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Properties

Label 2-162-9.4-c5-0-13
Degree $2$
Conductor $162$
Sign $0.939 + 0.342i$
Analytic cond. $25.9821$
Root an. cond. $5.09727$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−7.99 + 13.8i)4-s + (−33 + 57.1i)5-s + (−88 − 152. i)7-s − 63.9·8-s − 264·10-s + (−30 − 51.9i)11-s + (329 − 569. i)13-s + (352 − 609. i)14-s + (−128 − 221. i)16-s + 414·17-s + 956·19-s + (−528 − 914. i)20-s + (120 − 207. i)22-s + (300 − 519. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.590 + 1.02i)5-s + (−0.678 − 1.17i)7-s − 0.353·8-s − 0.834·10-s + (−0.0747 − 0.129i)11-s + (0.539 − 0.935i)13-s + (0.479 − 0.831i)14-s + (−0.125 − 0.216i)16-s + 0.347·17-s + 0.607·19-s + (−0.295 − 0.511i)20-s + (0.0528 − 0.0915i)22-s + (0.118 − 0.204i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(25.9821\)
Root analytic conductor: \(5.09727\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :5/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.446970679\)
\(L(\frac12)\) \(\approx\) \(1.446970679\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 \)
good5 \( 1 + (33 - 57.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (88 + 152. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (30 + 51.9i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-329 + 569. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 414T + 1.41e6T^{2} \)
19 \( 1 - 956T + 2.47e6T^{2} \)
23 \( 1 + (-300 + 519. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.78e3 - 4.82e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.79e3 + 3.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 8.45e3T + 6.93e7T^{2} \)
41 \( 1 + (-9.59e3 + 1.66e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (6.65e3 + 1.15e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (9.84e3 + 1.70e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 3.12e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.31e4 + 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.55e4 - 2.69e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.40e3 + 1.45e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.12e3T + 1.80e9T^{2} \)
73 \( 1 + 2.55e4T + 2.07e9T^{2} \)
79 \( 1 + (3.72e4 + 6.44e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.23e3 + 5.60e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 3.27e4T + 5.58e9T^{2} \)
97 \( 1 + (8.30e4 + 1.43e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−11.98875818054334684521571466455, −10.75131981621552018410509488516, −10.14450077903738874267086658423, −8.529197023935716213472784550273, −7.33626192498415116717211737082, −6.85952592027834732659941452985, −5.50639799335371246321401698355, −3.83726253818949210828191344914, −3.15687136856707435364339014561, −0.49019946487129206942234873435, 1.18268506728603258127753662652, 2.79572615860706246305250595709, 4.17053614048430523978843520804, 5.27988647598606773773556913638, 6.43993901506786695393637258436, 8.183547698941227998986595933953, 9.066480256802874639223227844858, 9.864808419257020752482010093121, 11.44599180217576130406331988773, 12.03862313276269817443236552260

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