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Properties

Label 2-639-71.20-c1-0-23
Degree $2$
Conductor $639$
Sign $-0.922 + 0.385i$
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  + (−0.320 − 1.40i)2-s + (−0.0703 + 0.0338i)4-s − 0.128·5-s + (0.484 − 2.12i)7-s + (−1.72 − 2.16i)8-s + (0.0411 + 0.180i)10-s + (1.22 − 1.53i)11-s + (−0.0643 − 0.0807i)13-s − 3.13·14-s + (−2.58 + 3.24i)16-s + 1.78·17-s + (−0.156 − 0.0752i)19-s + (0.00902 − 0.00434i)20-s + (−2.55 − 1.23i)22-s + (0.347 − 1.52i)23-s + ⋯
L(s)  = 1  + (−0.226 − 0.993i)2-s + (−0.0351 + 0.0169i)4-s − 0.0574·5-s + (0.183 − 0.801i)7-s + (−0.610 − 0.765i)8-s + (0.0130 + 0.0570i)10-s + (0.369 − 0.463i)11-s + (−0.0178 − 0.0223i)13-s − 0.838·14-s + (−0.646 + 0.811i)16-s + 0.433·17-s + (−0.0358 − 0.0172i)19-s + (0.00201 − 0.000972i)20-s + (−0.544 − 0.262i)22-s + (0.0725 − 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $-0.922 + 0.385i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ -0.922 + 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250115 - 1.24837i\)
\(L(\frac12)\) \(\approx\) \(0.250115 - 1.24837i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 7.82i)T \)
good2 \( 1 + (0.320 + 1.40i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + 0.128T + 5T^{2} \)
7 \( 1 + (-0.484 + 2.12i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-1.22 + 1.53i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.0643 + 0.0807i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + (0.156 + 0.0752i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.347 + 1.52i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.60 + 1.25i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (3.38 + 4.24i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-0.156 - 0.684i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (1.53 + 1.92i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (1.60 + 7.04i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (4.43 + 2.13i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-6.94 - 3.34i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (4.96 - 6.22i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (1.70 + 7.47i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-2.72 + 1.31i)T + (41.7 - 52.3i)T^{2} \)
73 \( 1 + (-1.16 - 5.10i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-7.20 - 9.03i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-6.75 + 8.47i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.54 + 0.745i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (1.43 - 1.79i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.31246395469523521092143541334, −9.632193969896406799289281483253, −8.657238230993946567125404255873, −7.59123187913897119991666460449, −6.66338117250755629931110505222, −5.66433846072726466992600509790, −4.18689197587821452457765845393, −3.39227875537437471889444103606, −2.06497488982932962006932491387, −0.74131252049136757479148371580, 1.97701134124856679879747651715, 3.29909639194069106860777476387, 4.83329016404128809925739994894, 5.72746886423398248420934912443, 6.53850785067512984867719987490, 7.44982558619790923950270210270, 8.209150694916813660422635024703, 9.012480986514480877703554010356, 9.808912738843186023629867571948, 11.03283337050265866706127662596

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