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Properties

Label 2-162-81.16-c1-0-1
Degree $2$
Conductor $162$
Sign $0.834 - 0.550i$
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  + (−0.973 − 0.230i)2-s + (−1.13 − 1.30i)3-s + (0.893 + 0.448i)4-s + (−1.92 + 2.58i)5-s + (0.807 + 1.53i)6-s + (2.72 + 1.79i)7-s + (−0.766 − 0.642i)8-s + (−0.403 + 2.97i)9-s + (2.46 − 2.06i)10-s + (4.37 − 0.511i)11-s + (−0.432 − 1.67i)12-s + (0.873 − 2.91i)13-s + (−2.23 − 2.37i)14-s + (5.55 − 0.433i)15-s + (0.597 + 0.802i)16-s + (0.670 + 3.80i)17-s + ⋯
L(s)  = 1  + (−0.688 − 0.163i)2-s + (−0.657 − 0.753i)3-s + (0.446 + 0.224i)4-s + (−0.859 + 1.15i)5-s + (0.329 + 0.625i)6-s + (1.02 + 0.677i)7-s + (−0.270 − 0.227i)8-s + (−0.134 + 0.990i)9-s + (0.779 − 0.654i)10-s + (1.31 − 0.154i)11-s + (−0.124 − 0.484i)12-s + (0.242 − 0.809i)13-s + (−0.598 − 0.633i)14-s + (1.43 − 0.112i)15-s + (0.149 + 0.200i)16-s + (0.162 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629751 + 0.189004i\)
\(L(\frac12)\) \(\approx\) \(0.629751 + 0.189004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.973 + 0.230i)T \)
3 \( 1 + (1.13 + 1.30i)T \)
good5 \( 1 + (1.92 - 2.58i)T + (-1.43 - 4.78i)T^{2} \)
7 \( 1 + (-2.72 - 1.79i)T + (2.77 + 6.42i)T^{2} \)
11 \( 1 + (-4.37 + 0.511i)T + (10.7 - 2.53i)T^{2} \)
13 \( 1 + (-0.873 + 2.91i)T + (-10.8 - 7.14i)T^{2} \)
17 \( 1 + (-0.670 - 3.80i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (1.07 - 6.08i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (3.43 - 2.26i)T + (9.10 - 21.1i)T^{2} \)
29 \( 1 + (-1.48 + 1.57i)T + (-1.68 - 28.9i)T^{2} \)
31 \( 1 + (-0.405 - 6.97i)T + (-30.7 + 3.59i)T^{2} \)
37 \( 1 + (3.18 + 1.15i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + (-2.37 + 0.562i)T + (36.6 - 18.4i)T^{2} \)
43 \( 1 + (-3.68 + 8.54i)T + (-29.5 - 31.2i)T^{2} \)
47 \( 1 + (-0.128 + 2.21i)T + (-46.6 - 5.45i)T^{2} \)
53 \( 1 + (6.74 + 11.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.21 - 1.07i)T + (57.4 + 13.6i)T^{2} \)
61 \( 1 + (8.22 - 4.13i)T + (36.4 - 48.9i)T^{2} \)
67 \( 1 + (4.38 + 4.65i)T + (-3.89 + 66.8i)T^{2} \)
71 \( 1 + (-2.09 + 1.75i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-7.71 - 6.47i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (1.48 + 0.351i)T + (70.5 + 35.4i)T^{2} \)
83 \( 1 + (-14.5 - 3.45i)T + (74.1 + 37.2i)T^{2} \)
89 \( 1 + (0.592 + 0.497i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (10.5 + 14.1i)T + (-27.8 + 92.9i)T^{2} \)
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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

−12.34762095807382704915370374223, −11.88203097799997663578860300575, −11.04164800332481992531602993597, −10.32678102885848067009233368178, −8.462484017506098408465126600815, −7.87222060725401825529324771963, −6.75610903991072841814997549717, −5.74348817526571793127247446078, −3.66054125779842465569190063376, −1.76251259182315566239163584104, 0.957095999527916182398819869034, 4.21690935587045545352773285085, 4.75237399343772646762894790183, 6.47551262984263645970623208367, 7.70752229997943352951391801581, 8.896427705416939206806377705711, 9.469409211463314903979520540896, 11.00215723918750138022174000222, 11.53694145422163392914701211382, 12.26427628542444682166091758909

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