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Properties

Label 2-162-9.2-c2-0-5
Degree $2$
Conductor $162$
Sign $0.939 - 0.342i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (3.67 − 2.12i)5-s + (2 − 3.46i)7-s + 2.82i·8-s + 6·10-s + (14.6 + 8.48i)11-s + (−4 − 6.92i)13-s + (4.89 − 2.82i)14-s + (−2.00 + 3.46i)16-s + 12.7i·17-s − 16·19-s + (7.34 + 4.24i)20-s + (12 + 20.7i)22-s + (14.6 − 8.48i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.734 − 0.424i)5-s + (0.285 − 0.494i)7-s + 0.353i·8-s + 0.600·10-s + (1.33 + 0.771i)11-s + (−0.307 − 0.532i)13-s + (0.349 − 0.202i)14-s + (−0.125 + 0.216i)16-s + 0.748i·17-s − 0.842·19-s + (0.367 + 0.212i)20-s + (0.545 + 0.944i)22-s + (0.638 − 0.368i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(4.41418\)
Root analytic conductor: \(2.10099\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :1),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.31063 + 0.407427i\)
\(L(\frac12)\) \(\approx\) \(2.31063 + 0.407427i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + (-3.67 + 2.12i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-14.6 - 8.48i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4 + 6.92i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 + (-14.6 + 8.48i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.67 - 2.12i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (22 + 38.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 34T + 1.36e3T^{2} \)
41 \( 1 + (40.4 - 23.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20 + 34.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (73.4 + 42.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 38.1iT - 2.80e3T^{2} \)
59 \( 1 + (29.3 - 16.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (25 - 43.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 16T + 5.32e3T^{2} \)
79 \( 1 + (-38 + 65.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-102. - 59.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 + (88 - 152. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.85376447201395744750820805326, −11.95787659005491752520343431562, −10.72142816059755379748956270824, −9.612768128500410158044132479966, −8.526418836284378476621410743676, −7.20640416515618152299617012085, −6.20889367143516985852749912517, −4.97318677650644879623271920543, −3.85086579071850667810935227774, −1.79877935892192256795035608831, 1.79964609406910535143879502568, 3.28770246934692631876822675452, 4.82377070063963437675771575990, 6.07535911693213322830452563462, 6.90486468172323226368728828864, 8.727287624815488222493534994112, 9.573560809867613060311278434282, 10.80248132310668171127907941617, 11.63082769185746127933212200686, 12.49205440797676868985275391665

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