Properties

Label 2-639-639.283-c0-0-5
Degree $2$
Conductor $639$
Sign $-0.318 - 0.947i$
Analytic cond. $0.318902$
Root an. cond. $0.564714$
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  + (0.988 + 1.71i)2-s + (0.826 − 0.563i)3-s + (−1.45 + 2.52i)4-s + (0.222 − 0.385i)5-s + (1.78 + 0.858i)6-s − 3.77·8-s + (0.365 − 0.930i)9-s + 0.880·10-s + (0.217 + 2.90i)12-s + (−0.0332 − 0.443i)15-s + (−2.28 − 3.95i)16-s + (1.95 − 0.294i)18-s − 1.97·19-s + (0.647 + 1.12i)20-s + (−3.12 + 2.12i)24-s + (0.400 + 0.694i)25-s + ⋯
L(s)  = 1  + (0.988 + 1.71i)2-s + (0.826 − 0.563i)3-s + (−1.45 + 2.52i)4-s + (0.222 − 0.385i)5-s + (1.78 + 0.858i)6-s − 3.77·8-s + (0.365 − 0.930i)9-s + 0.880·10-s + (0.217 + 2.90i)12-s + (−0.0332 − 0.443i)15-s + (−2.28 − 3.95i)16-s + (1.95 − 0.294i)18-s − 1.97·19-s + (0.647 + 1.12i)20-s + (−3.12 + 2.12i)24-s + (0.400 + 0.694i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(0.318902\)
Root analytic conductor: \(0.564714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :0),\ -0.318 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678585683\)
\(L(\frac12)\) \(\approx\) \(1.678585683\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.826 + 0.563i)T \)
71 \( 1 - T \)
good2 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.97T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 0.149T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−11.41050953841716007089127300360, −9.567496298094563017886995690503, −8.894200427786648573332681308082, −8.105235073985811442375955798623, −7.51827273289746739569477690467, −6.46705172246165748674070301529, −5.95191669458802514934196526539, −4.62095889893105193570256052863, −3.88592530580026676316155525678, −2.58020830461233674104833828243, 1.93465634187727637152683235773, 2.73106624625522479681594466815, 3.77749104335629209426548745712, 4.48892832504997331123894413218, 5.50432654087080233699564232828, 6.65932257878813131994816414391, 8.435083734318801161714176138328, 9.150443750739006771044681544744, 9.997303404164293737274005332907, 10.70506813363881751453444215608

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