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Properties

Label 2-639-639.70-c0-0-2
Degree $2$
Conductor $639$
Sign $0.698 - 0.715i$
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  + (−0.826 + 1.43i)2-s + (−0.733 + 0.680i)3-s + (−0.865 − 1.49i)4-s + (−0.623 − 1.07i)5-s + (−0.367 − 1.61i)6-s + 1.20·8-s + (0.0747 − 0.997i)9-s + 2.06·10-s + (1.65 + 0.510i)12-s + (1.19 + 0.367i)15-s + (−0.132 + 0.229i)16-s + (1.36 + 0.930i)18-s + 1.65·19-s + (−1.07 + 1.86i)20-s + (−0.885 + 0.821i)24-s + (−0.277 + 0.480i)25-s + ⋯
L(s)  = 1  + (−0.826 + 1.43i)2-s + (−0.733 + 0.680i)3-s + (−0.865 − 1.49i)4-s + (−0.623 − 1.07i)5-s + (−0.367 − 1.61i)6-s + 1.20·8-s + (0.0747 − 0.997i)9-s + 2.06·10-s + (1.65 + 0.510i)12-s + (1.19 + 0.367i)15-s + (−0.132 + 0.229i)16-s + (1.36 + 0.930i)18-s + 1.65·19-s + (−1.07 + 1.86i)20-s + (−0.885 + 0.821i)24-s + (−0.277 + 0.480i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.698 - 0.715i$
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: \((2,\ 639,\ (\ :0),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3826336361\)
\(L(\frac12)\) \(\approx\) \(0.3826336361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.733 - 0.680i)T \)
71 \( 1 - T \)
good2 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.65T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.24T + T^{2} \)
79 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - 1.91T + T^{2} \)
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

−10.56228285985964899052084877472, −9.708716020232581024708515358895, −9.048677262447020060960989468378, −8.301638137877622323218337641670, −7.43164152391498365174363419647, −6.50008932228611041564516325602, −5.41614843870445110554153932746, −4.96579868119190599678467285801, −3.73926449407821642176590990897, −0.74088366976534356196980207176, 1.29180947452822969782607743125, 2.71684208723707074168848505101, 3.53369430989747261285007271655, 5.06301908829835155045923876013, 6.44967208149114706062785086033, 7.36588555943792038792487744020, 8.033092080493714014240987128739, 9.197177603734747215780129076912, 10.17142603805933948969752007982, 10.87758226126822478115018449092

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