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Properties

Label 2-162-27.7-c1-0-0
Degree $2$
Conductor $162$
Sign $0.983 + 0.183i$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.177 + 1.00i)5-s + (2.04 + 1.71i)7-s + (0.500 − 0.866i)8-s + (0.510 + 0.884i)10-s + (0.720 − 4.08i)11-s + (−3.68 − 1.34i)13-s + (2.50 + 0.912i)14-s + (0.173 − 0.984i)16-s + (−0.925 − 1.60i)17-s + (−3.21 + 5.57i)19-s + (0.782 + 0.656i)20-s + (−0.720 − 4.08i)22-s + (−6.69 + 5.61i)23-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.383 − 0.321i)4-s + (0.0793 + 0.449i)5-s + (0.772 + 0.647i)7-s + (0.176 − 0.306i)8-s + (0.161 + 0.279i)10-s + (0.217 − 1.23i)11-s + (−1.02 − 0.371i)13-s + (0.669 + 0.243i)14-s + (0.0434 − 0.246i)16-s + (−0.224 − 0.388i)17-s + (−0.737 + 1.27i)19-s + (0.174 + 0.146i)20-s + (−0.153 − 0.870i)22-s + (−1.39 + 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.983 + 0.183i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64943 - 0.152370i\)
\(L(\frac12)\) \(\approx\) \(1.64943 - 0.152370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (-0.177 - 1.00i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.04 - 1.71i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.720 + 4.08i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (3.68 + 1.34i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.925 + 1.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.21 - 5.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.69 - 5.61i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.17 + 0.428i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.56 - 2.15i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4.58 + 7.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.53 - 1.28i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.536 + 3.04i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.11 - 1.77i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 0.231T + 53T^{2} \)
59 \( 1 + (0.613 + 3.48i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.405 - 0.339i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-7.67 - 2.79i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.03 - 6.98i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.57 + 2.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.43 + 0.886i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (7.55 - 2.74i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.51 - 8.57i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.70118283498717310568606281000, −11.88957933008119171476768607721, −11.03860942432582617536229109951, −10.07194377505328505706785466626, −8.685420691787276154031483093581, −7.56990106381495019524599459151, −6.11315354050672813721479667293, −5.24229688523698948603732589841, −3.68522913873082516312178388797, −2.21131508332172688628203670088, 2.14794964920307706180648382101, 4.36501419556530549543279003645, 4.83543078379614275494892395084, 6.57612713368289664724963996548, 7.47190299451493745656812940374, 8.652953142322801472795583671986, 9.965089922886714194556645245831, 11.03872743182778520268979062882, 12.18438802467002272515922534354, 12.81609169288401124590846302271

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