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Properties

Label 2-192-16.11-c10-0-7
Degree $2$
Conductor $192$
Sign $0.999 + 0.00124i$
Analytic cond. $121.988$
Root an. cond. $11.0448$
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  + (99.2 + 99.2i)3-s + (−3.61e3 − 3.61e3i)5-s − 1.77e4·7-s + 1.96e4i·9-s + (−1.66e5 + 1.66e5i)11-s + (−5.56e4 + 5.56e4i)13-s − 7.16e5i·15-s + 7.41e5·17-s + (−3.21e6 − 3.21e6i)19-s + (−1.76e6 − 1.76e6i)21-s − 1.08e7·23-s + 1.63e7i·25-s + (−1.95e6 + 1.95e6i)27-s + (−3.18e6 + 3.18e6i)29-s + 3.30e7i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−1.15 − 1.15i)5-s − 1.05·7-s + 0.333i·9-s + (−1.03 + 1.03i)11-s + (−0.149 + 0.149i)13-s − 0.943i·15-s + 0.522·17-s + (−1.30 − 1.30i)19-s + (−0.432 − 0.432i)21-s − 1.68·23-s + 1.66i·25-s + (−0.136 + 0.136i)27-s + (−0.155 + 0.155i)29-s + 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00124i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.999 + 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.999 + 0.00124i$
Analytic conductor: \(121.988\)
Root analytic conductor: \(11.0448\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :5),\ 0.999 + 0.00124i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.3871346998\)
\(L(\frac12)\) \(\approx\) \(0.3871346998\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-99.2 - 99.2i)T \)
good5 \( 1 + (3.61e3 + 3.61e3i)T + 9.76e6iT^{2} \)
7 \( 1 + 1.77e4T + 2.82e8T^{2} \)
11 \( 1 + (1.66e5 - 1.66e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (5.56e4 - 5.56e4i)T - 1.37e11iT^{2} \)
17 \( 1 - 7.41e5T + 2.01e12T^{2} \)
19 \( 1 + (3.21e6 + 3.21e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 1.08e7T + 4.14e13T^{2} \)
29 \( 1 + (3.18e6 - 3.18e6i)T - 4.20e14iT^{2} \)
31 \( 1 - 3.30e7iT - 8.19e14T^{2} \)
37 \( 1 + (6.14e7 + 6.14e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.25e7iT - 1.34e16T^{2} \)
43 \( 1 + (1.94e8 - 1.94e8i)T - 2.16e16iT^{2} \)
47 \( 1 + 2.05e8iT - 5.25e16T^{2} \)
53 \( 1 + (5.85e7 + 5.85e7i)T + 1.74e17iT^{2} \)
59 \( 1 + (2.34e8 - 2.34e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (-7.92e8 + 7.92e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-6.66e8 - 6.66e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.20e9T + 3.25e18T^{2} \)
73 \( 1 - 1.58e9iT - 4.29e18T^{2} \)
79 \( 1 + 5.80e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.82e8 - 3.82e8i)T + 1.55e19iT^{2} \)
89 \( 1 - 5.75e8iT - 3.11e19T^{2} \)
97 \( 1 - 1.54e10T + 7.37e19T^{2} \)
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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

−10.50882727556572537381428951485, −9.652614121103930596566596902393, −8.667797482034134850006815902938, −7.888977938679759071478843409685, −6.80891263647746174505317156847, −5.15686386119453541545923230311, −4.37127351896669880675272585645, −3.38380648830902362503323565307, −2.04697243959292724373079139535, −0.27015542629831252484346215186, 0.25700673165821060255467046630, 2.25184078608345194844473945687, 3.28913289777439048825210905297, 3.84437478253832361214916980106, 5.88021217641988237873854760289, 6.67134382943436973888147969130, 7.87256342029974696772113275331, 8.250254890565713872908287889919, 9.964750799634313577077053017473, 10.59244406478000017574876033746

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