Properties

Label 2-162-27.25-c1-0-0
Degree $2$
Conductor $162$
Sign $-0.913 - 0.407i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−2.42 + 2.03i)5-s + (−3.46 + 1.26i)7-s + (0.5 − 0.866i)8-s + (−1.58 − 2.74i)10-s + (1.75 + 1.46i)11-s + (0.538 + 3.05i)13-s + (−0.640 − 3.62i)14-s + (0.766 + 0.642i)16-s + (0.862 + 1.49i)17-s + (1.69 − 2.93i)19-s + (2.97 − 1.08i)20-s + (−1.75 + 1.46i)22-s + (−3.15 − 1.14i)23-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.469 − 0.171i)4-s + (−1.08 + 0.910i)5-s + (−1.30 + 0.476i)7-s + (0.176 − 0.306i)8-s + (−0.500 − 0.867i)10-s + (0.527 + 0.442i)11-s + (0.149 + 0.846i)13-s + (−0.171 − 0.970i)14-s + (0.191 + 0.160i)16-s + (0.209 + 0.362i)17-s + (0.389 − 0.674i)19-s + (0.665 − 0.242i)20-s + (−0.373 + 0.313i)22-s + (−0.657 − 0.239i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127318 + 0.598192i\)
\(L(\frac12)\) \(\approx\) \(0.127318 + 0.598192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 \)
good5 \( 1 + (2.42 - 2.03i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (3.46 - 1.26i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.75 - 1.46i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.538 - 3.05i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.862 - 1.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 2.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.14i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.101 + 0.576i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.35 - 1.58i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-3.65 - 6.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.22 - 6.97i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.27 - 1.06i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.61 - 1.31i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 2.58T + 53T^{2} \)
59 \( 1 + (-7.40 + 6.21i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (12.3 - 4.47i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.49 + 8.47i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.993 - 1.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.32 - 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.44 + 13.8i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.538 + 3.05i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (8.67 - 15.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.04 - 5.90i)T + (16.8 + 95.5i)T^{2} \)
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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

−13.40326832093900927559916927372, −12.23576834076110183387745689412, −11.43303813165779943071496101770, −10.07555764283520453050576966410, −9.203481837687280697673211763967, −7.961225277397418346897834350677, −6.83277803035884154232599971158, −6.27767437921047497786052764227, −4.35787094991922621909290758867, −3.12484185828243026803511606807, 0.62000674302271946184457691417, 3.31074701436373429543120237564, 4.16682949392549297956386198608, 5.80183549630411299666042653939, 7.42337204210966237754808303710, 8.432743428129058509263323340875, 9.472105849058975254077276749926, 10.40234787612470309138122024236, 11.64047223007495133768363241161, 12.36654473016410970581693161672

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