Properties

Label 2-162-81.25-c1-0-6
Degree $2$
Conductor $162$
Sign $0.994 + 0.103i$
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 + (1.02 + 1.39i)3-s + (0.597 − 0.802i)4-s + (0.664 − 2.21i)5-s + (1.54 + 0.787i)6-s + (−0.590 + 1.36i)7-s + (0.173 − 0.984i)8-s + (−0.897 + 2.86i)9-s + (−0.402 − 2.28i)10-s + (−4.28 − 1.01i)11-s + (1.73 + 0.0112i)12-s + (0.407 − 0.267i)13-s + (0.0866 + 1.48i)14-s + (3.77 − 1.34i)15-s + (−0.286 − 0.957i)16-s + (6.18 + 2.24i)17-s + ⋯
L(s)  = 1  + (0.631 − 0.317i)2-s + (0.591 + 0.805i)3-s + (0.298 − 0.401i)4-s + (0.297 − 0.992i)5-s + (0.629 + 0.321i)6-s + (−0.223 + 0.517i)7-s + (0.0613 − 0.348i)8-s + (−0.299 + 0.954i)9-s + (−0.127 − 0.721i)10-s + (−1.29 − 0.305i)11-s + (0.499 + 0.00325i)12-s + (0.113 − 0.0743i)13-s + (0.0231 + 0.397i)14-s + (0.975 − 0.347i)15-s + (−0.0717 − 0.239i)16-s + (1.49 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78856 - 0.0927608i\)
\(L(\frac12)\) \(\approx\) \(1.78856 - 0.0927608i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 - 1.39i)T \)
good5 \( 1 + (-0.664 + 2.21i)T + (-4.17 - 2.74i)T^{2} \)
7 \( 1 + (0.590 - 1.36i)T + (-4.80 - 5.09i)T^{2} \)
11 \( 1 + (4.28 + 1.01i)T + (9.82 + 4.93i)T^{2} \)
13 \( 1 + (-0.407 + 0.267i)T + (5.14 - 11.9i)T^{2} \)
17 \( 1 + (-6.18 - 2.24i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (6.76 - 2.46i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (1.15 + 2.67i)T + (-15.7 + 16.7i)T^{2} \)
29 \( 1 + (-0.382 + 6.57i)T + (-28.8 - 3.36i)T^{2} \)
31 \( 1 + (1.78 + 0.208i)T + (30.1 + 7.14i)T^{2} \)
37 \( 1 + (3.51 - 2.94i)T + (6.42 - 36.4i)T^{2} \)
41 \( 1 + (7.03 + 3.53i)T + (24.4 + 32.8i)T^{2} \)
43 \( 1 + (0.304 - 0.322i)T + (-2.50 - 42.9i)T^{2} \)
47 \( 1 + (-7.76 + 0.907i)T + (45.7 - 10.8i)T^{2} \)
53 \( 1 + (-0.984 - 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.3 + 2.93i)T + (52.7 - 26.4i)T^{2} \)
61 \( 1 + (-4.60 - 6.18i)T + (-17.4 + 58.4i)T^{2} \)
67 \( 1 + (0.221 + 3.81i)T + (-66.5 + 7.77i)T^{2} \)
71 \( 1 + (-1.34 - 7.62i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-1.74 + 9.92i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-15.0 + 7.53i)T + (47.1 - 63.3i)T^{2} \)
83 \( 1 + (4.65 - 2.33i)T + (49.5 - 66.5i)T^{2} \)
89 \( 1 + (0.537 - 3.05i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.403 - 1.34i)T + (-81.0 + 53.3i)T^{2} \)
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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

−12.90341313148274203515281728004, −12.14236989952110235051505957214, −10.60431272758753284434003098469, −10.03937825957337840587103295367, −8.760107063675585086702866037179, −8.022170273211190003221548709657, −5.85703196789303481507857350285, −5.09538102116409564961534786990, −3.80604515676556739339171093990, −2.34727298127201137508406041905, 2.42906702094086306517380386254, 3.52810737042808426261926807065, 5.42469448203298785662851439207, 6.75839044270831320023248518310, 7.32739046784730016265054868989, 8.425358957201750197978823064391, 10.00037476832761900652242490724, 10.90556515491489133933527325532, 12.27505588443795987814927401901, 13.07186155191622167666262719562

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