↑

Properties

Label 2-162-9.5-c2-0-4
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

Related objects

Downloads

Learn more

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: $\chi_{162} (53, \cdot )$
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 \( 1 + (88 + 152. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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