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Properties

Label 2-162-81.76-c1-0-0
Degree $2$
Conductor $162$
Sign $0.761 - 0.648i$
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 + (−1.60 + 0.658i)3-s + (0.893 − 0.448i)4-s + (2.07 + 2.78i)5-s + (−1.40 + 1.00i)6-s + (−1.72 + 1.13i)7-s + (0.766 − 0.642i)8-s + (2.13 − 2.10i)9-s + (2.65 + 2.23i)10-s + (1.29 + 0.151i)11-s + (−1.13 + 1.30i)12-s + (0.585 + 1.95i)13-s + (−1.41 + 1.50i)14-s + (−5.15 − 3.09i)15-s + (0.597 − 0.802i)16-s + (1.17 − 6.66i)17-s + ⋯
L(s)  = 1  + (0.688 − 0.163i)2-s + (−0.924 + 0.379i)3-s + (0.446 − 0.224i)4-s + (0.927 + 1.24i)5-s + (−0.574 + 0.412i)6-s + (−0.653 + 0.429i)7-s + (0.270 − 0.227i)8-s + (0.711 − 0.702i)9-s + (0.841 + 0.705i)10-s + (0.390 + 0.0455i)11-s + (−0.328 + 0.377i)12-s + (0.162 + 0.542i)13-s + (−0.379 + 0.402i)14-s + (−1.33 − 0.799i)15-s + (0.149 − 0.200i)16-s + (0.285 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30586 + 0.481000i\)
\(L(\frac12)\) \(\approx\) \(1.30586 + 0.481000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 - 0.658i)T \)
good5 \( 1 + (-2.07 - 2.78i)T + (-1.43 + 4.78i)T^{2} \)
7 \( 1 + (1.72 - 1.13i)T + (2.77 - 6.42i)T^{2} \)
11 \( 1 + (-1.29 - 0.151i)T + (10.7 + 2.53i)T^{2} \)
13 \( 1 + (-0.585 - 1.95i)T + (-10.8 + 7.14i)T^{2} \)
17 \( 1 + (-1.17 + 6.66i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-0.360 - 2.04i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (4.39 + 2.89i)T + (9.10 + 21.1i)T^{2} \)
29 \( 1 + (5.65 + 5.99i)T + (-1.68 + 28.9i)T^{2} \)
31 \( 1 + (-0.391 + 6.71i)T + (-30.7 - 3.59i)T^{2} \)
37 \( 1 + (5.04 - 1.83i)T + (28.3 - 23.7i)T^{2} \)
41 \( 1 + (-11.5 - 2.72i)T + (36.6 + 18.4i)T^{2} \)
43 \( 1 + (-0.530 - 1.23i)T + (-29.5 + 31.2i)T^{2} \)
47 \( 1 + (0.169 + 2.91i)T + (-46.6 + 5.45i)T^{2} \)
53 \( 1 + (-1.23 + 2.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-14.3 + 1.68i)T + (57.4 - 13.6i)T^{2} \)
61 \( 1 + (-3.25 - 1.63i)T + (36.4 + 48.9i)T^{2} \)
67 \( 1 + (-6.78 + 7.19i)T + (-3.89 - 66.8i)T^{2} \)
71 \( 1 + (0.218 + 0.183i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (12.7 - 10.6i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (7.68 - 1.82i)T + (70.5 - 35.4i)T^{2} \)
83 \( 1 + (9.75 - 2.31i)T + (74.1 - 37.2i)T^{2} \)
89 \( 1 + (2.63 - 2.21i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.0866 - 0.116i)T + (-27.8 - 92.9i)T^{2} \)
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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

−12.95971710305434908117569165642, −11.78838330142324529219629955225, −11.21658297140206791503403496833, −9.943293028889327886383819588744, −9.590224246049998104552360010546, −7.15552922412149139619397065221, −6.27006645060957711773535144763, −5.61997680836089196504093706429, −3.99017598974629230153517801293, −2.49200665585174521493340495172, 1.51769608085607746992580591471, 3.98003562871479888956450020658, 5.39251878330666399184993787330, 5.95809137578988971851497533005, 7.14980684281696518292887207058, 8.594321317257703173396068289341, 9.920950162703828529406282777118, 10.83790864894766794700544062592, 12.20903442554578131546863753753, 12.85168400490410698455196471324

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