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Properties

Label 2-639-71.57-c1-0-1
Degree $2$
Conductor $639$
Sign $0.545 + 0.837i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.782 + 2.40i)2-s + (−3.57 − 2.59i)4-s + (−1.75 + 1.27i)5-s + (−1.09 + 3.38i)7-s + (4.94 − 3.59i)8-s + (−1.69 − 5.22i)10-s + (−1.39 + 4.29i)11-s + (1.03 + 3.18i)13-s + (−7.28 − 5.29i)14-s + (2.05 + 6.33i)16-s + (−1.83 − 5.66i)17-s + (0.463 − 1.42i)19-s + 9.58·20-s + (−9.25 − 6.72i)22-s + 0.643·23-s + ⋯
L(s)  = 1  + (−0.553 + 1.70i)2-s + (−1.78 − 1.29i)4-s + (−0.785 + 0.570i)5-s + (−0.415 + 1.27i)7-s + (1.74 − 1.27i)8-s + (−0.537 − 1.65i)10-s + (−0.420 + 1.29i)11-s + (0.286 + 0.883i)13-s + (−1.94 − 1.41i)14-s + (0.514 + 1.58i)16-s + (−0.446 − 1.37i)17-s + (0.106 − 0.327i)19-s + 2.14·20-s + (−1.97 − 1.43i)22-s + 0.134·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.545 + 0.837i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.545 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269102 - 0.145837i\)
\(L(\frac12)\) \(\approx\) \(0.269102 - 0.145837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (-0.269 - 8.42i)T \)
good2 \( 1 + (0.782 - 2.40i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.75 - 1.27i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.09 - 3.38i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (1.39 - 4.29i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.03 - 3.18i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.83 + 5.66i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.463 + 1.42i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.643T + 23T^{2} \)
29 \( 1 + (1.83 - 1.33i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.48 + 7.65i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 - 9.34T + 41T^{2} \)
43 \( 1 + (-4.39 + 3.19i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (4.12 + 12.6i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.85 - 5.70i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.85 - 2.07i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.37 - 13.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.81 + 5.67i)T + (20.7 + 63.7i)T^{2} \)
73 \( 1 + (1.44 - 4.43i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.8 - 8.62i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.964 + 0.700i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-10.7 + 7.83i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + 5.92T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.31057257693147063041574283149, −9.935148201671161825813062206216, −9.247380627850640811204245289609, −8.675588884839712350304521070803, −7.39193209004009945062823139821, −7.20284899758897999018053131981, −6.16975746672586351150464726971, −5.22290842869970884753185754472, −4.27265493152007342685216411154, −2.52081688069602887549685701830, 0.22489078979506518634336816270, 1.25945253841102463780641107566, 3.12137229627990795089335593967, 3.72427011133913008010197936681, 4.61637101069825646760204208393, 6.18308145478300577862016537442, 7.84457052630275318818992365987, 8.217034857358299237676664872930, 9.117295832644142568272215953074, 10.21369177579288262581141088764

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