Properties

Degree 1
Conductor 71
Sign $-0.998 - 0.0614i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.134 + 0.990i)2-s + (−0.550 − 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.809 + 0.587i)5-s + (0.753 − 0.657i)6-s + (−0.691 + 0.722i)7-s + (−0.393 − 0.919i)8-s + (−0.393 + 0.919i)9-s + (−0.691 − 0.722i)10-s + (−0.995 − 0.0896i)11-s + (0.753 + 0.657i)12-s + (−0.995 + 0.0896i)13-s + (−0.809 − 0.587i)14-s + (0.936 + 0.351i)15-s + (0.858 − 0.512i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.134 + 0.990i)2-s + (−0.550 − 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.809 + 0.587i)5-s + (0.753 − 0.657i)6-s + (−0.691 + 0.722i)7-s + (−0.393 − 0.919i)8-s + (−0.393 + 0.919i)9-s + (−0.691 − 0.722i)10-s + (−0.995 − 0.0896i)11-s + (0.753 + 0.657i)12-s + (−0.995 + 0.0896i)13-s + (−0.809 − 0.587i)14-s + (0.936 + 0.351i)15-s + (0.858 − 0.512i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.998 - 0.0614i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.998 - 0.0614i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.998 - 0.0614i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (3, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ -0.998 - 0.0614i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.008491461097 + 0.2759367663i$
$L(\frac12,\chi)$  $\approx$  $0.008491461097 + 0.2759367663i$
$L(\chi,1)$  $\approx$  0.4291529565 + 0.2873547158i
$L(1,\chi)$  $\approx$  0.4291529565 + 0.2873547158i

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

−31.499788606305134835728523516200, −29.75755922579475365427360978049, −28.92070637951300771145086746604, −28.05648030039012528394053658126, −26.98452904640747474161479213902, −26.37825141852278295744031641272, −24.105937635606786284198643020179, −23.01963171750021747049867571804, −22.462988081239169851158524235346, −20.95801362686040666446526974589, −20.29857672627368689624711635269, −19.22275729563063604909547344036, −17.70901837551388970356797752669, −16.47337139084398181569330760365, −15.46954020518875210431298625861, −13.822935206725208228724375109135, −12.42604275053871375093326580146, −11.595136924649488047773037131056, −10.22792173997149533119256683489, −9.5070278467813976921723828042, −7.72514658810739839275565661843, −5.39745410591807751135533706819, −4.35492178234084544837278616608, −3.15558612100620440940967396263, −0.32178154833034830476105345051, 2.98788271166085887148448147673, 5.039750927886499711479134343042, 6.28842297157690522563675221763, 7.35085638166833273349472560407, 8.33109536802639074042623389610, 10.218977197777907768098299233863, 12.00682520951696089825061287647, 12.78975123218302581971007929945, 14.21179305932152680201959029044, 15.49782937985420702473707838510, 16.36290991353125768441600836597, 17.78677574336538086948009853349, 18.6737425348688368068411867132, 19.494826155110234818801676400340, 21.92881111153974459130123721868, 22.575273238634362313823355704814, 23.68547054327314600726533362190, 24.35465449794328554894856419103, 25.66916709462295543586953893120, 26.536848974661099551434170216010, 27.89390412447425495650003535803, 28.91307970551790025851937307317, 30.376593873477367222022009729465, 31.26681278219021498125863770892, 32.11814778403276082339621191153

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