Properties

Label 2-639-1.1-c1-0-6
Degree $2$
Conductor $639$
Sign $1$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.138·2-s − 1.98·4-s − 0.703·5-s + 3.84·7-s − 0.551·8-s − 0.0975·10-s − 0.606·11-s − 0.0975·13-s + 0.532·14-s + 3.88·16-s − 6.59·17-s + 8.14·19-s + 1.39·20-s − 0.0839·22-s + 8.05·23-s − 4.50·25-s − 0.0135·26-s − 7.61·28-s + 5.50·29-s + 5.85·31-s + 1.64·32-s − 0.913·34-s − 2.70·35-s + 1.62·37-s + 1.12·38-s + 0.388·40-s + 10.2·41-s + ⋯
L(s)  = 1  + 0.0979·2-s − 0.990·4-s − 0.314·5-s + 1.45·7-s − 0.195·8-s − 0.0308·10-s − 0.182·11-s − 0.0270·13-s + 0.142·14-s + 0.971·16-s − 1.59·17-s + 1.86·19-s + 0.311·20-s − 0.0179·22-s + 1.68·23-s − 0.900·25-s − 0.00264·26-s − 1.43·28-s + 1.02·29-s + 1.05·31-s + 0.290·32-s − 0.156·34-s − 0.457·35-s + 0.267·37-s + 0.183·38-s + 0.0613·40-s + 1.59·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.371127244\)
\(L(\frac12)\) \(\approx\) \(1.371127244\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 - 0.138T + 2T^{2} \)
5 \( 1 + 0.703T + 5T^{2} \)
7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 + 0.606T + 11T^{2} \)
13 \( 1 + 0.0975T + 13T^{2} \)
17 \( 1 + 6.59T + 17T^{2} \)
19 \( 1 - 8.14T + 19T^{2} \)
23 \( 1 - 8.05T + 23T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 - 5.85T + 31T^{2} \)
37 \( 1 - 1.62T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 - 8.86T + 47T^{2} \)
53 \( 1 + 8.68T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 3.45T + 89T^{2} \)
97 \( 1 + 1.91T + 97T^{2} \)
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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

−10.72801640978726301853006915305, −9.511907080470790142793492260253, −8.844461998306324429051663864126, −7.996745868499447377798883696651, −7.30137308582198836489580818247, −5.81337808085756186180673294718, −4.78841557446078871461224095426, −4.37072698857398647878510519292, −2.85812974291257741350270234132, −1.08592788243893316693695030542, 1.08592788243893316693695030542, 2.85812974291257741350270234132, 4.37072698857398647878510519292, 4.78841557446078871461224095426, 5.81337808085756186180673294718, 7.30137308582198836489580818247, 7.996745868499447377798883696651, 8.844461998306324429051663864126, 9.511907080470790142793492260253, 10.72801640978726301853006915305

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