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Properties

Label 2-639-213.59-c1-0-17
Degree $2$
Conductor $639$
Sign $0.989 - 0.147i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 + 1.64i)2-s + (0.165 − 3.68i)4-s + (2.39 − 3.29i)5-s + (4.18 + 0.566i)7-s + (2.64 + 3.02i)8-s + (1.30 + 9.61i)10-s + (−4.07 + 2.43i)11-s + (1.67 − 2.80i)13-s + (−8.14 + 5.91i)14-s + (−2.20 − 0.198i)16-s + (0.752 − 2.31i)17-s + (−1.41 + 2.13i)19-s + (−11.7 − 9.34i)20-s + (3.00 − 10.8i)22-s + (5.89 + 2.83i)23-s + ⋯
L(s)  = 1  + (−1.21 + 1.16i)2-s + (0.0826 − 1.84i)4-s + (1.06 − 1.47i)5-s + (1.58 + 0.214i)7-s + (0.934 + 1.07i)8-s + (0.411 + 3.03i)10-s + (−1.22 + 0.733i)11-s + (0.465 − 0.778i)13-s + (−2.17 + 1.58i)14-s + (−0.550 − 0.0495i)16-s + (0.182 − 0.561i)17-s + (−0.324 + 0.490i)19-s + (−2.62 − 2.09i)20-s + (0.641 − 2.32i)22-s + (1.22 + 0.591i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.989 - 0.147i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06904 + 0.0793537i\)
\(L(\frac12)\) \(\approx\) \(1.06904 + 0.0793537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.50 + 5.35i)T \)
good2 \( 1 + (1.72 - 1.64i)T + (0.0897 - 1.99i)T^{2} \)
5 \( 1 + (-2.39 + 3.29i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-4.18 - 0.566i)T + (6.74 + 1.86i)T^{2} \)
11 \( 1 + (4.07 - 2.43i)T + (5.21 - 9.68i)T^{2} \)
13 \( 1 + (-1.67 + 2.80i)T + (-6.16 - 11.4i)T^{2} \)
17 \( 1 + (-0.752 + 2.31i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.41 - 2.13i)T + (-7.46 - 17.4i)T^{2} \)
23 \( 1 + (-5.89 - 2.83i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (1.56 + 5.66i)T + (-24.8 + 14.8i)T^{2} \)
31 \( 1 + (-0.182 - 2.02i)T + (-30.5 + 5.53i)T^{2} \)
37 \( 1 + (-7.30 + 3.51i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-0.641 + 2.81i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (8.74 + 1.58i)T + (40.2 + 15.1i)T^{2} \)
47 \( 1 + (1.69 - 0.636i)T + (35.3 - 30.9i)T^{2} \)
53 \( 1 + (-0.112 - 2.50i)T + (-52.7 + 4.75i)T^{2} \)
59 \( 1 + (-0.611 + 1.43i)T + (-40.7 - 42.6i)T^{2} \)
61 \( 1 + (7.31 - 0.990i)T + (58.8 - 16.2i)T^{2} \)
67 \( 1 + (6.69 + 0.300i)T + (66.7 + 6.00i)T^{2} \)
73 \( 1 + (-6.33 - 6.62i)T + (-3.27 + 72.9i)T^{2} \)
79 \( 1 + (1.22 - 1.07i)T + (10.6 - 78.2i)T^{2} \)
83 \( 1 + (-12.9 - 5.53i)T + (57.3 + 59.9i)T^{2} \)
89 \( 1 + (2.03 - 0.0912i)T + (88.6 - 7.97i)T^{2} \)
97 \( 1 + (11.4 - 2.61i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.18412654866472727907986676319, −9.473270292642792537329861576228, −8.681976687698683295726602528231, −8.050218904659518820462263777695, −7.51479057651334053149627552769, −5.99925695605799773103367883663, −5.26392180860410918890461551140, −4.83880703591023645392753951831, −2.04805573551666971029845797258, −0.989665236761777954647549528525, 1.48314923680302294341282515038, 2.39967872743783140711657160061, 3.27552420628297210541522688586, 4.96099090468293007848072198126, 6.23613694119550726799849030670, 7.35074243148535811567933866126, 8.193165926505481014714072048122, 8.942954599279659873716380685807, 9.978746011010628414647190134653, 10.76196030473477793820589602985

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