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Properties

Label 2-639-213.53-c1-0-21
Degree $2$
Conductor $639$
Sign $0.656 + 0.754i$
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  + (0.0602 + 0.669i)2-s + (1.52 − 0.276i)4-s + (2.37 − 3.26i)5-s + (−2.27 − 3.81i)7-s + (0.634 + 2.30i)8-s + (2.33 + 1.39i)10-s + (−1.61 + 2.44i)11-s + (0.650 − 0.429i)13-s + (2.41 − 1.75i)14-s + (1.39 − 0.523i)16-s + (0.299 − 0.921i)17-s + (−2.84 − 2.97i)19-s + (2.71 − 5.62i)20-s + (−1.73 − 0.934i)22-s + (1.15 + 5.06i)23-s + ⋯
L(s)  = 1  + (0.0426 + 0.473i)2-s + (0.761 − 0.138i)4-s + (1.06 − 1.46i)5-s + (−0.861 − 1.44i)7-s + (0.224 + 0.813i)8-s + (0.737 + 0.440i)10-s + (−0.486 + 0.737i)11-s + (0.180 − 0.119i)13-s + (0.646 − 0.469i)14-s + (0.348 − 0.130i)16-s + (0.0725 − 0.223i)17-s + (−0.653 − 0.683i)19-s + (0.606 − 1.25i)20-s + (−0.370 − 0.199i)22-s + (0.241 + 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.656 + 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77400 - 0.807781i\)
\(L(\frac12)\) \(\approx\) \(1.77400 - 0.807781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.91 - 5.99i)T \)
good2 \( 1 + (-0.0602 - 0.669i)T + (-1.96 + 0.357i)T^{2} \)
5 \( 1 + (-2.37 + 3.26i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (2.27 + 3.81i)T + (-3.31 + 6.16i)T^{2} \)
11 \( 1 + (1.61 - 2.44i)T + (-4.32 - 10.1i)T^{2} \)
13 \( 1 + (-0.650 + 0.429i)T + (5.10 - 11.9i)T^{2} \)
17 \( 1 + (-0.299 + 0.921i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.84 + 2.97i)T + (-0.852 + 18.9i)T^{2} \)
23 \( 1 + (-1.15 - 5.06i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (3.39 - 1.82i)T + (15.9 - 24.2i)T^{2} \)
31 \( 1 + (0.455 - 1.21i)T + (-23.3 - 20.3i)T^{2} \)
37 \( 1 + (0.180 - 0.788i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-4.41 + 5.53i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-9.31 + 8.14i)T + (5.77 - 42.6i)T^{2} \)
47 \( 1 + (-1.27 - 9.38i)T + (-45.3 + 12.5i)T^{2} \)
53 \( 1 + (-7.74 - 1.40i)T + (49.6 + 18.6i)T^{2} \)
59 \( 1 + (0.0589 + 1.31i)T + (-58.7 + 5.28i)T^{2} \)
61 \( 1 + (-1.09 + 1.84i)T + (-28.9 - 53.7i)T^{2} \)
67 \( 1 + (-2.90 - 16.0i)T + (-62.7 + 23.5i)T^{2} \)
73 \( 1 + (-12.9 + 1.16i)T + (71.8 - 13.0i)T^{2} \)
79 \( 1 + (8.89 - 2.45i)T + (67.8 - 40.5i)T^{2} \)
83 \( 1 + (1.99 - 0.0897i)T + (82.6 - 7.44i)T^{2} \)
89 \( 1 + (-2.46 + 13.5i)T + (-83.3 - 31.2i)T^{2} \)
97 \( 1 + (11.4 - 9.14i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.32115620252444831363297271597, −9.636911780934246994380704656766, −8.791010054127168493494289465492, −7.54599582868366254802683558713, −6.97351994439384623516235258143, −5.89548593330472226387250845239, −5.16452261397105614812968283652, −4.06118400506630104663088891658, −2.36621554327941896608614923743, −1.05781407071139275112433354009, 2.18342545688912205440901527676, 2.66494132815544919570155374006, 3.56068006457218960624569698977, 5.74351414071605053515921618935, 6.14896413974573398115865759661, 6.81450670180106397291417681559, 8.070159253991779349228819535919, 9.263344519258012010024859688831, 10.00305334917338285381794575389, 10.76316777273519383002329457528

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