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Properties

Label 2-162-9.7-c7-0-20
Degree $2$
Conductor $162$
Sign $-0.173 + 0.984i$
Analytic cond. $50.6063$
Root an. cond. $7.11381$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (−232. − 403. i)5-s + (24.8 − 43.0i)7-s + 511.·8-s + 3.72e3·10-s + (151. − 262. i)11-s + (2.75e3 + 4.77e3i)13-s + (198. + 344. i)14-s + (−2.04e3 + 3.54e3i)16-s + 2.26e4·17-s + 5.28e4·19-s + (−1.49e4 + 2.58e4i)20-s + (1.21e3 + 2.09e3i)22-s + (1.00e4 + 1.73e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.833 − 1.44i)5-s + (0.0274 − 0.0474i)7-s + 0.353·8-s + 1.17·10-s + (0.0343 − 0.0594i)11-s + (0.348 + 0.602i)13-s + (0.0193 + 0.0335i)14-s + (−0.125 + 0.216i)16-s + 1.11·17-s + 1.76·19-s + (−0.416 + 0.721i)20-s + (0.0242 + 0.0420i)22-s + (0.171 + 0.297i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162\)    =    \(2 \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(50.6063\)
Root analytic conductor: \(7.11381\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{162} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 162,\ (\ :7/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.079392859\)
\(L(\frac12)\) \(\approx\) \(1.079392859\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 \)
good5 \( 1 + (232. + 403. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-24.8 + 43.0i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-151. + 262. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-2.75e3 - 4.77e3i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 2.26e4T + 4.10e8T^{2} \)
19 \( 1 - 5.28e4T + 8.93e8T^{2} \)
23 \( 1 + (-1.00e4 - 1.73e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-1.25e5 + 2.16e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (1.31e5 + 2.27e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + 4.60e5T + 9.49e10T^{2} \)
41 \( 1 + (1.20e5 + 2.08e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (2.84e5 - 4.92e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (-4.37e5 + 7.57e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + 8.11e5T + 1.17e12T^{2} \)
59 \( 1 + (6.46e5 + 1.11e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-8.14e5 + 1.41e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-9.21e5 - 1.59e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 4.56e6T + 9.09e12T^{2} \)
73 \( 1 + 1.91e6T + 1.10e13T^{2} \)
79 \( 1 + (-9.05e5 + 1.56e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-1.54e6 + 2.67e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 - 4.10e6T + 4.42e13T^{2} \)
97 \( 1 + (3.95e6 - 6.85e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−11.52976296270729593089423441973, −9.889254780282480578888992445317, −9.076367913145829603760104569092, −8.097419209972464317041651583500, −7.39572134703213001738992273912, −5.78930073222790034873443688987, −4.83622531096427063434486509579, −3.67903809351569460899624066067, −1.36296713629566567329000184238, −0.39368407787198132571757313991, 1.20154424740003537013402286102, 3.07005824133388019483695744360, 3.43478499253081713673226127234, 5.26678086408763781597638371877, 6.93014333809001464205309650449, 7.61923080625313212158563642673, 8.781376490125422194110153080580, 10.22907066688198203289088299043, 10.67584551759265000817913976184, 11.76046107497198455659334771892

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