Properties

Label 2-162-81.70-c1-0-2
Degree $2$
Conductor $162$
Sign $0.196 - 0.980i$
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  + (−0.686 + 0.727i)2-s + (0.224 + 1.71i)3-s + (−0.0581 − 0.998i)4-s + (2.24 + 0.262i)5-s + (−1.40 − 1.01i)6-s + (3.76 − 1.88i)7-s + (0.766 + 0.642i)8-s + (−2.89 + 0.771i)9-s + (−1.73 + 1.45i)10-s + (−1.42 + 3.29i)11-s + (1.70 − 0.323i)12-s + (−4.26 + 1.01i)13-s + (−1.20 + 4.03i)14-s + (0.0533 + 3.91i)15-s + (−0.993 + 0.116i)16-s + (−0.241 − 1.36i)17-s + ⋯
L(s)  = 1  + (−0.485 + 0.514i)2-s + (0.129 + 0.991i)3-s + (−0.0290 − 0.499i)4-s + (1.00 + 0.117i)5-s + (−0.572 − 0.414i)6-s + (1.42 − 0.714i)7-s + (0.270 + 0.227i)8-s + (−0.966 + 0.257i)9-s + (−0.547 + 0.459i)10-s + (−0.428 + 0.994i)11-s + (0.491 − 0.0935i)12-s + (−1.18 + 0.280i)13-s + (−0.322 + 1.07i)14-s + (0.0137 + 1.01i)15-s + (−0.248 + 0.0290i)16-s + (−0.0585 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.196 - 0.980i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860977 + 0.705693i\)
\(L(\frac12)\) \(\approx\) \(0.860977 + 0.705693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.686 - 0.727i)T \)
3 \( 1 + (-0.224 - 1.71i)T \)
good5 \( 1 + (-2.24 - 0.262i)T + (4.86 + 1.15i)T^{2} \)
7 \( 1 + (-3.76 + 1.88i)T + (4.18 - 5.61i)T^{2} \)
11 \( 1 + (1.42 - 3.29i)T + (-7.54 - 8.00i)T^{2} \)
13 \( 1 + (4.26 - 1.01i)T + (11.6 - 5.83i)T^{2} \)
17 \( 1 + (0.241 + 1.36i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-0.883 + 5.00i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (0.261 + 0.131i)T + (13.7 + 18.4i)T^{2} \)
29 \( 1 + (-2.13 - 7.13i)T + (-24.2 + 15.9i)T^{2} \)
31 \( 1 + (-6.75 + 4.44i)T + (12.2 - 28.4i)T^{2} \)
37 \( 1 + (2.90 + 1.05i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + (5.07 + 5.37i)T + (-2.38 + 40.9i)T^{2} \)
43 \( 1 + (-1.66 - 2.23i)T + (-12.3 + 41.1i)T^{2} \)
47 \( 1 + (0.834 + 0.548i)T + (18.6 + 43.1i)T^{2} \)
53 \( 1 + (5.96 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.66 + 8.49i)T + (-40.4 + 42.9i)T^{2} \)
61 \( 1 + (0.863 - 14.8i)T + (-60.5 - 7.08i)T^{2} \)
67 \( 1 + (1.60 - 5.34i)T + (-55.9 - 36.8i)T^{2} \)
71 \( 1 + (-4.88 + 4.10i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-4.66 - 3.91i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-8.31 + 8.81i)T + (-4.59 - 78.8i)T^{2} \)
83 \( 1 + (-2.12 + 2.25i)T + (-4.82 - 82.8i)T^{2} \)
89 \( 1 + (0.706 + 0.592i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.85 + 0.216i)T + (94.3 - 22.3i)T^{2} \)
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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

−13.52951646849528027988487835532, −11.78204588971976333806584346339, −10.67625613763018563661901398818, −9.997807513571883671086422882352, −9.189930189610492671919700858000, −7.952183288933059962937843141307, −6.91228500039942436959242725255, −5.15234379411811509904703396456, −4.68822441305766771718200806987, −2.23267783575360868593342997522, 1.60970847142071363158121886695, 2.69040187217130900566003472437, 5.15953566076887697480928732743, 6.17334945716835386011210924223, 7.909094436101060101991462376049, 8.303972612697615050120082459864, 9.564439437432719202855688696026, 10.71343076052354913184974299819, 11.82983516981398188580932045694, 12.39283013007837903043488158364

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