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Properties

Label 2-162-9.4-c1-0-3
Degree $2$
Conductor $162$
Sign $0.173 + 0.984i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (0.5 + 0.866i)7-s + 0.999·8-s − 3·10-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 2·19-s + (1.50 + 2.59i)20-s + (−1.5 + 2.59i)22-s + (−3 + 5.19i)23-s + (−2 − 3.46i)25-s − 3.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.188 + 0.327i)7-s + 0.353·8-s − 0.948·10-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.458·19-s + (0.335 + 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.625 + 1.08i)23-s + (−0.400 − 0.692i)25-s − 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789425 - 0.662406i\)
\(L(\frac12)\) \(\approx\) \(0.789425 - 0.662406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.75178558925765669159719781894, −11.63360400216876953770638969512, −10.61363673681156605231795676965, −9.549727085724035416741753721477, −8.684033332862363744896073032803, −7.86696656695305832468540019692, −5.88500719907871205888044515656, −4.98561863570386518508403707012, −3.18664569842764989663889374755, −1.32677919860022877648496519353, 2.26057032192534261335558596218, 4.26276781331185966554912181940, 5.87094293569908246181183890819, 6.79669319985080365079294648711, 7.67003970350535586772668150124, 9.051656500452837115222561683600, 10.12893922575711505368266899299, 10.72647343618202831450071557671, 12.00455734165583297223534542533, 13.51703282903545428007093410731

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