Properties

Label 2-1638-7.2-c1-0-25
Degree $2$
Conductor $1638$
Sign $0.386 + 0.922i$
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 + (1.5 − 2.59i)5-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (−1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + 13-s + (2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s − 3·20-s + 3·22-s + (3 − 5.19i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.474 − 0.821i)10-s + (0.452 + 0.783i)11-s + 0.277·13-s + (0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s − 0.670·20-s + 0.639·22-s + (0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.386 + 0.922i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.415036546\)
\(L(\frac12)\) \(\approx\) \(2.415036546\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−9.256641012689256851608678656865, −8.774867611674503932879902599317, −7.87264543978164493121379191625, −6.43074036615804138800314602138, −5.89779746362935082912521645757, −4.94117735804494608267835584105, −4.40262111055043961759429878924, −3.06127535050666841314399495950, −2.04132312773080753569056477495, −1.06755725453847286167940506226, 1.17240791147503843095174597719, 3.02667421671310850877187875558, 3.36747468089217585794957718983, 4.62321460050586810059880066670, 5.68381845720599514463471482957, 6.33032608211884014369495241872, 7.09623980941927222602735005639, 7.59049832618749718059399906596, 8.657381878749197627320158681074, 9.677286499342414638979079150218

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