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Properties

Label 2-639-213.41-c1-0-0
Degree $2$
Conductor $639$
Sign $-0.894 + 0.447i$
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  + (−1.14 + 2.38i)2-s + (−3.12 − 3.91i)4-s − 2.70i·5-s + (−1.69 − 3.51i)7-s + (7.75 − 1.76i)8-s + (6.46 + 3.11i)10-s + (−1.14 + 5.02i)11-s + (−3.19 + 0.729i)13-s + 10.3·14-s + (−2.45 + 10.7i)16-s + 4.05·17-s + (−2.54 + 3.19i)19-s + (−10.6 + 8.45i)20-s + (−10.6 − 8.51i)22-s + (−7.47 + 3.60i)23-s + ⋯
L(s)  = 1  + (−0.811 + 1.68i)2-s + (−1.56 − 1.95i)4-s − 1.21i·5-s + (−0.640 − 1.33i)7-s + (2.74 − 0.625i)8-s + (2.04 + 0.983i)10-s + (−0.346 + 1.51i)11-s + (−0.885 + 0.202i)13-s + 2.76·14-s + (−0.613 + 2.68i)16-s + 0.983·17-s + (−0.584 + 0.732i)19-s + (−2.37 + 1.89i)20-s + (−2.27 − 1.81i)22-s + (−1.55 + 0.750i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0424689 - 0.179825i\)
\(L(\frac12)\) \(\approx\) \(0.0424689 - 0.179825i\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.28 + 1.55i)T \)
good2 \( 1 + (1.14 - 2.38i)T + (-1.24 - 1.56i)T^{2} \)
5 \( 1 + 2.70iT - 5T^{2} \)
7 \( 1 + (1.69 + 3.51i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.14 - 5.02i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.19 - 0.729i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 - 4.05T + 17T^{2} \)
19 \( 1 + (2.54 - 3.19i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (7.47 - 3.60i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-3.02 + 2.41i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.89 - 0.660i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-5.60 - 2.70i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-2.01 - 8.84i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (3.79 + 1.82i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.07 - 1.34i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (7.12 - 8.93i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (1.72 - 7.56i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-0.287 + 0.597i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-12.0 + 9.57i)T + (14.9 - 65.3i)T^{2} \)
73 \( 1 + (-3.93 - 1.89i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (1.40 + 6.16i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-3.73 - 0.852i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (5.88 + 4.69i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (11.2 + 2.57i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.34189641228708647923354814411, −9.797414323527601330643438852500, −9.441429221432678440897346200853, −7.989481596613728251081448652868, −7.76482302697980161548932991777, −6.84959409221634657920758494989, −5.88482349313074736194992096954, −4.79649597384575674402958303850, −4.22130039487508665781459408170, −1.38329141292368255136827103078, 0.13724098250687915681095406753, 2.37729965100564737864467599261, 2.83252609759578443906411581111, 3.71111662655638031413586518376, 5.42930650806902943953807844410, 6.54740569431383943629335222060, 7.927732691352309679288818519359, 8.581768957507718068191333638657, 9.512562055373053157596955536924, 10.18073489122095576108584431932

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