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Properties

Label 2-1638-91.16-c1-0-13
Degree $2$
Conductor $1638$
Sign $0.322 - 0.946i$
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 + (−1.14 + 1.98i)5-s + (2.63 − 0.222i)7-s − 8-s + (1.14 − 1.98i)10-s + (0.439 − 0.760i)11-s + (−0.786 + 3.51i)13-s + (−2.63 + 0.222i)14-s + 16-s + 6.40·17-s + (−0.754 − 1.30i)19-s + (−1.14 + 1.98i)20-s + (−0.439 + 0.760i)22-s − 1.31·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.512 + 0.887i)5-s + (0.996 − 0.0839i)7-s − 0.353·8-s + (0.362 − 0.627i)10-s + (0.132 − 0.229i)11-s + (−0.218 + 0.975i)13-s + (−0.704 + 0.0593i)14-s + 0.250·16-s + 1.55·17-s + (−0.173 − 0.299i)19-s + (−0.256 + 0.443i)20-s + (−0.0936 + 0.162i)22-s − 0.274·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.322 - 0.946i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.216868580\)
\(L(\frac12)\) \(\approx\) \(1.216868580\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.222i)T \)
13 \( 1 + (0.786 - 3.51i)T \)
good5 \( 1 + (1.14 - 1.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.439 + 0.760i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 + (0.754 + 1.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.31T + 23T^{2} \)
29 \( 1 + (-0.669 - 1.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.94 + 3.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.38T + 37T^{2} \)
41 \( 1 + (1.80 + 3.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.95 - 8.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.188 + 0.327i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.22 - 2.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.96T + 59T^{2} \)
61 \( 1 + (2.40 + 4.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.87 - 8.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.02 - 1.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.432 - 0.749i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.18 - 7.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-4.40 + 7.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.579321899615441621132779062374, −8.668327559409616403668257151013, −7.84174399348415571886177777068, −7.39636038376847053692739515288, −6.55423837590290837897059190885, −5.59384434175451752025278756763, −4.46052040928516387077490614122, −3.47426844814855112311988686542, −2.40609925990034703241460623332, −1.20053654896759337676560165212, 0.69986941166867744824772523947, 1.71474185531684030200779045158, 3.09464767381316634660846610758, 4.26153533808272865601073199549, 5.17215262306307829324747129782, 5.85527592291762620296422033945, 7.17668435785846736808130337001, 8.003186882499539659463914495782, 8.199039281481400842907129327997, 9.101912383443893135419214874067

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