Properties

Degree 1
Conductor 67
Sign $0.533 + 0.845i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.995 + 0.0950i)2-s + (−0.841 − 0.540i)3-s + (0.981 + 0.189i)4-s + (0.142 + 0.989i)5-s + (−0.786 − 0.618i)6-s + (−0.580 + 0.814i)7-s + (0.959 + 0.281i)8-s + (0.415 + 0.909i)9-s + (0.0475 + 0.998i)10-s + (0.786 − 0.618i)11-s + (−0.723 − 0.690i)12-s + (−0.235 + 0.971i)13-s + (−0.654 + 0.755i)14-s + (0.415 − 0.909i)15-s + (0.928 + 0.371i)16-s + (0.981 − 0.189i)17-s + ⋯
L(s,χ)  = 1  + (0.995 + 0.0950i)2-s + (−0.841 − 0.540i)3-s + (0.981 + 0.189i)4-s + (0.142 + 0.989i)5-s + (−0.786 − 0.618i)6-s + (−0.580 + 0.814i)7-s + (0.959 + 0.281i)8-s + (0.415 + 0.909i)9-s + (0.0475 + 0.998i)10-s + (0.786 − 0.618i)11-s + (−0.723 − 0.690i)12-s + (−0.235 + 0.971i)13-s + (−0.654 + 0.755i)14-s + (0.415 − 0.909i)15-s + (0.928 + 0.371i)16-s + (0.981 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.533 + 0.845i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.533 + 0.845i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.533 + 0.845i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (2, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (1:\ ),\ 0.533 + 0.845i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.980550940 + 1.092203260i$
$L(\frac12,\chi)$  $\approx$  $1.980550940 + 1.092203260i$
$L(\chi,1)$  $\approx$  1.529775123 + 0.3799690314i
$L(1,\chi)$  $\approx$  1.529775123 + 0.3799690314i

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

−32.0860997608806226050259278790, −30.30068950046386923864193314244, −29.47278822209482229652693351553, −28.42722510903025563997917079800, −27.58184355808159933898790959084, −25.88039934923639137380630314232, −24.62496420530947158587926660656, −23.54754800828053375622160281914, −22.70012657728969058123643163728, −21.7601733188229683832018875236, −20.46546940964177642418990663136, −19.86199286284046161189282606117, −17.521616262782847864373366391039, −16.5708300756243728680137775805, −15.7477331192231501342074546393, −14.292972493014341419848864319288, −12.78511091196945487600506947073, −12.16534367589574757076654287071, −10.63969925556725098718943981492, −9.59114248648782310422035745627, −7.26381104590399578311814476482, −5.85616558480951726275085754062, −4.75944937935268326829004540346, −3.6233106332487323312219461875, −1.022137483119961865812362567622, 2.04398919337999714727731013531, 3.65720061357952602547949088261, 5.67964835852877766753341836384, 6.33525076195561171445705399303, 7.54039677409698998290086393963, 9.95946652463905708730601916223, 11.55968285006404015624994143185, 12.0012930559934220032315937342, 13.555276813978283060055852465292, 14.51729314772787004984055050257, 15.96566073145053090920051375645, 16.948668884896585568643520409213, 18.61828727789095674753277483931, 19.3026341885628123253881651962, 21.3419478461252987371739483732, 22.23811957260963812875099452279, 22.7809586211342197034806053849, 24.067242483582228762357168466724, 25.006239204212810210072227103533, 26.1037124335545414422583513953, 27.8120168297427308311091648409, 29.202090675547551245900768451263, 29.66370189343486775851562989407, 30.77591006059832307972793980947, 31.80403639149337697264809183404

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