Properties

Label 2-162-81.13-c1-0-1
Degree $2$
Conductor $162$
Sign $0.458 - 0.888i$
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  + (−0.893 − 0.448i)2-s + (0.674 + 1.59i)3-s + (0.597 + 0.802i)4-s + (0.195 + 0.653i)5-s + (0.113 − 1.72i)6-s + (0.386 + 0.895i)7-s + (−0.173 − 0.984i)8-s + (−2.09 + 2.15i)9-s + (0.118 − 0.671i)10-s + (−2.53 + 0.601i)11-s + (−0.877 + 1.49i)12-s + (4.17 + 2.74i)13-s + (0.0566 − 0.973i)14-s + (−0.910 + 0.752i)15-s + (−0.286 + 0.957i)16-s + (0.260 − 0.0947i)17-s + ⋯
L(s)  = 1  + (−0.631 − 0.317i)2-s + (0.389 + 0.921i)3-s + (0.298 + 0.401i)4-s + (0.0874 + 0.292i)5-s + (0.0464 − 0.705i)6-s + (0.145 + 0.338i)7-s + (−0.0613 − 0.348i)8-s + (−0.697 + 0.716i)9-s + (0.0374 − 0.212i)10-s + (−0.764 + 0.181i)11-s + (−0.253 + 0.431i)12-s + (1.15 + 0.760i)13-s + (0.0151 − 0.260i)14-s + (−0.235 + 0.194i)15-s + (−0.0717 + 0.239i)16-s + (0.0631 − 0.0229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814341 + 0.496254i\)
\(L(\frac12)\) \(\approx\) \(0.814341 + 0.496254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.893 + 0.448i)T \)
3 \( 1 + (-0.674 - 1.59i)T \)
good5 \( 1 + (-0.195 - 0.653i)T + (-4.17 + 2.74i)T^{2} \)
7 \( 1 + (-0.386 - 0.895i)T + (-4.80 + 5.09i)T^{2} \)
11 \( 1 + (2.53 - 0.601i)T + (9.82 - 4.93i)T^{2} \)
13 \( 1 + (-4.17 - 2.74i)T + (5.14 + 11.9i)T^{2} \)
17 \( 1 + (-0.260 + 0.0947i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.980 - 0.356i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-1.91 + 4.43i)T + (-15.7 - 16.7i)T^{2} \)
29 \( 1 + (-0.0947 - 1.62i)T + (-28.8 + 3.36i)T^{2} \)
31 \( 1 + (0.00909 - 0.00106i)T + (30.1 - 7.14i)T^{2} \)
37 \( 1 + (4.52 + 3.79i)T + (6.42 + 36.4i)T^{2} \)
41 \( 1 + (-7.48 + 3.75i)T + (24.4 - 32.8i)T^{2} \)
43 \( 1 + (8.21 + 8.70i)T + (-2.50 + 42.9i)T^{2} \)
47 \( 1 + (9.17 + 1.07i)T + (45.7 + 10.8i)T^{2} \)
53 \( 1 + (-1.58 + 2.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.2 - 3.13i)T + (52.7 + 26.4i)T^{2} \)
61 \( 1 + (-3.46 + 4.65i)T + (-17.4 - 58.4i)T^{2} \)
67 \( 1 + (0.747 - 12.8i)T + (-66.5 - 7.77i)T^{2} \)
71 \( 1 + (0.445 - 2.52i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-0.289 - 1.64i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (4.63 + 2.32i)T + (47.1 + 63.3i)T^{2} \)
83 \( 1 + (-10.6 - 5.32i)T + (49.5 + 66.5i)T^{2} \)
89 \( 1 + (-1.45 - 8.23i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-4.54 + 15.1i)T + (-81.0 - 53.3i)T^{2} \)
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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

−13.02416471620346262424153705945, −11.69976515166562739336631313224, −10.77697905865026616019071554013, −10.12738215311755002713929828479, −8.906803756302423886983080034992, −8.354482403901495122317681442862, −6.85348461429812293246965265614, −5.29214517058123610392854235665, −3.79792521579133636838640850084, −2.40994765852084329159636761783, 1.22381005831425234021513576589, 3.14157276157111112343335474881, 5.36049618057641945530898091366, 6.50868693362436326590719250075, 7.70574433001542289070411143827, 8.332413116175766628463911242945, 9.393172258787986367140924292921, 10.68509324358948644765235954611, 11.60040017471110401250989331294, 13.02357473099575398212058002801

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