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Properties

Label 2-1638-91.74-c1-0-22
Degree $2$
Conductor $1638$
Sign $0.835 + 0.549i$
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 + (0.441 + 0.764i)5-s + (0.369 − 2.61i)7-s − 8-s + (−0.441 − 0.764i)10-s + (0.775 + 1.34i)11-s + (2.13 − 2.90i)13-s + (−0.369 + 2.61i)14-s + 16-s + 7.17·17-s + (−2.37 + 4.10i)19-s + (0.441 + 0.764i)20-s + (−0.775 − 1.34i)22-s − 5.29·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.197 + 0.341i)5-s + (0.139 − 0.990i)7-s − 0.353·8-s + (−0.139 − 0.241i)10-s + (0.233 + 0.405i)11-s + (0.591 − 0.805i)13-s + (−0.0988 + 0.700i)14-s + 0.250·16-s + 1.74·17-s + (−0.543 + 0.942i)19-s + (0.0986 + 0.170i)20-s + (−0.165 − 0.286i)22-s − 1.10·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350285947\)
\(L(\frac12)\) \(\approx\) \(1.350285947\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 + 2.90i)T \)
good5 \( 1 + (-0.441 - 0.764i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.775 - 1.34i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.17T + 17T^{2} \)
19 \( 1 + (2.37 - 4.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + (3.87 - 6.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.330T + 37T^{2} \)
41 \( 1 + (-3.02 + 5.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.976 + 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.74 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.27T + 59T^{2} \)
61 \( 1 + (-7.17 + 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.75 - 6.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.00 + 8.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (-5.79 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.544168965636521124023672130568, −8.210536550993324582507395689033, −7.946407570367842396278500940270, −7.05004826297445306379034270567, −6.20490419717523930069796037755, −5.41865042259358608762164316344, −4.03769590473368127294708923487, −3.32121652258163894092631306555, −1.93960397986032415393729693512, −0.807562358209331671621075562810, 1.10861780422736724208706849367, 2.20038616441397662251642543388, 3.29110670230818779875389013050, 4.48824274472044644650783553501, 5.70131475488079387521291578458, 6.07718253324886014599606667275, 7.20053332526102318210979617631, 8.071690436578614044924649099691, 8.794922529211677993738219320618, 9.264886665686453512388090571765

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