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Properties

Label 2-162-81.70-c1-0-3
Degree $2$
Conductor $162$
Sign $0.796 - 0.604i$
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  + (−0.686 + 0.727i)2-s + (1.72 + 0.0885i)3-s + (−0.0581 − 0.998i)4-s + (2.21 + 0.258i)5-s + (−1.25 + 1.19i)6-s + (−2.58 + 1.30i)7-s + (0.766 + 0.642i)8-s + (2.98 + 0.306i)9-s + (−1.70 + 1.43i)10-s + (2.24 − 5.19i)11-s + (−0.0121 − 1.73i)12-s + (−3.63 + 0.860i)13-s + (0.830 − 2.77i)14-s + (3.80 + 0.643i)15-s + (−0.993 + 0.116i)16-s + (1.29 + 7.36i)17-s + ⋯
L(s)  = 1  + (−0.485 + 0.514i)2-s + (0.998 + 0.0511i)3-s + (−0.0290 − 0.499i)4-s + (0.989 + 0.115i)5-s + (−0.510 + 0.488i)6-s + (−0.978 + 0.491i)7-s + (0.270 + 0.227i)8-s + (0.994 + 0.102i)9-s + (−0.539 + 0.452i)10-s + (0.675 − 1.56i)11-s + (−0.00351 − 0.499i)12-s + (−1.00 + 0.238i)13-s + (0.222 − 0.741i)14-s + (0.982 + 0.166i)15-s + (−0.248 + 0.0290i)16-s + (0.314 + 1.78i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.796 - 0.604i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21309 + 0.407851i\)
\(L(\frac12)\) \(\approx\) \(1.21309 + 0.407851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.686 - 0.727i)T \)
3 \( 1 + (-1.72 - 0.0885i)T \)
good5 \( 1 + (-2.21 - 0.258i)T + (4.86 + 1.15i)T^{2} \)
7 \( 1 + (2.58 - 1.30i)T + (4.18 - 5.61i)T^{2} \)
11 \( 1 + (-2.24 + 5.19i)T + (-7.54 - 8.00i)T^{2} \)
13 \( 1 + (3.63 - 0.860i)T + (11.6 - 5.83i)T^{2} \)
17 \( 1 + (-1.29 - 7.36i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.140 - 0.794i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (4.42 + 2.22i)T + (13.7 + 18.4i)T^{2} \)
29 \( 1 + (1.91 + 6.38i)T + (-24.2 + 15.9i)T^{2} \)
31 \( 1 + (-2.05 + 1.34i)T + (12.2 - 28.4i)T^{2} \)
37 \( 1 + (6.41 + 2.33i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + (-2.55 - 2.70i)T + (-2.38 + 40.9i)T^{2} \)
43 \( 1 + (6.77 + 9.10i)T + (-12.3 + 41.1i)T^{2} \)
47 \( 1 + (-3.20 - 2.10i)T + (18.6 + 43.1i)T^{2} \)
53 \( 1 + (1.66 + 2.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.15 + 2.67i)T + (-40.4 + 42.9i)T^{2} \)
61 \( 1 + (0.409 - 7.02i)T + (-60.5 - 7.08i)T^{2} \)
67 \( 1 + (0.117 - 0.394i)T + (-55.9 - 36.8i)T^{2} \)
71 \( 1 + (-0.709 + 0.595i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-3.44 - 2.89i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.27 + 1.34i)T + (-4.59 - 78.8i)T^{2} \)
83 \( 1 + (2.38 - 2.53i)T + (-4.82 - 82.8i)T^{2} \)
89 \( 1 + (-13.5 - 11.3i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.60 + 0.421i)T + (94.3 - 22.3i)T^{2} \)
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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

−13.29657782089538123682566428933, −12.16548893044779171735533958380, −10.40854271390047671177477663356, −9.736836182917140443524877312246, −8.899156652253444455135509913446, −8.039291630797219150794614974137, −6.48962964948932158255571519743, −5.84839820626803782236146836631, −3.71603956015917593546583110871, −2.14775614097493780472675474139, 1.90722814677625890108583616024, 3.19445685065880598706248438687, 4.76827573541535342193368172569, 6.84509753965290058436802801260, 7.48915955756677231747045614861, 9.168481762573844797516360116376, 9.753851020936964128122916771141, 10.09143470122248724556829583718, 12.01374752389441100302801882675, 12.75970999482772435047884917872

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