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Properties

Label 2-639-213.56-c1-0-9
Degree $2$
Conductor $639$
Sign $0.0900 - 0.995i$
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  + (2.23 + 0.956i)2-s + (2.70 + 2.83i)4-s + (−1.60 − 0.521i)5-s + (3.14 + 3.60i)7-s + (1.63 + 4.36i)8-s + (−3.09 − 2.70i)10-s + (−3.16 + 0.873i)11-s + (0.682 − 2.47i)13-s + (3.59 + 11.0i)14-s + (−0.157 + 3.50i)16-s + (0.780 − 0.567i)17-s + (3.71 + 6.90i)19-s + (−2.86 − 5.95i)20-s + (−7.91 − 1.07i)22-s + (0.573 − 2.51i)23-s + ⋯
L(s)  = 1  + (1.58 + 0.676i)2-s + (1.35 + 1.41i)4-s + (−0.717 − 0.233i)5-s + (1.19 + 1.36i)7-s + (0.579 + 1.54i)8-s + (−0.977 − 0.854i)10-s + (−0.953 + 0.263i)11-s + (0.189 − 0.685i)13-s + (0.961 + 2.95i)14-s + (−0.0393 + 0.876i)16-s + (0.189 − 0.137i)17-s + (0.852 + 1.58i)19-s + (−0.641 − 1.33i)20-s + (−1.68 − 0.228i)22-s + (0.119 − 0.523i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0900 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0900 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.0900 - 0.995i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.0900 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53152 + 2.31296i\)
\(L(\frac12)\) \(\approx\) \(2.53152 + 2.31296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.48 - 3.87i)T \)
good2 \( 1 + (-2.23 - 0.956i)T + (1.38 + 1.44i)T^{2} \)
5 \( 1 + (1.60 + 0.521i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.14 - 3.60i)T + (-0.939 + 6.93i)T^{2} \)
11 \( 1 + (3.16 - 0.873i)T + (9.44 - 5.64i)T^{2} \)
13 \( 1 + (-0.682 + 2.47i)T + (-11.1 - 6.66i)T^{2} \)
17 \( 1 + (-0.780 + 0.567i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-3.71 - 6.90i)T + (-10.4 + 15.8i)T^{2} \)
23 \( 1 + (-0.573 + 2.51i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-4.96 + 0.672i)T + (27.9 - 7.71i)T^{2} \)
31 \( 1 + (2.19 - 0.0985i)T + (30.8 - 2.77i)T^{2} \)
37 \( 1 + (2.06 + 9.04i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (3.14 + 3.94i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (1.39 + 0.125i)T + (42.3 + 7.67i)T^{2} \)
47 \( 1 + (6.57 - 1.19i)T + (44.0 - 16.5i)T^{2} \)
53 \( 1 + (-9.06 + 9.48i)T + (-2.37 - 52.9i)T^{2} \)
59 \( 1 + (-4.93 - 7.47i)T + (-23.1 + 54.2i)T^{2} \)
61 \( 1 + (-7.45 + 8.53i)T + (-8.18 - 60.4i)T^{2} \)
67 \( 1 + (3.23 - 3.09i)T + (3.00 - 66.9i)T^{2} \)
73 \( 1 + (-2.59 + 6.06i)T + (-50.4 - 52.7i)T^{2} \)
79 \( 1 + (-0.412 + 0.154i)T + (59.4 - 51.9i)T^{2} \)
83 \( 1 + (8.23 - 5.43i)T + (32.6 - 76.3i)T^{2} \)
89 \( 1 + (8.89 + 8.50i)T + (3.99 + 88.9i)T^{2} \)
97 \( 1 + (-0.579 - 0.461i)T + (21.5 + 94.5i)T^{2} \)
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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

−11.25532382069500716623458601460, −10.07674524489279306514987201806, −8.434442355471215470531069291746, −8.055516107512279255936830507574, −7.20463667391895545297947785562, −5.69392871618664835404666431641, −5.44724439711535417950123197404, −4.51471753628805764642803849571, −3.40983625506714157408860628890, −2.24222379497291947423366796645, 1.33525187992560712109781773042, 2.87532777746063827391137977029, 3.83764646523129766442958466526, 4.68608329985588776144189043534, 5.26695773232246784705655997062, 6.75960501909975167428590086542, 7.46240964862554305955782912241, 8.422995674745331335795438175873, 10.05958779233183907122336521507, 10.80088302740646050465937559774

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