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Properties

Label 2-162-27.4-c1-0-0
Degree $2$
Conductor $162$
Sign $0.286 - 0.957i$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.439 + 2.49i)5-s + (−1.79 + 1.50i)7-s + (−0.500 − 0.866i)8-s + (1.26 − 2.19i)10-s + (0.745 + 4.22i)11-s + (−0.713 + 0.259i)13-s + (2.20 − 0.802i)14-s + (0.173 + 0.984i)16-s + (2.46 − 4.26i)17-s + (3.62 + 6.27i)19-s + (−1.93 + 1.62i)20-s + (0.745 − 4.22i)22-s + (0.233 + 0.196i)23-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.196 + 1.11i)5-s + (−0.679 + 0.570i)7-s + (−0.176 − 0.306i)8-s + (0.400 − 0.693i)10-s + (0.224 + 1.27i)11-s + (−0.197 + 0.0719i)13-s + (0.589 − 0.214i)14-s + (0.0434 + 0.246i)16-s + (0.596 − 1.03i)17-s + (0.831 + 1.44i)19-s + (−0.433 + 0.363i)20-s + (0.158 − 0.900i)22-s + (0.0487 + 0.0409i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.286 - 0.957i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1/2),\ 0.286 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579940 + 0.431749i\)
\(L(\frac12)\) \(\approx\) \(0.579940 + 0.431749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 \)
good5 \( 1 + (0.439 - 2.49i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.79 - 1.50i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.745 - 4.22i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.713 - 0.259i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.46 + 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.62 - 6.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.233 - 0.196i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.91 + 1.06i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (6.58 + 5.52i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.78 + 6.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.60 + 1.67i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.283 + 1.60i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.39 + 1.16i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 0.573T + 53T^{2} \)
59 \( 1 + (0.950 - 5.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-8.46 + 7.10i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.0393 - 0.0143i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.10 - 3.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.54 - 9.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.92 - 2.52i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.41 - 2.33i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-3.96 - 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.570 + 3.23i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.71749144336518778768054513659, −11.98478921250038695573976522778, −11.01211788920716916255781954718, −9.815620735409105391534243872198, −9.426458773736395475948834139948, −7.66671403148015728602356138260, −7.06864243300658888459624370570, −5.72777640943653494195939986064, −3.66901602202065051517885507611, −2.36715758806242702386775584580, 0.892424071567505690977637293974, 3.43266242464513517182798164566, 5.10067758148170325622425824768, 6.34269798896863724004544120113, 7.60243153257811029199748570975, 8.671127342945080909874793287009, 9.371533387987808887549179115564, 10.56140639470251944972940754056, 11.54936479833203810209123599860, 12.74231525872092187005580850773

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