Properties

Degree 1
Conductor 71
Sign $-0.809 + 0.586i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.809 + 0.586i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.809 + 0.586i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.809 + 0.586i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (54, \cdot )$
Sato-Tate  :  $\mu(5)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 71,\ (0:\ ),\ -0.809 + 0.586i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2974799917 + 0.9177013403i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2974799917 + 0.9177013403i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6986263381 + 0.8009032803i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6986263381 + 0.8009032803i\)

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

−30.925519612913097969183744693998, −30.23153083076018423443950361718, −29.51594104369282481386478896374, −28.25332594881897827597508935663, −26.8904551860661989669528120037, −26.17407785718693884588130584813, −24.0517658405000330562886212288, −23.72018266739959417747479084289, −22.56046102111023124572537638481, −21.1763625507074818310924855043, −19.87336433252962294469143743976, −19.234397234252233101571805702374, −18.35358203491721237597861963068, −16.93360603853131472167600354235, −14.84007173038876433723245487607, −13.96849738145078313089766039401, −12.9828863127665789505144155198, −11.49480199586583148993119832595, −10.98233413886806417607432941097, −9.02810248030520918818407323062, −7.70442989448004983546879258074, −6.313883352341728818566169892729, −4.154601741612806550822808282955, −3.00918260645587926930266165330, −1.16632299339860574045361892331, 3.27526540442796136980518357982, 4.64827047902065854909872442234, 5.52669789147003023966384543223, 7.61541582264587777157557126419, 8.61472145040845506246531181564, 9.64361169879213407340333752563, 11.65117241690968121374018134417, 12.79685706117192022938737428197, 14.49648311516042901080852136404, 15.3139953220971887837190211858, 16.02677991522209118319243258432, 17.1956044738936899529411353078, 18.60289000493752789175951252196, 20.25651781441391918541220777499, 21.15215637192293311191319668666, 22.54277729925778067201225488844, 23.15868206456786392862310139650, 24.96249828278937319827960967812, 25.177032448232243538822786609336, 26.83775036886913405610573497883, 27.54614711448784879065864067882, 28.30655318752459069895599718210, 30.62374414424634006299856221809, 31.42831548319946581081502180689, 32.10018650011734475736139928174

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