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Properties

Label 2-162-81.22-c1-0-4
Degree $2$
Conductor $162$
Sign $0.491 - 0.870i$
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 + (1.65 + 0.524i)3-s + (−0.0581 + 0.998i)4-s + (−1.97 + 0.230i)5-s + (0.750 + 1.56i)6-s + (0.868 + 0.436i)7-s + (−0.766 + 0.642i)8-s + (2.44 + 1.73i)9-s + (−1.52 − 1.27i)10-s + (−1.12 − 2.61i)11-s + (−0.619 + 1.61i)12-s + (1.19 + 0.284i)13-s + (0.278 + 0.931i)14-s + (−3.37 − 0.654i)15-s + (−0.993 − 0.116i)16-s + (0.944 − 5.35i)17-s + ⋯
L(s)  = 1  + (0.485 + 0.514i)2-s + (0.952 + 0.303i)3-s + (−0.0290 + 0.499i)4-s + (−0.881 + 0.103i)5-s + (0.306 + 0.637i)6-s + (0.328 + 0.164i)7-s + (−0.270 + 0.227i)8-s + (0.816 + 0.577i)9-s + (−0.480 − 0.403i)10-s + (−0.340 − 0.789i)11-s + (−0.178 + 0.466i)12-s + (0.332 + 0.0788i)13-s + (0.0745 + 0.248i)14-s + (−0.871 − 0.168i)15-s + (−0.248 − 0.0290i)16-s + (0.229 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.491 - 0.870i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.491 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45162 + 0.847406i\)
\(L(\frac12)\) \(\approx\) \(1.45162 + 0.847406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 0.524i)T \)
good5 \( 1 + (1.97 - 0.230i)T + (4.86 - 1.15i)T^{2} \)
7 \( 1 + (-0.868 - 0.436i)T + (4.18 + 5.61i)T^{2} \)
11 \( 1 + (1.12 + 2.61i)T + (-7.54 + 8.00i)T^{2} \)
13 \( 1 + (-1.19 - 0.284i)T + (11.6 + 5.83i)T^{2} \)
17 \( 1 + (-0.944 + 5.35i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (0.724 + 4.10i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (5.83 - 2.92i)T + (13.7 - 18.4i)T^{2} \)
29 \( 1 + (-1.00 + 3.37i)T + (-24.2 - 15.9i)T^{2} \)
31 \( 1 + (-7.08 - 4.66i)T + (12.2 + 28.4i)T^{2} \)
37 \( 1 + (3.74 - 1.36i)T + (28.3 - 23.7i)T^{2} \)
41 \( 1 + (6.52 - 6.91i)T + (-2.38 - 40.9i)T^{2} \)
43 \( 1 + (-6.25 + 8.40i)T + (-12.3 - 41.1i)T^{2} \)
47 \( 1 + (3.54 - 2.33i)T + (18.6 - 43.1i)T^{2} \)
53 \( 1 + (-0.380 + 0.659i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0874 - 0.202i)T + (-40.4 - 42.9i)T^{2} \)
61 \( 1 + (-0.258 - 4.44i)T + (-60.5 + 7.08i)T^{2} \)
67 \( 1 + (-2.08 - 6.95i)T + (-55.9 + 36.8i)T^{2} \)
71 \( 1 + (-11.3 - 9.56i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (5.51 - 4.63i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.53 + 1.62i)T + (-4.59 + 78.8i)T^{2} \)
83 \( 1 + (-8.03 - 8.52i)T + (-4.82 + 82.8i)T^{2} \)
89 \( 1 + (-0.108 + 0.0910i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (10.5 + 1.23i)T + (94.3 + 22.3i)T^{2} \)
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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

−13.53935094769532686268670228763, −12.04496513728150456248498628009, −11.25674585865913416248341347014, −9.850410113339606625143391998423, −8.554184953732941809888324645686, −7.968118847566985557649500692504, −6.88750254760227010162326521078, −5.19798498205092344202966631134, −4.01089878708677680260998672117, −2.85980349738214025919337645690, 1.89490617403707246793688727530, 3.59764613107670411759621143482, 4.44087754689297421693581637800, 6.29207584689613414820938105187, 7.78612259382889194701970425405, 8.327709574628478996488534233037, 9.823031135869409313948021757264, 10.68432217331025230928636191156, 12.13382643755425537118441060096, 12.51499996975749255213638294960

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