Properties

Label 2-162-9.2-c6-0-14
Degree $2$
Conductor $162$
Sign $0.939 - 0.342i$
Analytic cond. $37.2687$
Root an. cond. $6.10481$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (−150. + 86.9i)5-s + (242 − 419. i)7-s + 181. i·8-s − 984·10-s + (1.16e3 + 670. i)11-s + (−1.68e3 − 2.91e3i)13-s + (2.37e3 − 1.36e3i)14-s + (−512. + 886. i)16-s − 12.7i·17-s + 5.74e3·19-s + (−4.82e3 − 2.78e3i)20-s + (3.79e3 + 6.56e3i)22-s + (2.92e3 − 1.68e3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.20 + 0.695i)5-s + (0.705 − 1.22i)7-s + 0.353i·8-s − 0.983·10-s + (0.872 + 0.503i)11-s + (−0.766 − 1.32i)13-s + (0.864 − 0.498i)14-s + (−0.125 + 0.216i)16-s − 0.00259i·17-s + 0.837·19-s + (−0.602 − 0.347i)20-s + (0.356 + 0.616i)22-s + (0.240 − 0.138i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(37.2687\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :3),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.598207004\)
\(L(\frac12)\) \(\approx\) \(2.598207004\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.89 - 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (150. - 86.9i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-242 + 419. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-1.16e3 - 670. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (1.68e3 + 2.91e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 12.7iT - 2.41e7T^{2} \)
19 \( 1 - 5.74e3T + 4.70e7T^{2} \)
23 \( 1 + (-2.92e3 + 1.68e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-2.54e4 - 1.46e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-1.98e4 - 3.44e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 5.25e4T + 2.56e9T^{2} \)
41 \( 1 + (-3.20e4 + 1.85e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (1.90e3 - 3.29e3i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-6.65e4 - 3.83e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 2.38e5iT - 2.21e10T^{2} \)
59 \( 1 + (-2.16e5 + 1.24e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (6.62e3 - 1.14e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (8.44e4 + 1.46e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 5.31e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.36e5T + 1.51e11T^{2} \)
79 \( 1 + (-1.75e4 + 3.04e4i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (9.50e3 + 5.48e3i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 1.29e5iT - 4.96e11T^{2} \)
97 \( 1 + (-1.60e5 + 2.78e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−11.83420020351124353544162047054, −10.99732356218261132213437846764, −10.06249281884288094783441628962, −8.220869518635133166171206790474, −7.43872737615214771500091752394, −6.81732794773635628134895952984, −5.01630530401509216395095038427, −4.05628146208289659304046691655, −3.03117855075388977782798148846, −0.857360581248139349664254015392, 0.974702470519451748452545281303, 2.47834121020123300726526553451, 4.07260889787333175308090272135, 4.82963678373416413121488419196, 6.11885008651945158997091741295, 7.58726822478093512114565848737, 8.683619810380128394899485499931, 9.503173716956654058995307638188, 11.33150400763226096692522321342, 11.86495595429433786855972211989

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