Properties

Label 2-162-81.5-c2-0-0
Degree $2$
Conductor $162$
Sign $-0.982 - 0.185i$
Analytic cond. $4.41418$
Root an. cond. $2.10099$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.326 + 1.37i)2-s + (1.73 − 2.44i)3-s + (−1.78 + 0.897i)4-s + (−6.68 + 4.97i)5-s + (3.93 + 1.59i)6-s + (−10.2 + 6.73i)7-s + (−1.81 − 2.16i)8-s + (−2.94 − 8.50i)9-s + (−9.03 − 7.57i)10-s + (1.17 − 10.0i)11-s + (−0.915 + 5.92i)12-s + (3.87 + 12.9i)13-s + (−12.6 − 11.9i)14-s + (0.533 + 25.0i)15-s + (2.38 − 3.20i)16-s + (−9.14 − 1.61i)17-s + ⋯
L(s)  = 1  + (0.163 + 0.688i)2-s + (0.579 − 0.814i)3-s + (−0.446 + 0.224i)4-s + (−1.33 + 0.995i)5-s + (0.655 + 0.266i)6-s + (−1.46 + 0.962i)7-s + (−0.227 − 0.270i)8-s + (−0.327 − 0.944i)9-s + (−0.903 − 0.757i)10-s + (0.106 − 0.910i)11-s + (−0.0763 + 0.494i)12-s + (0.297 + 0.995i)13-s + (−0.901 − 0.850i)14-s + (0.0355 + 1.66i)15-s + (0.149 − 0.200i)16-s + (−0.537 − 0.0948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.982 - 0.185i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.982 - 0.185i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0552781 + 0.592149i\)
\(L(\frac12)\) \(\approx\) \(0.0552781 + 0.592149i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.326 - 1.37i)T \)
3 \( 1 + (-1.73 + 2.44i)T \)
good5 \( 1 + (6.68 - 4.97i)T + (7.17 - 23.9i)T^{2} \)
7 \( 1 + (10.2 - 6.73i)T + (19.4 - 44.9i)T^{2} \)
11 \( 1 + (-1.17 + 10.0i)T + (-117. - 27.9i)T^{2} \)
13 \( 1 + (-3.87 - 12.9i)T + (-141. + 92.8i)T^{2} \)
17 \( 1 + (9.14 + 1.61i)T + (271. + 98.8i)T^{2} \)
19 \( 1 + (-4.22 - 23.9i)T + (-339. + 123. i)T^{2} \)
23 \( 1 + (9.36 - 14.2i)T + (-209. - 485. i)T^{2} \)
29 \( 1 + (-39.9 + 37.6i)T + (48.8 - 839. i)T^{2} \)
31 \( 1 + (1.02 - 17.5i)T + (-954. - 111. i)T^{2} \)
37 \( 1 + (13.3 - 4.86i)T + (1.04e3 - 879. i)T^{2} \)
41 \( 1 + (12.3 - 52.2i)T + (-1.50e3 - 754. i)T^{2} \)
43 \( 1 + (-9.85 - 22.8i)T + (-1.26e3 + 1.34e3i)T^{2} \)
47 \( 1 + (61.2 - 3.56i)T + (2.19e3 - 256. i)T^{2} \)
53 \( 1 + (41.0 + 23.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-0.286 - 2.44i)T + (-3.38e3 + 802. i)T^{2} \)
61 \( 1 + (-51.1 - 25.6i)T + (2.22e3 + 2.98e3i)T^{2} \)
67 \( 1 + (57.4 - 60.9i)T + (-261. - 4.48e3i)T^{2} \)
71 \( 1 + (-17.4 + 20.8i)T + (-875. - 4.96e3i)T^{2} \)
73 \( 1 + (-29.6 + 24.8i)T + (925. - 5.24e3i)T^{2} \)
79 \( 1 + (-17.8 + 4.23i)T + (5.57e3 - 2.80e3i)T^{2} \)
83 \( 1 + (13.6 + 57.7i)T + (-6.15e3 + 3.09e3i)T^{2} \)
89 \( 1 + (-28.5 - 34.0i)T + (-1.37e3 + 7.80e3i)T^{2} \)
97 \( 1 + (46.5 - 62.5i)T + (-2.69e3 - 9.01e3i)T^{2} \)
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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

−13.24551890007949851444934034165, −12.08632394835148441202017391892, −11.58237710915477148954803228749, −9.783945684293737362108588306381, −8.635243456287403043946834381174, −7.86415845198662460611402746160, −6.57560814786983746728821834453, −6.26472391798498571634502898146, −3.76788923450797538845372620274, −2.96196754670037062948637278536, 0.32036510107022947003319185117, 3.13033285640920146237851100459, 4.05787328235848446197845890175, 4.89441572217922131443614391053, 7.04561578892716968071786912671, 8.310606947592594032446347706543, 9.220583419316095954912584629850, 10.18215917167046593842609196223, 11.00081007362296286254367764268, 12.38538314108321974890788057926

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