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Properties

Label 2-639-639.283-c0-0-3
Degree $2$
Conductor $639$
Sign $0.995 - 0.0995i$
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  + (−0.0747 − 0.129i)2-s + (0.955 + 0.294i)3-s + (0.488 − 0.846i)4-s + (−0.623 + 1.07i)5-s + (−0.0332 − 0.145i)6-s − 0.295·8-s + (0.826 + 0.563i)9-s + 0.186·10-s + (0.716 − 0.664i)12-s + (−0.914 + 0.848i)15-s + (−0.466 − 0.808i)16-s + (0.0111 − 0.149i)18-s + 0.149·19-s + (0.609 + 1.05i)20-s + (−0.282 − 0.0871i)24-s + (−0.277 − 0.480i)25-s + ⋯
L(s)  = 1  + (−0.0747 − 0.129i)2-s + (0.955 + 0.294i)3-s + (0.488 − 0.846i)4-s + (−0.623 + 1.07i)5-s + (−0.0332 − 0.145i)6-s − 0.295·8-s + (0.826 + 0.563i)9-s + 0.186·10-s + (0.716 − 0.664i)12-s + (−0.914 + 0.848i)15-s + (−0.466 − 0.808i)16-s + (0.0111 − 0.149i)18-s + 0.149·19-s + (0.609 + 1.05i)20-s + (−0.282 − 0.0871i)24-s + (−0.277 − 0.480i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.995 - 0.0995i$
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: \((2,\ 639,\ (\ :0),\ 0.995 - 0.0995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196862196\)
\(L(\frac12)\) \(\approx\) \(1.196862196\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 - 0.294i)T \)
71 \( 1 - T \)
good2 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 0.149T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.46T + T^{2} \)
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

−10.73764774723415803349258150058, −9.986529913304886318374210933572, −9.267346669847288227253321738818, −8.110439487580216848364179142922, −7.28402295212302327269034607678, −6.56492313559452679166015695026, −5.31525590479459578407360196401, −3.97423411067458324121381778986, −3.01692340490899560829368850522, −1.98466437418230637296344547236, 1.73298430463114228998140204531, 3.18213300119091276440730186859, 3.95671922851171183810807975119, 5.13087054930520200888550504809, 6.71683948265948095805391674596, 7.40902391012058716060429053649, 8.304938171985212803448429846746, 8.684061272569509674486379167829, 9.565089236286128764594916948448, 10.87201080392646522294618863493

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