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Properties

Label 2-1638-273.68-c1-0-26
Degree $2$
Conductor $1638$
Sign $0.844 - 0.534i$
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  + i·2-s − 4-s + (1.64 − 2.85i)5-s + (1.61 + 2.09i)7-s − i·8-s + (2.85 + 1.64i)10-s + (3.22 + 1.85i)11-s + (3.56 + 0.505i)13-s + (−2.09 + 1.61i)14-s + 16-s − 2.34·17-s + (1.72 − 0.994i)19-s + (−1.64 + 2.85i)20-s + (−1.85 + 3.22i)22-s − 1.90i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.736 − 1.27i)5-s + (0.609 + 0.792i)7-s − 0.353i·8-s + (0.901 + 0.520i)10-s + (0.971 + 0.560i)11-s + (0.990 + 0.140i)13-s + (−0.560 + 0.430i)14-s + 0.250·16-s − 0.568·17-s + (0.395 − 0.228i)19-s + (−0.368 + 0.637i)20-s + (−0.396 + 0.686i)22-s − 0.398i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.844 - 0.534i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.844 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207004984\)
\(L(\frac12)\) \(\approx\) \(2.207004984\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-1.61 - 2.09i)T \)
13 \( 1 + (-3.56 - 0.505i)T \)
good5 \( 1 + (-1.64 + 2.85i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.22 - 1.85i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + (-1.72 + 0.994i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.90iT - 23T^{2} \)
29 \( 1 + (-1.62 + 0.937i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.35 - 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.66T + 37T^{2} \)
41 \( 1 + (2.22 + 3.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.71 - 8.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.69 + 6.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.17 + 2.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.98T + 59T^{2} \)
61 \( 1 + (3.68 - 2.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.05 - 1.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.45 - 4.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.59 + 3.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.318 - 0.550i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (-1.25 - 0.727i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.111929302048453152874771714401, −8.756974306836817896526159258844, −8.116079347169079877016645850669, −6.88870864894245455530305908863, −6.17638512267576899690735914028, −5.32477393978040008336864345964, −4.77497476551737402484908158727, −3.82614073403538609712828354375, −2.09020353144317912875745679665, −1.15185327119240970531283679738, 1.14906637085588964100907514698, 2.14823139397715594487062191892, 3.38808915791181563374088968809, 3.88132127369163356490927785655, 5.16135639927268888797659029576, 6.18907806060109516321940949069, 6.77749142293947839806954518671, 7.74020151909089615966956211911, 8.697422921977784002075446906955, 9.446064806612431188588909414899

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