Properties

Label 2-162-81.65-c2-0-17
Degree $2$
Conductor $162$
Sign $-0.789 - 0.613i$
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 + (−1.18 − 2.75i)3-s + (−1.78 − 0.897i)4-s + (−3.75 − 2.79i)5-s + (−4.17 + 0.735i)6-s + (1.03 + 0.679i)7-s + (−1.81 + 2.16i)8-s + (−6.18 + 6.54i)9-s + (−5.07 + 4.26i)10-s + (−0.479 − 4.10i)11-s + (−0.351 + 5.98i)12-s + (−1.23 + 4.12i)13-s + (1.27 − 1.20i)14-s + (−3.24 + 13.6i)15-s + (2.38 + 3.20i)16-s + (−7.03 + 1.24i)17-s + ⋯
L(s)  = 1  + (0.163 − 0.688i)2-s + (−0.395 − 0.918i)3-s + (−0.446 − 0.224i)4-s + (−0.751 − 0.559i)5-s + (−0.696 + 0.122i)6-s + (0.147 + 0.0971i)7-s + (−0.227 + 0.270i)8-s + (−0.686 + 0.726i)9-s + (−0.507 + 0.426i)10-s + (−0.0436 − 0.373i)11-s + (−0.0292 + 0.499i)12-s + (−0.0950 + 0.317i)13-s + (0.0909 − 0.0857i)14-s + (−0.216 + 0.911i)15-s + (0.149 + 0.200i)16-s + (−0.414 + 0.0730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.202553 + 0.590534i\)
\(L(\frac12)\) \(\approx\) \(0.202553 + 0.590534i\)
\(L(2)\) 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 \( 1 + (1.18 + 2.75i)T \)
good5 \( 1 + (3.75 + 2.79i)T + (7.17 + 23.9i)T^{2} \)
7 \( 1 + (-1.03 - 0.679i)T + (19.4 + 44.9i)T^{2} \)
11 \( 1 + (0.479 + 4.10i)T + (-117. + 27.9i)T^{2} \)
13 \( 1 + (1.23 - 4.12i)T + (-141. - 92.8i)T^{2} \)
17 \( 1 + (7.03 - 1.24i)T + (271. - 98.8i)T^{2} \)
19 \( 1 + (1.18 - 6.73i)T + (-339. - 123. i)T^{2} \)
23 \( 1 + (18.9 + 28.8i)T + (-209. + 485. i)T^{2} \)
29 \( 1 + (8.94 + 8.43i)T + (48.8 + 839. i)T^{2} \)
31 \( 1 + (0.301 + 5.16i)T + (-954. + 111. i)T^{2} \)
37 \( 1 + (13.6 + 4.98i)T + (1.04e3 + 879. i)T^{2} \)
41 \( 1 + (17.2 + 72.6i)T + (-1.50e3 + 754. i)T^{2} \)
43 \( 1 + (15.2 - 35.3i)T + (-1.26e3 - 1.34e3i)T^{2} \)
47 \( 1 + (-5.69 - 0.331i)T + (2.19e3 + 256. i)T^{2} \)
53 \( 1 + (-69.6 + 40.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-6.72 + 57.5i)T + (-3.38e3 - 802. i)T^{2} \)
61 \( 1 + (-2.15 + 1.08i)T + (2.22e3 - 2.98e3i)T^{2} \)
67 \( 1 + (-80.9 - 85.7i)T + (-261. + 4.48e3i)T^{2} \)
71 \( 1 + (-3.83 - 4.56i)T + (-875. + 4.96e3i)T^{2} \)
73 \( 1 + (73.3 + 61.5i)T + (925. + 5.24e3i)T^{2} \)
79 \( 1 + (-71.4 - 16.9i)T + (5.57e3 + 2.80e3i)T^{2} \)
83 \( 1 + (25.0 - 105. i)T + (-6.15e3 - 3.09e3i)T^{2} \)
89 \( 1 + (-16.7 + 19.9i)T + (-1.37e3 - 7.80e3i)T^{2} \)
97 \( 1 + (-22.3 - 30.0i)T + (-2.69e3 + 9.01e3i)T^{2} \)
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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

−12.04322683645886178394370373714, −11.38250776614451568945638602645, −10.30484778432653847767720845004, −8.717356946013546745590230451003, −8.032966553650317206644706293810, −6.62873935332851936383065726119, −5.31835396907437968445908925785, −4.01897963890798918863078977792, −2.15022842218353781080498431944, −0.37993220247405788032882113091, 3.35647543028664019252449917996, 4.45325092498610733408004785475, 5.61076093415613673677296525005, 6.89799054356866924384563878891, 7.930267552960560679669837771523, 9.171596491837364725726773923372, 10.21952415454348107853853007191, 11.25651531691406789584790687922, 12.04097798432595943323717945245, 13.40289669962308828687837573797

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