Properties

Label 2-1638-91.16-c1-0-6
Degree $2$
Conductor $1638$
Sign $0.538 - 0.842i$
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  − 2-s + 4-s + (0.228 − 0.395i)5-s + (−0.369 − 2.61i)7-s − 8-s + (−0.228 + 0.395i)10-s + (−1.91 + 3.32i)11-s + (−3.13 − 1.78i)13-s + (0.369 + 2.61i)14-s + 16-s − 1.55·17-s + (1.44 + 2.49i)19-s + (0.228 − 0.395i)20-s + (1.91 − 3.32i)22-s − 3.24·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.102 − 0.176i)5-s + (−0.139 − 0.990i)7-s − 0.353·8-s + (−0.0721 + 0.124i)10-s + (−0.578 + 1.00i)11-s + (−0.869 − 0.494i)13-s + (0.0988 + 0.700i)14-s + 0.250·16-s − 0.376·17-s + (0.330 + 0.572i)19-s + (0.0510 − 0.0883i)20-s + (0.409 − 0.708i)22-s − 0.676·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8512548379\)
\(L(\frac12)\) \(\approx\) \(0.8512548379\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (0.369 + 2.61i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (-0.228 + 0.395i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.91 - 3.32i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.55T + 17T^{2} \)
19 \( 1 + (-1.44 - 2.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 + (-2.20 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.80 - 8.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.280T + 37T^{2} \)
41 \( 1 + (-3.57 - 6.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.93 + 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.550 + 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.36T + 59T^{2} \)
61 \( 1 + (-5.55 - 9.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.894 + 1.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.06 + 8.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.40 - 2.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.70 + 4.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.35T + 83T^{2} \)
89 \( 1 - 0.179T + 89T^{2} \)
97 \( 1 + (-4.73 + 8.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.687176113737454821900750781716, −8.710280975242832391053412276925, −7.84494513237856641130609031748, −7.24920042078738461746094524702, −6.62541691923194753641390554011, −5.35145547179792597209562261430, −4.61688994040348901810339772507, −3.42357646938269551810873524277, −2.31279785327506104618756346892, −1.07092933651168723351438510913, 0.47018327084703032839245528283, 2.34126497343175523204695245607, 2.71615788006818997510404638669, 4.21009238195853237658865674174, 5.36398338696956523485272397111, 6.11586113880167531617562610605, 6.84496106509830942385230264926, 7.943195870499171872919423687461, 8.406670419322570246015093639405, 9.354226719777153482132538855867

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