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Properties

Label 2-1638-13.9-c1-0-9
Degree $2$
Conductor $1638$
Sign $0.642 - 0.766i$
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-s + (0.5 + 0.866i)7-s − 0.999·8-s + (1 − 1.73i)10-s + (−2.58 + 4.48i)11-s + (−1.5 + 3.27i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + (−1.58 − 2.75i)19-s + (−0.999 − 1.73i)20-s + (2.58 + 4.48i)22-s + (−4.08 + 7.08i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (0.188 + 0.327i)7-s − 0.353·8-s + (0.316 − 0.547i)10-s + (−0.780 + 1.35i)11-s + (−0.416 + 0.909i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + (−0.364 − 0.631i)19-s + (−0.223 − 0.387i)20-s + (0.552 + 0.956i)22-s + (−0.852 + 1.47i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778924481\)
\(L(\frac12)\) \(\approx\) \(1.778924481\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (1.5 - 3.27i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + (2.58 - 4.48i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.08 - 7.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 - 2.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.58 - 6.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.08 + 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.17T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (6.08 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.91 + 3.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.08 + 1.88i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 6.17T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.17 - 8.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.849891857068225994299761051673, −8.940232451468262740977360627850, −7.935302343772086698076207861526, −7.01228270921548622540309619377, −6.06894147523107826736365550050, −5.26595917651851717156947008494, −4.58362102781566403867079944599, −3.48567260685993006009686981508, −2.11148824890994330368054093516, −1.81458932478449420559799585771, 0.56609538189557048231061896857, 2.37742152256085101369719529336, 3.23475308991306656247137067730, 4.43235479723775227348584287202, 5.46139131301414730624069823083, 5.83123635864447720890694604105, 6.71426592744920205484950982638, 7.87278049727842482252896268688, 8.158618129975142297417712651657, 9.195075344687032275850227158612

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