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Properties

Label 2-162-9.5-c10-0-28
Degree $2$
Conductor $162$
Sign $0.939 + 0.342i$
Analytic cond. $102.927$
Root an. cond. $10.1453$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.5 − 11.3i)2-s + (255. − 443. i)4-s + (3.14e3 + 1.81e3i)5-s + (1.16e4 + 2.01e4i)7-s − 1.15e4i·8-s + 8.21e4·10-s + (5.40e4 − 3.12e4i)11-s + (8.50e4 − 1.47e5i)13-s + (4.55e5 + 2.62e5i)14-s + (−1.31e5 − 2.27e5i)16-s − 2.66e6i·17-s + 7.66e5·19-s + (1.60e6 − 9.29e5i)20-s + (7.06e5 − 1.22e6i)22-s + (−1.21e6 − 7.01e5i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.00 + 0.580i)5-s + (0.690 + 1.19i)7-s − 0.353i·8-s + 0.821·10-s + (0.335 − 0.193i)11-s + (0.229 − 0.396i)13-s + (0.846 + 0.488i)14-s + (−0.125 − 0.216i)16-s − 1.87i·17-s + 0.309·19-s + (0.503 − 0.290i)20-s + (0.137 − 0.237i)22-s + (−0.188 − 0.109i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(102.927\)
Root analytic conductor: \(10.1453\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :5),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.842925397\)
\(L(\frac12)\) \(\approx\) \(4.842925397\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-19.5 + 11.3i)T \)
3 \( 1 \)
good5 \( 1 + (-3.14e3 - 1.81e3i)T + (4.88e6 + 8.45e6i)T^{2} \)
7 \( 1 + (-1.16e4 - 2.01e4i)T + (-1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 + (-5.40e4 + 3.12e4i)T + (1.29e10 - 2.24e10i)T^{2} \)
13 \( 1 + (-8.50e4 + 1.47e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + 2.66e6iT - 2.01e12T^{2} \)
19 \( 1 - 7.66e5T + 6.13e12T^{2} \)
23 \( 1 + (1.21e6 + 7.01e5i)T + (2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (-4.18e6 + 2.41e6i)T + (2.10e14 - 3.64e14i)T^{2} \)
31 \( 1 + (-2.09e7 + 3.62e7i)T + (-4.09e14 - 7.09e14i)T^{2} \)
37 \( 1 - 5.01e7T + 4.80e15T^{2} \)
41 \( 1 + (-1.29e8 - 7.46e7i)T + (6.71e15 + 1.16e16i)T^{2} \)
43 \( 1 + (-9.93e7 - 1.72e8i)T + (-1.08e16 + 1.87e16i)T^{2} \)
47 \( 1 + (1.34e8 - 7.75e7i)T + (2.62e16 - 4.55e16i)T^{2} \)
53 \( 1 - 4.21e7iT - 1.74e17T^{2} \)
59 \( 1 + (2.52e8 + 1.46e8i)T + (2.55e17 + 4.42e17i)T^{2} \)
61 \( 1 + (-2.65e8 - 4.59e8i)T + (-3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (2.61e8 - 4.52e8i)T + (-9.11e17 - 1.57e18i)T^{2} \)
71 \( 1 - 5.71e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.18e9T + 4.29e18T^{2} \)
79 \( 1 + (9.82e8 + 1.70e9i)T + (-4.73e18 + 8.19e18i)T^{2} \)
83 \( 1 + (1.89e9 - 1.09e9i)T + (7.75e18 - 1.34e19i)T^{2} \)
89 \( 1 + 2.38e8iT - 3.11e19T^{2} \)
97 \( 1 + (-4.42e9 - 7.65e9i)T + (-3.68e19 + 6.38e19i)T^{2} \)
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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

−11.24988277147065928701764443868, −9.900217788493559402919547536641, −9.206723187186624907255325066001, −7.79095552656537644941938910463, −6.33136269872921900492230615402, −5.64261126349044829229092373281, −4.61061822440641248821529825045, −2.84631191375019740568930283120, −2.33062069580958328047142372683, −0.967715519517559463640498780249, 1.10927112072433337759021881651, 1.91563874876337797355540455166, 3.74634646868112316449145125616, 4.60255662290978280473843182090, 5.72440080288015595206312718184, 6.69814110635918449714703541914, 7.86543918766505767854117264772, 8.892938262611265020415635695450, 10.15615750623862710357979264894, 10.98993465908002490891466796973

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