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Properties

Label 2-639-213.56-c1-0-15
Degree $2$
Conductor $639$
Sign $0.481 - 0.876i$
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  + (2.16 + 0.924i)2-s + (2.43 + 2.55i)4-s + (2.31 + 0.751i)5-s + (−0.903 − 1.03i)7-s + (1.26 + 3.36i)8-s + (4.30 + 3.76i)10-s + (0.807 − 0.222i)11-s + (−0.838 + 3.03i)13-s + (−0.997 − 3.06i)14-s + (−0.0625 + 1.39i)16-s + (0.263 − 0.191i)17-s + (−0.510 − 0.948i)19-s + (3.72 + 7.73i)20-s + (1.95 + 0.264i)22-s + (0.341 − 1.49i)23-s + ⋯
L(s)  = 1  + (1.52 + 0.653i)2-s + (1.21 + 1.27i)4-s + (1.03 + 0.336i)5-s + (−0.341 − 0.390i)7-s + (0.446 + 1.18i)8-s + (1.36 + 1.18i)10-s + (0.243 − 0.0671i)11-s + (−0.232 + 0.843i)13-s + (−0.266 − 0.820i)14-s + (−0.0156 + 0.348i)16-s + (0.0639 − 0.0464i)17-s + (−0.117 − 0.217i)19-s + (0.832 + 1.72i)20-s + (0.416 + 0.0563i)22-s + (0.0711 − 0.311i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.481 - 0.876i$
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: \((2,\ 639,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.26576 + 1.93208i\)
\(L(\frac12)\) \(\approx\) \(3.26576 + 1.93208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.40 - 0.555i)T \)
good2 \( 1 + (-2.16 - 0.924i)T + (1.38 + 1.44i)T^{2} \)
5 \( 1 + (-2.31 - 0.751i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.903 + 1.03i)T + (-0.939 + 6.93i)T^{2} \)
11 \( 1 + (-0.807 + 0.222i)T + (9.44 - 5.64i)T^{2} \)
13 \( 1 + (0.838 - 3.03i)T + (-11.1 - 6.66i)T^{2} \)
17 \( 1 + (-0.263 + 0.191i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.510 + 0.948i)T + (-10.4 + 15.8i)T^{2} \)
23 \( 1 + (-0.341 + 1.49i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (6.01 - 0.815i)T + (27.9 - 7.71i)T^{2} \)
31 \( 1 + (1.18 - 0.0530i)T + (30.8 - 2.77i)T^{2} \)
37 \( 1 + (-1.50 - 6.60i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (5.40 + 6.78i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-5.56 - 0.501i)T + (42.3 + 7.67i)T^{2} \)
47 \( 1 + (-2.55 + 0.464i)T + (44.0 - 16.5i)T^{2} \)
53 \( 1 + (7.47 - 7.82i)T + (-2.37 - 52.9i)T^{2} \)
59 \( 1 + (5.77 + 8.75i)T + (-23.1 + 54.2i)T^{2} \)
61 \( 1 + (-1.54 + 1.77i)T + (-8.18 - 60.4i)T^{2} \)
67 \( 1 + (0.181 - 0.173i)T + (3.00 - 66.9i)T^{2} \)
73 \( 1 + (1.91 - 4.47i)T + (-50.4 - 52.7i)T^{2} \)
79 \( 1 + (-16.2 + 6.09i)T + (59.4 - 51.9i)T^{2} \)
83 \( 1 + (2.31 - 1.52i)T + (32.6 - 76.3i)T^{2} \)
89 \( 1 + (-2.87 - 2.75i)T + (3.99 + 88.9i)T^{2} \)
97 \( 1 + (-2.13 - 1.70i)T + (21.5 + 94.5i)T^{2} \)
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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

−10.88466470454534669928076105157, −9.867598686359005583356847121878, −9.061436613113887260408896906085, −7.60487615496717099906167092619, −6.74915088395463827230667553149, −6.21572136907450088466831887292, −5.30381256085149519963178029389, −4.32522688584628875695374014576, −3.32083557265023443642698332589, −2.10412927183530678918417686088, 1.67934413001179624067090714510, 2.71293861149927855147013235518, 3.75580615500400748293055392198, 4.94274108264984876004697890821, 5.72214310498461183106654389034, 6.21656393535422960176270934693, 7.59598635724794215204505653434, 8.993940250887577217551477601432, 9.774812552972226199176862541707, 10.62354873375385188334684987610

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