963 − 0.266i)13-s + (0.309 + 0.951i)14-s + (0.473 − 0.880i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.854 + 0.519i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.854 + 0.519i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.854 + 0.519i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (50, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.854 + 0.519i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9927453351 + 0.2783634664i$
$L(\frac12,\chi)$  $\approx$  $0.9927453351 + 0.2783634664i$
$L(\chi,1)$  $\approx$  1.066175206 + 0.2762380642i
$L(1,\chi)$  $\approx$  1.066175206 + 0.2762380642i

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

−31.126078240607613544093499685, −30.67880477212795118796422101817, −29.54559992000993890816595490694, −28.55176362915648332676796969011, −26.84864647272310643241579637761, −26.6115190549968675876749706508, −25.334650036803291857735925977017, −24.11811243637060771197914591823, −22.29422908633571533296557748510, −21.47785622935370785037454371875, −20.523109862993696214718679608136, −19.28310518139032290527779650533, −18.41418919715919329983822563549, −17.68960823755824659286422319066, −15.55503096251957239545891963328, −14.31639283983521187938730870197, −13.403578444191167274331090113116, −11.96274799591227401760271039781, −10.6830594824132624638797425953, −9.51355386464829816963675254730, −8.34027184706800161099769419037, −7.169119829775527861309235338786, −4.75436770074589340832488575315, −2.84300216236873777990348007172, −2.22393705951275504771069381705, 1.73690650769953652870517947618, 4.28346597972935073180901778619, 5.33393889035050047291984954365, 7.469759417185134334006374857165, 8.155295595201770133737080008589, 9.41186386867819854535254350736, 10.411831036329084101119863521181, 12.85524735074896637353611279393, 13.81754634364142539534421181004, 14.88677716102963617366936726276, 15.97104870536622840184699563144, 17.14907904688218069005362530692, 18.162831103870807539816931863379, 19.70666740947048815690914235846, 20.48085527893486629591152239503, 21.74304228073683392752523748746, 23.58557984178042227797379602342, 24.369501467713579195629568793, 25.20158029305844605950874043670, 26.35274422161759878373489990299, 27.13213789515530215541384010184, 28.19504883424701036658844760794, 29.57866871878770169460618984742, 31.308854795476399747177124061807, 31.76819159998185472439448524712

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