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Properties

Label 2-1638-13.3-c1-0-33
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: $\chi_{1638} (757, \cdot )$
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

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

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