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Properties

Label 2-1638-7.2-c1-0-4
Degree $2$
Conductor $1638$
Sign $0.605 - 0.795i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−2.5 − 4.33i)11-s − 13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (3.5 + 6.06i)17-s + (−3.5 + 6.06i)19-s − 5·22-s + (1 − 1.73i)23-s + (2.5 + 4.33i)25-s + (−0.5 + 0.866i)26-s + (1.99 + 1.73i)28-s + 9·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−0.753 − 1.30i)11-s − 0.277·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.848 + 1.47i)17-s + (−0.802 + 1.39i)19-s − 1.06·22-s + (0.208 − 0.361i)23-s + (0.5 + 0.866i)25-s + (−0.0980 + 0.169i)26-s + (0.377 + 0.327i)28-s + 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.020451030\)
\(L(\frac12)\) \(\approx\) \(1.020451030\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.764717050405730391949446399441, −8.481213636942509685412672167072, −8.362730255356323489181195412855, −6.92656342783635747472832860635, −5.91383208341202130680338410728, −5.64903030720755033003214975341, −4.30431630219931507322813451246, −3.34940118302461705559251281734, −2.75902094013391765351294794265, −1.29561055762111638574518192723, 0.36097349834419984743586074329, 2.47461252438090593952243785767, 3.21109064944812983287736315743, 4.64080503531638618810366248364, 4.90140638607968982471553293470, 6.14654595655446289822978818093, 7.07025445252110664646804122261, 7.25946836630580690033140738911, 8.387192099684624494284264448779, 9.295491744358619932618581458772

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