Properties

Label 2-639-71.70-c0-0-2
Degree $2$
Conductor $639$
Sign $1$
Analytic cond. $0.318902$
Root an. cond. $0.564714$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.80·2-s + 2.24·4-s − 1.24·5-s + 2.24·8-s − 2.24·10-s + 1.80·16-s − 1.80·19-s − 2.80·20-s + 0.554·25-s + 0.445·29-s + 1.00·32-s − 1.80·37-s − 3.24·38-s − 2.80·40-s + 1.24·43-s + 49-s + 0.999·50-s + 0.801·58-s − 71-s + 1.24·73-s − 3.24·74-s − 4.04·76-s + 1.24·79-s − 2.24·80-s + 1.80·83-s + 2.24·86-s + 0.445·89-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s − 1.24·5-s + 2.24·8-s − 2.24·10-s + 1.80·16-s − 1.80·19-s − 2.80·20-s + 0.554·25-s + 0.445·29-s + 1.00·32-s − 1.80·37-s − 3.24·38-s − 2.80·40-s + 1.24·43-s + 49-s + 0.999·50-s + 0.801·58-s − 71-s + 1.24·73-s − 3.24·74-s − 4.04·76-s + 1.24·79-s − 2.24·80-s + 1.80·83-s + 2.24·86-s + 0.445·89-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $1$
Analytic conductor: \(0.318902\)
Root analytic conductor: \(0.564714\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (496, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.039039874\)
\(L(\frac12)\) \(\approx\) \(2.039039874\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + T \)
good2 \( 1 - 1.80T + T^{2} \)
5 \( 1 + 1.24T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.445T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
73 \( 1 - 1.24T + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 - 1.80T + T^{2} \)
89 \( 1 - 0.445T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.01117648416661450458079418458, −10.53576404954067268792347439840, −8.836934329516558101851519555806, −7.84966148329591903118429176352, −6.96469621423864126178713937736, −6.17296074601218658421437252310, −5.03167812273964799938864423182, −4.18420663369478119619908585192, −3.55182342283842946945061968422, −2.28751464686665026535167388429, 2.28751464686665026535167388429, 3.55182342283842946945061968422, 4.18420663369478119619908585192, 5.03167812273964799938864423182, 6.17296074601218658421437252310, 6.96469621423864126178713937736, 7.84966148329591903118429176352, 8.836934329516558101851519555806, 10.53576404954067268792347439840, 11.01117648416661450458079418458

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