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Properties

Label 2-192-24.11-c9-0-58
Degree $2$
Conductor $192$
Sign $0.998 + 0.0619i$
Analytic cond. $98.8868$
Root an. cond. $9.94418$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (133. + 44.6i)3-s + 1.42e3·5-s − 268. i·7-s + (1.56e4 + 1.18e4i)9-s − 3.57e3i·11-s − 2.44e4i·13-s + (1.88e5 + 6.34e4i)15-s − 4.61e5i·17-s + 2.81e5·19-s + (1.19e4 − 3.57e4i)21-s + 8.13e4·23-s + 6.59e4·25-s + (1.55e6 + 2.27e6i)27-s + 6.98e6·29-s − 3.97e6i·31-s + ⋯
L(s)  = 1  + (0.948 + 0.318i)3-s + 1.01·5-s − 0.0422i·7-s + (0.797 + 0.603i)9-s − 0.0735i·11-s − 0.237i·13-s + (0.963 + 0.323i)15-s − 1.34i·17-s + 0.495·19-s + (0.0134 − 0.0400i)21-s + 0.0606·23-s + 0.0337·25-s + (0.564 + 0.825i)27-s + 1.83·29-s − 0.772i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0619i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.998 + 0.0619i$
Analytic conductor: \(98.8868\)
Root analytic conductor: \(9.94418\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :9/2),\ 0.998 + 0.0619i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.342062814\)
\(L(\frac12)\) \(\approx\) \(4.342062814\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-133. - 44.6i)T \)
good5 \( 1 - 1.42e3T + 1.95e6T^{2} \)
7 \( 1 + 268. iT - 4.03e7T^{2} \)
11 \( 1 + 3.57e3iT - 2.35e9T^{2} \)
13 \( 1 + 2.44e4iT - 1.06e10T^{2} \)
17 \( 1 + 4.61e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.81e5T + 3.22e11T^{2} \)
23 \( 1 - 8.13e4T + 1.80e12T^{2} \)
29 \( 1 - 6.98e6T + 1.45e13T^{2} \)
31 \( 1 + 3.97e6iT - 2.64e13T^{2} \)
37 \( 1 - 6.19e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.56e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.58e7T + 5.02e14T^{2} \)
47 \( 1 + 1.87e7T + 1.11e15T^{2} \)
53 \( 1 + 3.23e7T + 3.29e15T^{2} \)
59 \( 1 + 8.09e7iT - 8.66e15T^{2} \)
61 \( 1 - 9.71e7iT - 1.16e16T^{2} \)
67 \( 1 - 6.32e7T + 2.72e16T^{2} \)
71 \( 1 + 2.04e8T + 4.58e16T^{2} \)
73 \( 1 - 2.16e7T + 5.88e16T^{2} \)
79 \( 1 - 1.79e8iT - 1.19e17T^{2} \)
83 \( 1 - 4.03e8iT - 1.86e17T^{2} \)
89 \( 1 + 9.79e8iT - 3.50e17T^{2} \)
97 \( 1 - 6.36e8T + 7.60e17T^{2} \)
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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

−10.52912521130889712322845326694, −9.747304786083655221326153206005, −9.068989190627393996597800022099, −7.953622538494024138116969801984, −6.88381298348363628723658446235, −5.56206845274588535158475889085, −4.48918797740378333182064077027, −3.06142553496274274657819949294, −2.24293732541746835456988686375, −0.925431460676783944818113025957, 1.13548449999011470494706150083, 2.03280443287394843757977846269, 3.10156435956809452242258006095, 4.40778506442364268903280711867, 5.87950641695073532660269939703, 6.78906020302676347798313610155, 8.004751479605334485146063308394, 8.900813066956876896391866762803, 9.789893903280617463708444987610, 10.59230330222214744595187629451

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