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Properties

Label 1-67-67.34-r1-0-0
Degree $1$
Conductor $67$
Sign $0.533 - 0.845i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980550940 - 1.092203260i\)
\(L(\frac12)\) \(\approx\) \(1.980550940 - 1.092203260i\)
\(L(1)\) \(\approx\) \(1.529775123 - 0.3799690314i\)
\(L(1)\) \(\approx\) \(1.529775123 - 0.3799690314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.995 - 0.0950i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (-0.580 - 0.814i)T \)
11 \( 1 + (0.786 + 0.618i)T \)
13 \( 1 + (-0.235 - 0.971i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (0.580 - 0.814i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.327 + 0.945i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.0475 + 0.998i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (0.981 - 0.189i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (0.928 - 0.371i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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