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Properties

Label 2-639-71.48-c1-0-11
Degree $2$
Conductor $639$
Sign $-0.971 + 0.237i$
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  + (−1.54 − 1.94i)2-s + (−0.929 + 4.07i)4-s − 1.04·5-s + (−0.718 + 0.900i)7-s + (4.87 − 2.34i)8-s + (1.61 + 2.02i)10-s + (−0.0304 − 0.0146i)11-s + (2.20 − 1.06i)13-s + 2.86·14-s + (−4.59 − 2.21i)16-s + 3.81·17-s + (−1.30 − 5.73i)19-s + (0.968 − 4.24i)20-s + (0.0186 + 0.0818i)22-s + (−2.53 + 3.18i)23-s + ⋯
L(s)  = 1  + (−1.09 − 1.37i)2-s + (−0.464 + 2.03i)4-s − 0.465·5-s + (−0.271 + 0.340i)7-s + (1.72 − 0.830i)8-s + (0.510 + 0.639i)10-s + (−0.00917 − 0.00441i)11-s + (0.612 − 0.294i)13-s + 0.765·14-s + (−1.14 − 0.553i)16-s + 0.925·17-s + (−0.300 − 1.31i)19-s + (0.216 − 0.948i)20-s + (0.00398 + 0.0174i)22-s + (−0.529 + 0.663i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.971 + 0.237i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0606343 - 0.503124i\)
\(L(\frac12)\) \(\approx\) \(0.0606343 - 0.503124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.30 - 4.20i)T \)
good2 \( 1 + (1.54 + 1.94i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + 1.04T + 5T^{2} \)
7 \( 1 + (0.718 - 0.900i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.0304 + 0.0146i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.20 + 1.06i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + (1.30 + 5.73i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (2.53 - 3.18i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-0.560 + 2.45i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-5.75 + 2.77i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (3.49 + 4.38i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.549 - 0.264i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (7.06 + 8.85i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.55 + 11.1i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (2.06 + 9.06i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (2.87 + 1.38i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-0.804 - 1.00i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.40 + 6.16i)T + (-60.3 - 29.0i)T^{2} \)
73 \( 1 + (-10.5 - 13.1i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (10.6 - 5.13i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-4.83 - 2.32i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.12 + 4.94i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (2.13 + 1.02i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.08381854088261226648998844520, −9.565886166940172697722882321468, −8.527027069625962463326763468998, −8.052568485023236524714986112217, −6.93180516667898388403203250494, −5.53146485051406046876087996147, −3.99708740592514839546050888107, −3.17888511754699891540091687096, −2.00654873839261258599925812258, −0.44387185622232584775189621053, 1.29943806893494936396130731485, 3.52691964361339473558897655676, 4.81629490618364388987732737911, 6.08325597250589166733779347893, 6.53282312929548739719330486462, 7.77393811855176440380436989766, 8.064676111833824653733991038181, 9.016352044327357279056812902779, 10.01796573435060365535911899974, 10.42578792599581102294373213259

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