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Properties

Label 2-639-639.70-c0-0-6
Degree $2$
Conductor $639$
Sign $-0.939 + 0.342i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−1 − 1.73i)5-s − 0.999·6-s + 8-s + (−0.499 + 0.866i)9-s − 1.99·10-s + (−0.999 + 1.73i)15-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s − 19-s + (−0.500 − 0.866i)24-s + (−1.49 + 2.59i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + (0.999 + 1.73i)30-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−1 − 1.73i)5-s − 0.999·6-s + 8-s + (−0.499 + 0.866i)9-s − 1.99·10-s + (−0.999 + 1.73i)15-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s − 19-s + (−0.500 − 0.866i)24-s + (−1.49 + 2.59i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + (0.999 + 1.73i)30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(0.318902\)
Root analytic conductor: \(0.564714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :0),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9054947318\)
\(L(\frac12)\) \(\approx\) \(0.9054947318\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 2T + T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
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

−10.94810329710770023372894674935, −9.601889766075401680398273533297, −8.273198157775566864001504648664, −8.054068979675368782004748867734, −6.93168592279735855617674122175, −5.56942463759524912869939188026, −4.62902339608169143428894644117, −3.93741213411494658494488744051, −2.30704217810534993027887754360, −0.999886217357332673956077748159, 2.81768141325513181605848721396, 3.95908717339340273953204206022, 4.67377130531210673298863397406, 6.06147144854335427000590045934, 6.48973965145159029043071871756, 7.38653167087342957204503546604, 8.238218180185442911372692183309, 9.701370160213133382693910587771, 10.60663244245285010480949968391, 10.96122332063516037263032534162

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