Properties

Label 2-192-8.5-c3-0-11
Degree $2$
Conductor $192$
Sign $-0.965 + 0.258i$
Analytic cond. $11.3283$
Root an. cond. $3.36576$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 10.3i·5-s − 3.46·7-s − 9·9-s − 55.4i·13-s − 31.1·15-s − 90·17-s + 116i·19-s + 10.3i·21-s − 103.·23-s + 17·25-s + 27i·27-s − 259. i·29-s − 301.·31-s + 36i·35-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.929i·5-s − 0.187·7-s − 0.333·9-s − 1.18i·13-s − 0.536·15-s − 1.28·17-s + 1.40i·19-s + 0.107i·21-s − 0.942·23-s + 0.136·25-s + 0.192i·27-s − 1.66i·29-s − 1.74·31-s + 0.173i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(11.3283\)
Root analytic conductor: \(3.36576\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 192,\ (\ :3/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.120766 - 0.917310i\)
\(L(\frac12)\) \(\approx\) \(0.120766 - 0.917310i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
good5 \( 1 + 10.3iT - 125T^{2} \)
7 \( 1 + 3.46T + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + 55.4iT - 2.19e3T^{2} \)
17 \( 1 + 90T + 4.91e3T^{2} \)
19 \( 1 - 116iT - 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 + 259. iT - 2.43e4T^{2} \)
31 \( 1 + 301.T + 2.97e4T^{2} \)
37 \( 1 + 34.6iT - 5.06e4T^{2} \)
41 \( 1 + 54T + 6.89e4T^{2} \)
43 \( 1 + 20iT - 7.95e4T^{2} \)
47 \( 1 - 394.T + 1.03e5T^{2} \)
53 \( 1 + 488. iT - 1.48e5T^{2} \)
59 \( 1 + 324iT - 2.05e5T^{2} \)
61 \( 1 - 575. iT - 2.26e5T^{2} \)
67 \( 1 + 116iT - 3.00e5T^{2} \)
71 \( 1 - 1.10e3T + 3.57e5T^{2} \)
73 \( 1 - 1.10e3T + 3.89e5T^{2} \)
79 \( 1 + 148.T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 - 918T + 7.04e5T^{2} \)
97 \( 1 - 190T + 9.12e5T^{2} \)
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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

−11.86666870918728871999911308416, −10.68040472437362072982510926194, −9.549438467199066509943249325939, −8.444558517412407775253868476438, −7.71557812046204282866431922030, −6.28404861270782720116298964256, −5.30427207067229797905608829478, −3.83949570730201878639250192108, −2.03181003938150826439498077059, −0.38459347818424065981239101587, 2.32070273420525050174313754597, 3.70082701931104018595896119667, 4.91639952454929353037505620538, 6.46874386512334420223438416424, 7.16050717905370848945451251612, 8.811460502961056568587749967068, 9.482338789018893215748026467942, 10.87964217928275438876686715944, 11.10969092885400219690465149831, 12.46453461442351283062180575682

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